ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2017

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1494

Hydropower plants and powersystems

Dynamic processes and control for stable andefficient operation

WEIJIA YANG

ISSN 1651-6214ISBN 978-91-554-9871-9urn:nbn:se:uu:diva-318470

Dissertation presented at Uppsala University to be publicly examined in Polhemsalen,Ångtröm 10134, Lägerhyddsvägen 1, Ångströmlaboratoriet, Uppsala, Friday, 19 May 2017 at13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English.Faculty examiner: Doctor Christophe Nicolet (Swiss Federal Institute of Technology inLausanne (EPFL)).

AbstractYang, W. 2017. Hydropower plants and power systems. Dynamic processes and control forstable and efficient operation. Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1494. 140 pp. Uppsala: Acta UniversitatisUpsaliensis. ISBN 978-91-554-9871-9.

As the largest global renewable source, hydropower shoulders a large portion of the regulationduty in many power systems. New challenges are emerging from variable renewable energy(VRE) sources, the increasing scale and complexity of hydropower plants (HPPs) and powergrid. Stable and efficient operation of HPPs and their interaction with power systems are ofgreat importance.

Theoretical analysis, numerical simulation and on-site measurement are adopted as mainstudy methods in this thesis. Various numerical models of HPPs are established, with differentdegrees of complexity for different purposes. The majority of the analysis and results are basedon eight HPPs in Sweden and China.

Stable operation (frequency stability and rotor angle stability) and efficient operation aretwo important goals. Regarding the stable operation, various operating conditions are analysed;the response time of primary frequency control (PFC) and the system stability of isolatedoperation are investigated. A fundamental study on hydraulic-mechanical-electrical couplingmechanisms for small signal stability of HPPs is conducted. A methodology is proposed toquantify the contribution to the damping of low frequency oscillations from hydraulic turbines.The oscillations, with periods ranging from less than one up to hundreds of seconds, areanalysed.

Regarding the efficient operation, a description and an initial analysis of wear and tear ofturbines are presented; a controller filter is proposed as a solution for wear reduction of turbinesand maintaining the frequency quality of power systems; then the study is further extended byproposing a framework that combines technical plant operation with economic indicators, toobtain relative values of regulation burden and performance of PFC.

Weijia Yang, Department of Engineering Sciences, Electricity, Box 534, Uppsala University,SE-75121 Uppsala, Sweden.

© Weijia Yang 2017

ISSN 1651-6214ISBN 978-91-554-9871-9urn:nbn:se:uu:diva-318470 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-318470)

Dedicated to my parents and my love

There is no elevator to success.You have to take stairs.

一步一个脚印

4

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

Stable operation regarding frequency stability:

I Weijia Yang, Jiandong Yang, Wencheng Guo, Wei Zeng, Chao Wang, Linn Saarinen, Per Norrlund. A mathematical model and its application for hydro power units under different operating conditions, Energies, 2015, 8(9), 10260-10275. DOI: 10.3390/en80910260

II Weijia Yang, Jiandong Yang, Wencheng Guo, Per Norrlund. Response time for primary frequency control of hydroelectric generating unit, International Journal of Electrical Power and Energy Systems, 74(2016):16–24. DOI: 10.1016/j.ijepes.2015.07.003

III Weijia Yang, Jiandong Yang, Wencheng Guo, Per Norrlund. Frequency stability of isolated hydropower plant with surge tank under different turbine control modes, Electric Power Compo-nents and Systems, 43(15): 1707-1716. DOI: 10.1080/15325008.2015.1049722

IV Wencheng Guo, Jiandong Yang, Weijia Yang, Jieping Chen, Yi Teng. Regulation quality for frequency response of turbine regu-lating system of isolated hydroelectric power plant with surge tank. International Journal of Electrical Power & Energy Sys-tems, 2015, 73: 528-538. DOI: 10.1016/j.ijepes.2015.05.043

V Wencheng Guo, Jiandong Yang, Jieping Chen, Weijia Yang, Yi Teng, Wei Zeng. Time response of the frequency of hydroelectric generator unit with surge tank under isolated operation based on turbine regulating modes. Electric Power Components and Sys-tems, 2015, 43(20), 2341-2355. DOI: 10.1080/15325008.2015.1082681

VI Wei Zeng, Jiandong Yang, Weijia Yang. Instability analysis of pumped-storage stations at no-load conditions using a parameter-varying model. Renewable Energy, 90 (2016): 420-429. DOI: 10.1016/j.renene.2016.01.024

5

VII Wei Zeng, Jiandong Yang, Renbo Tang, Weijia Yang. Extreme water-hammer pressure during one-after-another load shedding in pumped-storage stations. Renewable Energy, 99 (2016): 35-44. DOI: 10.1016/j.renene.2016.06.030

VIII Jiandong Yang, Huang Wang, Wencheng Guo, Weijia Yang, Wei Zeng. Simulation of wind speed in the ventilation tunnel for surge tank in transient process. Energies, 9.2 (2016): 95. DOI: 10.3390/en9020095

Stable operation regarding rotor angle stability:

IX Weijia Yang, Per Norrlund, Chi Yung Chung, Jiandong Yang, Urban Lundin. Eigen-analysis of hydraulic-mechanical-electrical coupling mechanism for small signal stability of hydropower plant, Submitted to: Renewable Energy, under review, 2017.

X Weijia Yang, Per Norrlund, Johan Bladh, Jiandong Yang, Urban Lundin. Hydraulic damping on rotor angle oscillations: quantifi-cation using a numerical hydropower plant model, Submitted to: IEEE Transactions on Energy Conversion, under review, 2017.

Efficient operation and balancing renewable power systems:

XI Weijia Yang, Per Norrlund, Linn Saarinen, Jiandong Yang, Wencheng Guo, Wei Zeng. Wear and tear on hydro power tur-bines – influence from primary frequency control, Renewable Energy, 87(2015) 88-95. DOI: 10.1016/j.renene.2015.10.009

XII Weijia Yang, Per Norrlund, Linn Saarinen, Jiandong Yang, Wei Zeng, Urban Lundin. Wear reduction for hydro power turbines considering frequency quality of power systems: a study on con-troller filters, IEEE Transactions on Power Systems, 2016. DOI: 10.1109/TPWRS.2016.2590504

XIII Weijia Yang, Per Norrlund, Jiandong Yang. Analysis on regula-tion strategies for extending service life of hydro power turbines, IOP Conference Series: Earth and Environmental Science. Vol. 49. No. 5. IOP Publishing, 2016. DOI: 10.1088/1755-1315/49/5/052013

XIV Weijia Yang, Per Norrlund, Linn Saarinen, Adam Witt, Brennan Smith, Jiandong Yang, Urban Lundin. Burden on hydropower units for balancing renewable power systems. Submitted to: Nature Energy, under formal peer review, 2017.

XV Linn Saarinen, Per Norrlund, Weijia Yang, Urban Lundin. Allo-cation of frequency control reserves and its impact on wear on a hydropower fleet, Revised and resubmitted to: IEEE Transac-tions on Power Systems, 2016.

6

XVI Linn Saarinen, Per Norrlund, Weijia Yang, Urban Lundin. Lin-ear synthetic inertia for improved frequency quality and reduced hydropower wear and tear, Submitted to: IEEE Transactions on Power Systems, under review, 2017.

Reprints were made with permission from the respective publishers. The au-thor has further contributed to the following papers, not included in this the-sis:

1 Weijia Yang, Jiandong Yang, Wencheng Guo, Per Norrlund.

Time-domain modeling and a case study on regulation and oper-ation of hydropower plants, Modeling and Dynamic Behavior of Hydro Power Plants, Institution of Engineering and Technology (IET), 2017. (Book chapter) ISBN: 978-1-78561-195-7

2 Wencheng Guo, Jiandong Yang, Weijia Yang. Modeling and stability analysis of turbine governing system of hydro power plant, Modeling and Dynamic Behavior of Hydro Power Plants, Institution of Engineering and Technology (IET), 2017. (Book chapter) ISBN: 978-1-78561-195-7

3 Jiandong Yang, Wei Zeng, Weijia Yang, Shangwu Yao, Wencheng Guo, Runaway stabilities of pump-turbines and its correlations with s-shaped characteristic curves (In Chinese), Transactions of the Chinese Society for Agricultural Machinery, 46(2015): 59-64. DOI: 10.6041/j.issn.1000-1298.2015.04.010

7

Contents

1  Introduction ......................................................................................... 17 1.1  Power system stability .................................................................... 18 1.2  Features of hydropower generating systems ................................... 18 

1.2.1  Hydraulic – mechanical – electrical coupling system ............ 18 1.2.2  Problems of oscillations ......................................................... 20 

1.3  Previous research ............................................................................ 21 1.3.1  Dynamic processes and modelling of hydropower plants ..... 21 1.3.2  Regulation quality and operating stability ............................. 23 1.3.3  Efficient operation: wear, efficiency and financial impacts .. 24 1.3.4  Brief summary ....................................................................... 24 

1.4  Hydropower research at Uppsala University .................................. 25 1.5  Scope of this thesis .......................................................................... 25 1.6  Outline of this thesis ....................................................................... 26 

2  Methods and theory ............................................................................. 28 2.1  Principles of methods ...................................................................... 28 

2.1.1  Numerical simulation ............................................................. 28 2.1.2  On-site measurement ............................................................. 28 2.1.3  Theoretical derivation ............................................................ 30 

2.2  Engineering cases: HPPs in Sweden and China .............................. 35 

3  Various hydropower plant models ....................................................... 36 3.1  Numerical models in TOPSYS ....................................................... 38 

3.1.1  Model 1 .................................................................................. 38 3.1.2  Model 4 and 4-S ..................................................................... 43 

3.2  Numerical models in MATLAB ..................................................... 45 3.2.1  Model 2-L (in Simulink) ........................................................ 45 3.2.2  Model 5 and 5-S (in SPS) ...................................................... 46 

3.3  Models for theoretical derivation .................................................... 48 3.3.1  Model 3-F .............................................................................. 49 3.3.2  Model 3-L .............................................................................. 50 3.3.3  Model 6 .................................................................................. 52 

3.4  Numerical models in MATLAB for HPPs with Kaplan turbines (Model 2-K).............................................................................................. 53 

3.4.1  System components ............................................................... 54 3.4.2  Turbine characteristic from measurements ............................ 57 

8

4  Stable operation regarding frequency stability .................................... 59 4.1  Case studies on different operating conditions ............................... 59 

4.1.1  Comparison of simulations and measurements ...................... 60 4.1.2  Discussion .............................................................................. 62 

4.2  Response time for primary frequency control ................................. 63 4.2.1  Specifications of response of PFC ......................................... 64 4.2.2  Formula and simulation of response time .............................. 64 

4.3  Frequency stability of isolated operation ........................................ 65 4.3.1  Theoretical derivation with the Hurwitz criterion.................. 66 4.3.2  Numerical simulation ............................................................. 67 

5  Stable operation regarding rotor angle stability ................................... 68 5.1  Hydraulic – mechanical – electrical coupling mechanism: eigen-analysis ..................................................................................................... 68 

5.1.1  Influence of water column elasticity (Te) ............................... 68 5.1.2  Influence of mechanical components of governor (Ty) .......... 70 5.1.3  Influence of water inertia (Tw) ............................................... 71 5.1.4  Influence on tuning of PSS .................................................... 73 

5.2  Quantification of hydraulic damping: numerical simulation .......... 74 5.2.1  Method and model ................................................................. 75 5.2.2  Quantification of the damping coefficient ............................. 78 5.2.3  Influence and significance of the damping coefficient .......... 80 

5.3  Discussion on quick response of hydraulic – mechanical subsystem ................................................................................................. 82 

6  Efficient operation and balancing renewable power systems .............. 83 6.1  Wear and tear due to frequency control .......................................... 83 

6.1.1  Description and definition ..................................................... 83 6.1.2  Cause ...................................................................................... 85 6.1.3  Analysis on influencing factors ............................................. 86 

6.2  Controller filters for wear reduction considering frequency quality of power systems .......................................................................... 88 

6.2.1  Method and model ................................................................. 89 6.2.2  On-site measurements ............................................................ 89 6.2.3  Time domain simulation ........................................................ 90 6.2.4  Frequency domain analysis: stability of the system .............. 92 6.2.5  Concluding comparison between different filters .................. 93 

6.3  Framework for evaluating the regulation of hydropower units ....... 94 6.3.1  The framework ....................................................................... 95 6.3.2  Methods ................................................................................. 96 6.3.3  Burden quantification .......................................................... 101 6.3.4  Regulation performance ....................................................... 104 6.3.5  Regulation payment ............................................................. 104 

7  Summary of results ............................................................................ 106 

9

8  Conclusion and discussion ................................................................. 109 

9  Future work ........................................................................................ 110 

10  Summary of papers ............................................................................ 111 

11  Acknowledgements............................................................................ 117 

12  Svensk sammanfattning ..................................................................... 120 

13  中文概要 (Summary in Chinese) ...................................................... 122 

14  Appendices ........................................................................................ 124 14.1  Appendix A .............................................................................. 124 14.2  Appendix B ............................................................................... 128 

15  References ......................................................................................... 131 

10

Abbreviations and Symbols

Abbreviation Description AVR automatic voltage regulator GV guide vane GVO guide vane opening HPP hydropower plant OF opening feedback PF power feedback PFC primary frequency control PID proportional-integral-derivative PI proportional-integral PJM PJM Interconnection LLC PSAT Power System Analysis Toolbox PSS power system stabilizer RB runner blade RBA runner blade angle SPS SimPowerSystems SISO single-input and single-output SvK Svenska Kraftnät TSO transmission system operator VRE variable renewable energy 0-D zero dimensional 1-D one dimensional 2-D two dimensional 3-D three dimensional

11

Latin symbols:

Symbol Unit Description a [pu] runner blade angle aw [m/s] velocity of pressure wave A [m2] cross section area of pipeline AP [m2] cross section area of turbine inlet AS [m2] cross section area of turbine outlet atk [pu] runner blade angle at time step tk BLa [pu] runner backlash BLgv [pu] guide vane backlash BM, BP, [m2/s] intermediate variables of method of characteristic CM, CP [m3/s] intermediate variables of method of characteristic bp [pu] governor droop bp2 [pu] governor droop of the rest of the units in the grid bp3 [pu] governor droop in Model 3 c [m/s] pressure propagation speed in penstock D [pu] common damping coefficient D1 [m] diameter of runner Dp [m] inner diameter of the pipe

Dt [pu] equivalent hydraulic turbine damping coefficient (“the damping coefficient”)

[pu] d-axis component of the sub-transient internal emf Efd [pu] excitation EMF eg [pu] coefficient of load damping

[pu] q-axis component of the transient internal emf

[pu] q-axis component of the sub-transient internal emf

eqy, eqω, eqh

[pu] partial derivative of turbine discharge with respect to guide vane opening, speed and head

ey ,eω ,eh [pu] partial derivative of turbine power output with respect to guide vane opening, speed and head

f [pu] frequency or turbine rotational speed fD [pu] Darcy-Weisbach coefficient of friction resistance f0 [Hz] rated frequency of power system (50 Hz in this thesis) fc [Hz] given frequency fg [Hz] generator frequency fi [Hz] frequency of oscillation corresponding to an eigenvalue fp [pu] frictional coefficient of penstock ft [pu] frictional coefficient of tunnel G [pu] comprehensive gate opening g [m/s2] gravitational acceleration

12

G1 [pu] gain from frequency deviation to power deviation for the Kaplan unit

G2 [pu] gain from frequency deviation to power deviation for the lumped hydropower plant

GF [pu] fitting function of the comprehensive gate opening Gg [pu] transfer function describing the grid

GP [pu] transfer function describing the head variation due to the dis-charge deviation in the penstock

GPI [pu] gain from GVO deviation to frequency deviation for the PI controller

GS [pu] transfer function describing the head variation due to the dis-charge deviation in the surge tank

Gt [pu] transfer function describing the Francis turbine and waterway system

h [pu] water head H [m] water head in the pipeline h0 [pu] initial water head H0 [m] net head of turbine h1 [s-1] derivative of water head with respect to time Hp [m] water head at turbine inlet Hs [m] water head at turbine outlet hy0 [pu] head loss of draw water tunnel Id, Iq [pu] d- and q-axis component of the armature current J [kg⋅m2] moment of inertia K1 [pu] scaling factor in Model 1 K2 [pu] scaling factor of the lumped HPP in Model 2-K-2 K3 [pu] scaling factor of the lumped HPP in Model 2-K-3 Ka [pu] gain of exitation system (automatic voltage regulator) Kd [s] governor parameter for the proportional term Ki [s-1] governor parameter for the integral term Kp [pu] governor parameter for the derivative term Ks [pu] gain of power system stabilizer Kω, KPe [pu] gain of power system stabilizer for selecting different input L [m] length of penstock M [s] system inertia M11 [N/m2.5] unit mechanical torque Mg [N⋅m] resistance torque of generator mg [pu] relative resistance torque of generator MR [MW] regulation mileage MR-base [MW] base value of regulation mileage Mt [N⋅m] mechanical torque mt [pu] relative mechanical torque n [rpm] rotational speed

13

n11 [m0.5/s] unit rotational speed nc [rpm] given rotational speed nr [rpm] rated rotational speed

PA,i, PB,i [pu] absolute value of a local maximum or local minimum of speed deviation

Paymile [pu] amount of mileage payment Paystrength [pu] amount of strength payment pc [MW] given power Pe, Pm [pu] electromagnetic active power and mechanical power pg [pu] generator power pl [pu] load pm [pu] active power pm, k [pu] active power at time step k pm0 [pu] initial active power pm2 [pu] active power of the lumped HPP Pm-rated [MW] rated power of generating unit pr [MW] rated power of generating unit PRMSE [pu] a root mean square error used for quantifying Dt

Pstep [MW] increase in output power caused by a frequency step change from 50 Hz to 49.9 Hz

q [pu] discharge q0 [pu] initial discharge Q11 [m0.5/s] unit discharge [m0.5/s] Qe, Qg [pu] reactive power of generator Qp [m3/s] discharge of turbine inlet Qs [m3/s] discharge of turbine outlet qt [pu] discharge of turbine qy [pu] discharge of draw water tunnel s [s-1] complex variable in Laplace transform SR [MW/Hz] regulation strength SR1, SR1-pu [pu] regulation strength of the Kaplan unit SR2, SR2-pu [pu] regulation strength of the lumped hydropower plant SR-base [MW/Hz] base value of regulation strength SRT [pu] regulation strength of all the units in the grid t [s] time T0, T1, T2 [s] parameters of power system stabilizer

, [s] open-circuit d-axis transient and sub-transient time constants Tdel-a [s] delay time in runner control Tdel-gv [s] delay time in guide vane control Te [s] time constant of water column elasticity, Te = L/c Tf [s] period of frequency oscillation TF [s] surge tank time constant Tj [s] mechanical time constant

14

Tj [s] mechanical time constant tk / number of time step tp [s] time constant in grid inverse model

[s] open-circuit d-axis sub-transient time constants

Tr [s] time constant in exitation system (automatic voltage regulator) Ts [s] time constant of surge Tw [s] water starting time constant Twp [s] water starting time constant of penstock Twt [s] water starting time constant of tunnel Twy [s] water starting time constant of draw water tunnel Ty [s] time constant of guide vane servo Tya [s] time constant of runner blade servo V [m/s] average flow velocity of pipeline section

V1 [pu] signal between washout and phase compensation block in power system stabilizer

Vg [pu] voltage at the generator terminal

Vgd, Vgq [pu] d- and q-axis component of the voltage at the generator terminal

VPSS [pu] output signal of power system stabilizer Vs [pu] infinite bus voltage Vsd, Vsq [pu] d- and q-axis component of the infinite bus voltage x [m] position

Xd, , [pu] d-axis synchronous, transient and sub-transient reactance of generator

, [pu] ;

xf [pu] relative value of speed (frequency) deviation, xf = (fg – fc)/fc Xq, [pu] q-axis synchronous and sub-transient reactance of generator

Xs [pu] total reactance of transmission line (between generator and infinite bus)

y [pu] guide vane opening yc [pu] given value of guide vane opening YGV, dist [pu] movement distance of guide vane yPI [pu] guide vane opening signal between PI terms and servo yPID [pu] guide vane opening signal after PID terms YRB, dist [pu] movement distance of runner blade yservo [pu] guide vane opening signal after the servo z [pu] relative change value of water level in surge tank

15

Greek symbols: Symbol Unit Description α, αp [pu] elasticity coefficient of penstock αHP [m-2] correlation coefficient of kinetic energy at turbine inlet αHS [m-2] correlation coefficient of kinetic energy at turbine outlet δ [rad] power (or rotor) angle Δ / stands for a deviation from a steady state value Δf [pu] frequency deviation from set-point value Δh [pu] water head deviation from initial value

ΔH [m] 22 2

-2 2

HSHPP

P S

H QgA gA

Δhp [pu] water head deviation from initial value due to hydraulic dy-namics in penstock

Δhs [pu] water head deviation from initial value due to hydraulic dy-namics in surge tank

Δn [pu] speed deviation Δq [pu] discharge deviation from initial value Δt [s] time step in simulation Δy [pu] guide vane opening deviation from set-point value

Z [m] absolute change value of water level in surge tank Δη [pu] efficiency change η [pu] turbine efficiency ηI [pu] interpolation function of the turbine efficiency

ηSj [pu] average value of the instantaneous efficiency during the op-eration period under a specific strategy (Sj)

ηst [pu] on-cam steady state efficiency θ [rad] angle between axis of pipeline and horizontal plane ξ / damping ratio of an oscillation φ [rad] power factor angle at the generator terminal ω [pu] angular velocity of the generator

ω0 [rad/s] synchronous angular velocity in electrical radians (equals to 2πf0)

Note that the symbols in subsection 2.1.3 of mathematical variables for theory introduction are explained within the text, and they are not listed here.

17

1 Introduction

Hydropower has played an important role in the safe, stable and efficient op-eration of electric power systems for a long time. Hydropower not only gen-erates electricity as the largest global renewable source, but also shoulders a large portion of the regulation and balancing duty in many power systems all over the world.

Hydropower technology is relatively mature, but new challenges are still emerging. First, with current trends toward de-carbonization in the electricity sector [1], the amount of electricity generated by variable renewable energy (VRE) sources has been constantly growing [2, 3]. Dealing with generation intermittency of VRE in an effective and efficient manner is a growing re-search field [3-6]. High VRE integration [7] and fewer heavy synchronously connected generators, which imply less inertia [8], lead to crucial conse-quences for power system stability. Second, a hydropower generation system is a complex nonlinear power system including hydraulic, mechanical and electrical subsystems (details in section 1.2). The generator size and the com-plexity of waterway systems in hydropower plants (HPPs) have been increas-ing. Especially in China [9, 10] dozens of HPPs with at least 1000 MW capac-ity are being planned, designed, constructed or operated. Third, many large HPPs are located far away from load centres, forming many hydro-dominant power systems, such as the cases in Sweden [11] and China [12].

In recent years, there has been a tendency that the new turbines experience fatigue to a greater extent than what seem to be the case for new runners dec-ades ago [13], and the maintenance needs at HPPs are affected [14], due to more regulation movements caused by increasingly more integration of VRE. In some countries, as in Sweden, primary frequency control (PFC) is a service that the transmission system operator (TSO) buys from the power producers. In other countries, as in Norway and China, there is also an obligation for the producers to deliver this service, free of charge. However, there are costs re-lated to this, e.g. due to design constraints and auxiliary equipment when pur-chasing a new unit or system, due to wear and tear that affects the expected life time and maintenance intervals, and due to efficiency loss when a unit operates in a condition that deviates from the best efficiency point, etc.

Based on the aforementioned aspects, the demand on the quality of regula-tion emanating from hydropower units has been increasing. Stable and effi-cient operation of HPPs and their interaction with power systems is of great importance.

18

1.1 Power system stability Power system stability is generally defined as a property of a power system, and it enables the system to remain in a stable operating state under normal operating conditions and to restore an equilibrium after a disturbance [15]. Three forms of power system stability are defined as follows [16].

(1) Frequency stability refers to the ability of a power system to maintain steady frequency after a severe system disturbance leading to a significant im-balance between generation and load. It depends on the ability to maintain and restore equilibrium between system generation and load, with minimum unin-tentional loss of load [16]. (2) Rotor angle stability refers to the ability of syn-chronous machines in a power system to remain synchronized after a disturb-ance. It depends on the ability to maintain and regain stability between elec-tromagnetic torque and mechanical torque of each synchronous machine in the system [16]. (3) Voltage stability means the capability of a power system to maintain steady voltages at all buses in the system after a disturbance from a given initial operating condition. It is determined by the ability to maintain and restore equilibrium between load demand and load supply from the power system [16].

In this thesis, the main focus is on the frequency stability and the rotor angle stability of power systems. In order to maintain frequency stability, generating units change their power output automatically according to the change of grid frequency, to make the active power balanced again. This is the PFC. PFC of electrical power grids is commonly performed by units in HPPs, because of the great rapidity and amplitude of the power regulation. PFC supplied by hy-dropower units is a core content of this thesis.

It is worth noting that the term “stability” is also used with respect to con-trol theory, which is introduced in subsection 2.1.3.

1.2 Features of hydropower generating systems In this section, important features of hydropower generating systems are high-lighted, as main analysis objects of the works throughout the thesis.

1.2.1 Hydraulic – mechanical – electrical coupling system A HPP is a complex nonlinear system integrating hydraulic – mechanical – electrical subsystems, as shown in Figure 1.1.

19

Figure 1.1.1 Simple illustration of a HPP: a hydraulic – mechanical – electrical cou-pling system.

Figure 1.2. Common ranges of some standard time constants in the hydropower sys-tem, indicating the interactions among the multiple variables. The definitions of the symbols are in the section Abbreviations and symbols. Time constants regarding the electrical subsystem (generator) are in red, and time constants regarding hydraulic and mechanical subsystems are in blue.

A core scientific challenge is to reveal the coupling mechanisms and oscilla-tion characteristics of diverse physical variables within multiple subsystems. For describing transient processes in HPPs, there are several common time constants, such as: transient and sub-transient time constants of generator

1 https://water.usgs.gov/edu/wuhy.html (accessed on March 14th, 2017)

20

( , , ), mechanical time constant (Tj) regarding the electrical subsys-tem; water starting time constant (Tw), time constants of water column elastic-ity (Te), servo (Ty), and surge (Ts) in surge tank or gate shaft, etc. regarding the hydraulic and mechanical subsystems. The common ranges of these time con-stants are presented in Figure 1.2, indicating the interactions among the mul-tiple variables.

1.2.2 Problems of oscillations There are different oscillation issues with various periods existing in hydro-power generating systems, as illustrated by the measured data in Figure 1.3 (350 s oscillation), Figure 1.4 (60 s oscillation) and Figure 1.5 (1 s oscillation).

In terms of different categories of stability studied in this thesis, oscillation periods regarding frequency stability (> 20 s) is normally larger than the ones regarding rotor angle stability (< 5 s). It is worth noting that in the power sys-tem field, a rotor angle oscillation with a frequency between 0.1 – 2.0 Hz is defined as a “low frequency oscillation” [17]; however in this thesis, this fre-quency range is relatively high. Hence, the oscillations regarding frequency stability is called “very low frequency oscillation”, and the term has already been used in [18].

In this thesis, the “very low frequency oscillation” regarding frequency sta-bility is investigated in chapter 4 and chapter 6; then the “low frequency os-cillation” regarding rotor angle stability is studied in chapter 5.

Figure 1.3. Oscillations with a period around 350 s: measurements of guide vane opening (GVO) and power output under a load step disturbance in a Chinese HPP with a surge tank.

21

Figure 1.4. Oscillation with a period around 60 s: measured frequency of the Nordic power system

Figure 1.5. Oscillation with a period around 1 s: measured electromagnetic power under disturbances of excitation voltage in a Swedish HPP.

1.3 Previous research

1.3.1 Dynamic processes and modelling of hydropower plants Previous studies have obtained many meaningful achievements regarding the dynamic processes and modelling of HPPs, and these works could be roughly divided into the following three categories.

(1) The first category mainly works on the electrical perspective (genera-tors and power grids) by simplifying the hydraulic and mechanical systems. This is a standard approach for small signal stability analysis of power systems in the classical books, such as in [15] and [19]. Power system stabilization was investigated based on hydropower generators through excitation control in several representative early works [20-22]. New controllers of reactive power were proposed for grid-connected operation [23, 24] and the emergency start-

22

up process [25]. Besides, in studies on inter-area mode oscillations, the hydro turbine model is normally very simple, such as in [26-29].

(2) The second category is to simplify the electrical sub-system by applying the first-order swing equation of generating units, focusing on the hydraulic-mechanical subsystem. Accuracy of second order hydraulic turbine models was studied [30, 31]. Hydraulic-turbine and turbine control-models for system dynamic studies were proposed [32, 33]. Field tests were conducted to validate a hydro turbine-governor model structure and parameters [34], and modelling and controller tuning of a HPP with units sharing a common penstock section were presented [35]. A nonlinear model of penstock and a hydraulic turbine model were proposed for simulation of hydraulic transients [36]. A basic sim-ulation tool for analysis of hydraulic transients in HPPs was established [37]. Modelling of the dynamic response and control of Francis turbines were con-ducted [38] [39]. A one-dimensional and three-dimensional (1D-3D) coupling method was proposed for hydraulic system transient simulations [40]. A phys-ics based hydraulic turbine model was built for system dynamics studies [41]. For HPPs with Kaplan turbines, a nonlinear digital simulation model was pro-posed [42], and differential evolution-based identification of a nonlinear model was conducted [43]. HPP models and control were reviewed in [44], and modelling and dynamic behaviour of HPPs was introduced in the two books [45] and [46].

(3) In recent years, the works on the coupling of the hydraulic-mechanical-electrical subsystems with relatively detailed models on all three subsystems have been emerging. Based on the simulation software SIMSEN, high-order modelling of HPPs in islanded power networks was conducted [47], and dy-namical behaviour of variable speed and synchronous machines with power system stabilizer (PSS) were compared [48]. Based on the simulation software TOPSYS, transient processes for hydraulic, mechanical and electrical cou-pling system were studied [49, 50]. Hydro turbine and governor models in a free and open source software package (Power System Analysis Toolbox, PSAT) were developed and implemented [51]. There are other software pack-ages for analysis of dynamic processes of HPPs, e.g. LVTrans [52], the Mod-elica Hydro Power Library2, and Alab3, etc. An object-oriented framework was applied to the study of electromechanical oscillations at a HPP [53]. An advanced-model of synchronous generators for HPP numerical simulations was built [54]. Nonlinear dynamic analysis and robust controller design were conducted for turbine regulating system with a straight-tube surge tank [55].

2 http://www.modelon.com/products/modelica-libraries/hydro-power-library/ (accessed on March 22nd, 2017) 3 https://alabdocs.atlassian.net/wiki/display/Public/Alab+-+The+Hydropower+Workbench (accessed on March 22nd, 2017)

23

1.3.2 Regulation quality and operating stability Previous research on the regulation and control of hydropower units is intro-duced here, in terms of frequency stability and rotor angle stability.

(1) On frequency stability The hydraulic turbine regulation was comprehensively introduced in a

book [56]. Oscillatory behaviour of a British HPP was reproduced by numer-ical simulation [57]. Hydro turbine-governor model validation in Pacific Northwest was conducted [58]. The effects of governor settings on the stabil-ity of the Turkish power system frequency were investigated [59]. Analysis of very low frequency oscillations in hydro-dominant power systems (the Co-lombian power system) and design of robust control to damp oscillatory modes were conducted [18, 60]. Speed and active power control of HPPs was studied [61], and impact from hydraulic transients [62] and a dynamic model of a Kaplan turbine regulating system [63] on power system dynamic stability were discussed. Load–frequency regulation of HPPs in isolated systems was studied [64-66]. Influence of the hydraulic system layout [67], the elastic property of penstock [68] and the effect of surge tank throttling [69] on the system stability were analysed. Stability analysis of governor-turbine-hydrau-lic system was conducted by state space method and graph theory [70]. A fuzzy sliding mode controller was designed via input state feedback lineariza-tion method [71].

A series of nonlinear modelling and stability analysis of hydro-turbine gov-erning system were conducted [72-76]. Comprehensive theoretical analysis on response and stability of HPPs based on various forms of surge tanks and tun-nels were carried on [77-80].

(2) On rotor angle stability Various controllers were proposed or optimized for enhancing rotor angle

stability of hydropower generating systems. An application of a multivariable feedback linearization scheme was presented [81]. An approach was proposed for the damping of local modes of oscillations resulting from large hydraulic transients [82]. Nonlinear decentralized robust governor control [83], nonlin-ear coordinated control of excitation and governor [84] and dynamic extend-ing nonlinear H∞ control [85] for hydropower units were studied. Damping low frequency oscillations with hydro governors was investigated [86], and the effect of governor settings on low frequency inter area oscillations was assessed [87]. Eigenvalue analysis on the stability of HPPs was performed based on phase variables a, b, c instead of d, q-components [88, 89]. An eigen-analysis was conducted for the oscillatory instability of a HPP, and the influ-ence of the water conduit dynamics was studied [90]. The second-order oscil-lation mode of hydropower systems was studied in [91] based on a linear elas-tic model and modal analysis.

24

1.3.3 Efficient operation: wear, efficiency and financial impacts Previous studies have obtained a considerable amount of illuminating achieve-ments regarding the wear and tear of hydropower turbines. From the point of view of hydraulics, the hydro turbine failure mechanisms [92] and the fatigue damage mechanism [93] were comprehensively reviewed; the fatigue design and life of Francis turbine runners were investigated [94-96]; the pressure at discharge control and frequency control was analysed for the life time of high head runners and low head runners [14]; consequences of PFC to the residual service life of Kaplan runners were investigated [97]; dynamic loads on Fran-cis runners and their impact on fatigue life were examined [13, 98]. Within the tribology research field, the wear on bearing materials of GVs in hydro turbines is a main topic [99-101]. Wear and tear is also mentioned in studies on the design and tuning of controllers of hydropower units [102-104], but it is not a main objective.

In terms of reducing efficiency loss, investigations were conducted on tur-bine design [105, 106] and strategies of operation and power dispatch [107-113]. Kaplan turbine efficiency improvement was proposed, e.g. through draft tube design [114, 115] and control methods optimizations [116, 117]. Operat-ing performance enhancements of Kaplan turbines were comprehensively de-scribed [118].

From the aspect of research regarding scheduling and financial feasibility, the hydropower unit start-up cost [119-121] was investigated. The costs and financial impacts of operation, production and maintenance of HPP were stud-ied [122-125].

1.3.4 Brief summary Many meaningful studies have been carried out on the topics above, however there are still limitations in these works. In terms of stable operation, most of the previous works only conducted numerical simulations and did not theoret-ically explain the oscillation mechanism. While in case of works that include theoretical analysis, the hydraulic-mechanical-electrical subsystems is still relatively simple, and only limited factors are discussed. Hence, a fundamental and comprehensive study on the stability of the multi-variable hydropower systems is very important.

Regarding the efficient operation, to the author’s knowledge, there have been few publications on wear and tear issue in the field of hydropower regu-lation and control. It is necessary to study the wear and tear from aspects of regulation strategies. Besides, in front of producers and TSOs, there are new challenges that are to minimize regulation burden, maintain good regulation performance, and to agree on reasonable compensation structures. Therefore, a systematic study quantifying and evaluating the trade-off between burden and performance of hydropower regulation is needed.

25

1.4 Hydropower research at Uppsala University The hydropower research at the Division of Electricity in the Department of Engineering Sciences at Uppsala University was initiated in 2003. Four PhD theses have been published until now. Ranlöf conducted electromagnetic anal-ysis of hydroelectric generators [126]; Bladh studied the interaction between hydropower generators and power systems [127]; Wallin focused on measure-ment and modelling of unbalanced magnetic pull in hydropower generators [128]; Saarinen investigated hydropower and grid frequency control [129].

Presently, the main topics of the hydropower research at the department are axial magnetic leakage flux in hydropower generators, actively controlled magnetic bearings, brushless exciter systems, frequency control and dynamic processes in HPPs.

1.5 Scope of this thesis In the title of this thesis, stable operation and efficient operation are high-lighted as two important goals. Regarding the stable operation, the frequency stability and the rotor angle stability are studied. For the frequency stability, the main focus is on PFC; secondary frequency control is also involved, such as in Paper XV, but it is not explicitly presented in the following context. For the rotor angle stability, the main objective is the local modes [15]. Regarding the efficient operation, here it is a general term that refers to a good operating condition or status with respect to overall economic performance; more ex-actly, it relates to wear and tear of units, generating efficiency, etc. Here, the term “economic operation” is not adopted, since the study is from a physical perspective by applying some indicators, and no practical profit or economic value is explicitly demonstrated.

In terms of common mathematical methods in the hydropower study field, they can be divided into the following three categories, from an aspect of mod-elling sophistication of hydropower units: 3-D (three-dimensional), 1-D (one-dimensional), and “0-D” (zero-dimensional) methods. Here, “0-D” means that a generating unit is simplified as a node in a model, and it is normally applied in analyses of complex systems. In this thesis, all the works are conducted from the 1-D (one-dimensional) perspective, as demonstrated in Figure 1.6. A goal of this thesis is starting to build a bridge between studies of detailed com-ponents (e.g. turbines, generators, etc.) with 3-D methods and studies of large systems (e.g. power systems, cascade HPPs, etc.) with “0-D” methods.

26

Figure 1.6 4. Scope of mathematical method in this thesis: 1-D method

1.6 Outline of this thesis In chapter 2, the methods and theory of the research in this thesis are briefly introduced. In chapter 3, various HPP models for different study purposes are developed and presented.

In chapter 4, the stable operation of HPPs regarding frequency stability of power systems is analysed, mainly based on Papers I – III. In chapter 5, the stable operation of HPPs regarding rotor angle stability of power systems is studied, mainly based on Paper IX and Paper X. In chapter 6, the efficient operation of HPPs during balancing actions for renewable power systems is investigated, mainly based on Papers XI – XIV.

In chapter 7, the results of the thesis are summarized. In chapter 8, the con-clusion and discussion are condensed, and future work is suggested in chapter 9.

4 Parts of Figure 1.6 are from figures on the following websites (accessed on March 17th, 2017): http://www.cfdsupport.com/francis-turbine-cfd-study.html http://naturstacken.com/2016/05/12/fritt-rinnande-vatten-ett-ord-fran-det-forgangna-vs-pro-jekt-umealven/ http://icseg.iti.illinois.edu/ieee-39-bus-system/

27

A brief summary of the papers, acknowledgements and summaries in Swe-dish and in Chinese are included in the thesis. At last, appendices are supplied for detailed information.

Categorization and relation of the main works and papers in this thesis are illustrated in Figure 1.7.

Figure 1.7. Categorization and relation of the main works in this thesis. The Roman numerals indicate the corresponding paper numbers.

28

2 Methods and theory

In this chapter, the methods and theory of this thesis are briefly displayed. In section 2.1, principles of three main methods used in this thesis are presented. In section 2.2, practical engineering cases adopted in this thesis are intro-duced.

2.1 Principles of methods In this section, principles of numerical simulation, on-site measurement, and theoretical derivation applied in the thesis are concisely introduced.

2.1.1 Numerical simulation Time-domain numerical simulation is a core method of this thesis. Various numerical models of HPPs are established, with different degrees of complex-ity for different purposes. The implementations are based on VC++, MATLAB / Simulink and MATLAB / SimPowerSystems (SPS); the PSAT is also applied to conduct basic power system simulations.

Details of the numerical models are introduced in section 3.

2.1.2 On-site measurement On-site measurement in HPPs in Sweden and China is an important approach of this thesis, for observing practical engineering problems and validating the numerical simulations. The majority of measurement data can be directly ob-tained from the measurement system installed in HPPs. Additional measure-ments were also conducted, and part of the measurement devices in Swedish HPPs are illustrated in Figure 2.1 and Figure 2.2.

Detailed information regarding settings and results of measurements are included in the following sections.

29

(a) (b)

Figure 2.1. Measuring device for turbine actuator movement for a Francis unit. (a) A distance transducer of wire type, which measures position of servo link; (b) An angu-lar transducer which measures GVO.

(a)

(b)

Figure 2.2. Measurement devices for a Kaplan unit. (a) Guide vane (GV) servo with feedback transducer, connected with orange cables, on its right. (b) Feedback de-vice on the top of the generator shaft, for runner blade angle (RBA). The red device on top is the transducer. The rod is close to 14 m long and ends in the runner hub.

30

2.1.3 Theoretical derivation First, the basic theory of the Laplace transform and transfer functions is pre-sented, and they are applied in the governor equation analysis in the studies of the response time (section 4.2) and wear of the turbine (section 6.1). Second, the Routh–Hurwitz criterion is introduced, and it is adopted to analyse the frequency stability in section 4.3. Third, the Nyquist stability criterion and the describing function method are introduced, and they are utilized for discussing the frequency stability in section 6.2. The content in these three parts are based on two books [130, 131]. Lastly, the state matrix and eigen-analysis are briefly presented based on [15], and they are applied for analysing the small signal stability in section 5.1.

2.1.3.1 Laplace transform and transfer function This part is based on [131], for introducing the basic theory of the Laplace transform and the transfer function.

● Laplace transform and inverse Laplace transform: Solving linear ordinary differential equations is a main purpose of the La-

place transform, through the following three steps: (1) transforming the dif-ferential equations into an algebraic equation in s-domain; (2) dealing with the simple algebraic equation; (3) taking the inverse Laplace transform to obtain the solution. The mathematical definition of the Laplace transform and the inverse Laplace transform are found in [131].

● Transfer function for a single-input and single-output (SISO) system: For a linear time-invariant SISO system, the definition of a transfer func-

tion of it is

G s L g t , (2.1)

and all the initial conditions set to zero. Here, L is the Laplace transform, and g(t) means the impulse response of the system with the scalar functions input u(t) and output y(t); s is a complex variable in the Laplace transform. The relationship between the Laplace transforms is

Y sG s

U s , (2.2)

with all the initial conditions set to zero. Here, Y(s) and U(s) are the Laplace transforms of y(t) and u(t), respectively.

For an nth-order differential equation with constant real coefficients, as presented in

1

1 1 01

1

1 1 01

...

...

n n

nn n

m m

m mm m

d y t d y t dy ta a a y t

dt dt dt

d u t d u t du tb b b b u t

dt dt dt

, (2.3)

31

by simply taking the Laplace transform of both sides of the equation and as-suming zero initial conditions, the transfer function of the linear time-invariant system is obtained, as shown in

11 1 0

11 1 0

...

...

n nn

m mm m

s a s a s a Y s

b s b s b s b U s

. (2.4)

The transfer function from u(t) to y(t) is

11 1 0

11 1 0

...

...

m mm m

n nn

Y s b s b s b s bG s

U s s a s a s a

. (2.5)

2.1.3.2 Routh–Hurwitz stability criterion This section is based on [131], for presenting the basic theory of the Routh–Hurwitz stability criterion. In order to determine the stability of linear time-invariant SISO systems, a simple way is to study the location of the roots of the characteristic equation of the system. The stable and unstable regions in the s-plane are shown in Figure 2.3 [131].

Figure 2.3. Stable and unstable regions in the s-plane (s=σ+jω)

For a linear time-invariant system that has a characteristic equation with con-stant coefficients, the Routh-Hurwitz criterion can be used to investigate the stability by simply manipulating arithmetic operations. For a linear time-var-iant SISO system of which the characteristic equation is

11 1 0... 0n n

n nF s a s a s a s a , (2.6)

the analysis process of Routh-Hurwitz criterion is presented in the following steps.

(1) Step one is forming the coefficients of the equation in (2.6) into the following tabulation:

2 4 6

1 3 5 7

...

...n n n n

n n n n

a a a a

a a a a

32

(2) Step two is to obtain the Routh's tabulation (or Routh's array); which is illustrated in Figure 2.4 for a sixth-order equation:

6 5

6 5 1 0... 0a s a s a s a . (2.7) 6

6 4 2 0

55 3 1

4 5 4 6

0

s a a a a

s a a a

a a a as

3 5 2 6 1 5 0 60

5 5 5

3 3 5 1 5 0 5

2 00

1 0

0 0

0 0 0 0

0 0 0 0 0

0

a a a a a a aA B a

a a a

Aa a B Aa a a A as C D

A A ACa ABC AD C A

s E aC C C

ED Cas F

E

0 00

0 0

0 0 0 0

Fa Es a

F

Figure 2.4. The Routh's tabulation for a sixth-order equation

(3) Step three is to determine the location of the roots by studying the signs of the coefficients in the first column of the tabulation. If the signs of all the elements of the first column are the same, the roots of the equation are all in the left half of the s-plane that is the stable region. Besides, the number of roots in the right half of the s-plane (unstable region) equals the number of changes of signs in the elements of the first column.

2.1.3.3 Nyquist stability criterion This part is based on [131], for introducing the basic theory of the Nyquist stability criterion.

For a SISO system with the closed-loop transfer function that is

1

G sM s

G s H s

, (2.8)

the characteristic equation roots must satisfy

1 1 0s G s H s L s . (2.9)

Here, L(s) is the loop transfer function. The stability of a closed-loop system can be determined by the Nyquist

criterion, through analysing the frequency-domain plots (the Nyquist plot) of the loop transfer function L(s).

● Absolute stability: There are two types of absolute stability as follows. (1) Open-loop stability: A system is open-loop stable, if all the poles of L(s) are in the left-half s-plane. (2) Closed-loop stability: A system is closed-loop stable, if all the zeros of the 1+ L(s) or all the poles of the closed-loop transfer function are in the left-half s-plane.

33

● Relative stability: The relative stability indicates how stable the system is, and it is measured by how close the Nyquist plot (or Nyquist curve) of L(s) is to the (-1, j0) point, in the frequency domain. The gain margin and phase margin are frequently used criteria to reflect relative stability of control sys-tems. The detailed definitions of the two types of margin can be found in [131].

2.1.3.4 Describing function method for non-linear systems This part is based on [130], to introduce the basic theory of the stability anal-ysis for non-linear systems and the describing function method.

The describing function method is commonly used together with the Nyquist criterion to analyse the stability of non-linear systems. Here, a system without external signals is taken as an example, the block diagram is shown in Figure 2.5 [130].

Figure 2.5. Block diagram for the describing function method

In the system, G(s) is the transfer function of a linear part, the function f de-scribes a static nonlinearity, and the input and the output signal of f are as-sumed to be

sin t

sin t

e t C

w t f C

. (2.10)

By expanding this function into a Fourier series, the function w can be written in the form

0

2 2 3 3

sin t sin

+ sin 2 sin 3 ...

w t f C f C A C t C

A C t C A C t C

(2.11)

For the fundamental component (with the angular frequency ω), the gain is

G i and the change in phase is argG i .

Then, a complex number is defined as

i C

f

A C eY C

C

, (2.12)

which is the describing function of the nonlinearity. By introducing the Fou-rier coefficients

34

2

0

2

0

1sin cos sin

1sin sin cos

a C f C d A C C

b C f C d A C C

, (2.13)

the complex number can be described as

f

b C ia CY C

C

. (2.14)

The equations of the system can be presented in an equation with the two unknowns C and ω

1fY C G i . (2.15)

The solutions of the equation are the amplitude and frequency of a correspond-ing oscillation. No solution of the equation indicates that the system likely does not oscillate.

By rewriting (2.15) as

1

f

G iY C

, (2.16)

it is clearer that the solution can be interpreted as an intersection of the fol-lowing two curves in the complex plane: the Nyquist curve G(iω) plotted as a function of ω and the -1/Yf(C) curve plotted as a function of C. An intersection of these two curves indicates the possible presence of an oscillation in the system [130], and time domain simulations might be needed to obtain clearer results.

2.1.3.5 State matrix and eigen-analysis This part is based on [15], to introduce the basic theory of the state matrix and eigen-analysis.

The eigenvalues of a matrix are the values of the scalar parameter λ for which the following equation has non-trivial solutions (i.e., other than Ф = 0)

A . (2.17) Here, A is an n×n matrix (e.g., for a power system), and Ф is an n×1 vector. The n solutions of λ (i.e., λ1, λ2,…, λn) are eigenvalues of A.

The free motion of a dynamic system can be described by a linearized form x A x , (2.18)

where the column vector x means the state vector, and the derivative of the state variable with respect to time is ; A is the state matrix of the system. The eigenvalues are applied to determine the stability of the system as follows.

● Real eigenvalue: A real eigenvalue indicates a non-oscillatory mode. A positive real eigenvalue corresponds to an aperiodic instability, and a negative real eigenvalue represents a decaying mode. A larger magnitude of a negative eigenvalue means a faster decay.

35

● Complex eigenvalue: Complex eigenvalues appear in conjugate pairs (for a state matrix A of which all the elements are real), and each pair indicates an oscillatory mode. More exactly, for a pair of complex eigenvalues

j , (2.19)

the frequency (in Hz) of the corresponding oscillation is

2

f

, (2.20)

which stands for the actual or damped frequency. The damping ratio is de-scribed as

2 2

, (2.21)

which determines the rate of decay of the oscillation amplitude. More detailed introduction of eigen-analysis and its application to power

system analysis can be found in [15].

2.2 Engineering cases: HPPs in Sweden and China In this thesis, the majority of the analysis and results are based on real engi-neering cases. Eight HPPs in Sweden and China are applied as study cases in different parts of the thesis for various purposes. Table 2.1 presents the basic information of these HPPs, and more details are given in the following sec-tions and Appendices. The Swedish HPPs are owned by Vattenfall, the largest hydropower owner and operator in Sweden.

Table 2.1. Basic information of the engineering cases of this thesis. The values of each HPP are rated values for a single unit.

HPP Location Type of turbine

Active power [MW]

Water head [m]

Discharge [m3/s]

Rotational speed

[r/min]

1 Sweden Francis 169.2 135.0 135.0 187.5

2 China Francis 610.0 288.0 228.6 166.7

3 China Francis 51.3 46.0 122.3 136.4

4 China Francis 256.5 128.0 225.0 166.7

5 Sweden Francis 185.4 73.0 275.0 115.4

6 Sweden Kaplan 52.0 30.0 170.0 150.0

7 Sweden Kaplan 42.3 24.0 180.0 125.0

8 Sweden Kaplan 149.0 42.8 385.0 115.4

36

3 Various hydropower plant models

In this thesis, various HPP models for different study purposes are developed. For a clearer demonstration and classification of models, they are distin-guished from two aspects: (1) the implementation and (2) the generator mod-elling approach, as shown in Figure 3.1. There are three categories regarding the implementations: TOPSYS (a software for transient processes in HPPs) [50, 132], MATLAB and theoretical derivation; models in TOPSYS and MATLAB are for numerical simulation. In terms of the generator modelling approach, the first-order model is used in HPP models for studies regarding frequency stability, and high-order generator models are built in HPP models to investigate rotor angle stability.

Figure 3.1. Various HPP models in this thesis, and the models are distinguished from two aspects: implementation and generator modelling approach. “F”, “K”, “L” and “S” mean “Francis”, “Kaplan”, “Lumped” and “Simplified” respectively. There are totally ten models in six categories. The arrows pointing to “Sophisticated” generally indicates the degree of complexity of among these categories.

As shown in Figure 3.1, there are totally ten models in six categories. “F”, “K”, “L” and “S” mean “Francis”, “Kaplan”, “lumped” and “simplified” re-spectively. Generally speaking, the degree of complexity in TOPSYS models

37

is higher than in the MATLAB models, and the theoretical models are further simplified, as indicated by the arrows in Figure 3.1. Table 3.1 shows the main differences and characteristics of the ten models.

Table 3.1. Ten models built in this thesis. The descriptions of abbreviations are as follows: “Char”. – “Characteristics”, “F” – “Francis”, “K” – “Kaplan”, “L” – “lumped”, “S” – “simplified”. Pm is the turbine mechanical power. The implementa-tions in MATLAB are divided into Simulink and SPS. The column “Chapter/Section” shows the corresponding parts in the thesis for main applications of the models.

Model Implementa-

tion Generator

model Turbine

type

Main corresponding

papers

Chapter/ Section

1 TOPSYS 1st-order Francis I, II, III, XI 4, 6.1

2-K Simulink 1st-order Kaplan XIV 6.3

2-L Simulink 1st-order Francis XII 6.2

3-F Theoretical 1st-order Francis III 4.3

3-L Theoretical 1st-order Francis XII 6.2

4 TOPSYS 5th-order Francis X 5.2

4-S TOPSYS 5th-order None X 5.2

5 SPS 7th-order Francis IX 5.1

5-S SPS 7th-order None X 5.2

6 Theoretical 5th-order Francis IX 5.1

Model Turbine

characteristic Water

column Surge tank Grid

Swing equation

1 Char. curves Elastic Yes Various General

2-K Specific model Elastic Yes Nordic grid General

2-L Ideal Rigid No Nordic grid General

3-F Linear Rigid Yes Isolated General

3-L Ideal Rigid No Nordic grid General

4 Char. curves Elastic Yes SMIB General

4-S None None No SMIB Constant Pm

5 Linear Elastic No SMIB General

5-S None None No SMIB Constant Pm

6 Linear Elastic No SMIB General

Details of these ten HPP models are introduced in the following sections. It is worth noting that there are several other models extended or slightly trans-formed from these ten models for various purposes, and they are presented in the corresponding parts in section 4 through section 6. Additionally, some

38

power system models are also applied in the thesis but not presented here, such as a model implemented in the PSAT in section 5.

3.1 Numerical models in TOPSYS In this section, Model 1, Model 4 and Model 4-S are presented. These models are based on the software TOPSYS [50, 132] that is developed by applying Visual C++ for scientific studies and consultant analyses of transient pro-cesses in HPPs. These models are now implemented in TOPSYS as an exten-sion version.

The graphical user interface of TOPSYS is shown in Figure 3.2. In the basic version of TOPSYS, the model of waterway systems and hydraulic turbines has the following merits. (1) Equations for compressible flow are utilized in the draw water tunnel and penstock, considering the elasticity of water and pipe wall. (2) Various types of surge tanks and tunnels are included in the model library. (3) Turbine characteristic curves are used, instead of applying a linearized model with transmission coefficients. These characteristics lay a solid foundation for this work to achieve efficient and accurate simulation re-sults.

Figure 3.2. Graphical user interface of TOPSYS and a model of a Swedish HPP (HPP 1 in this thesis)

3.1.1 Model 1 Based on Paper I, a mathematical HPP model, especially a governor system model for different operating conditions, is presented in this subsection. Model 1 is applied for case studies on different operating conditions of HPPs

39

and the main results are presented in section 4, including comparisons with on-site measurements. Model 1 is also utilized for investigating the influence of PFC on wear and tear in section 6.1.

3.1.1.1 Generator and power grid As shown in Figure 3.1, the main difference of Model 1 and Model 4 is the generator modelling approach. Model 1 adopts the first-order swing equation to describe the whole generator. Equations for three grid-connection condi-tions are presented as follows.

(1) For the single-machine isolated operation, the equation has the general form, as shown in

2

30J

30g r

t gr

e pdnM M n

dt n

. (3.1)

(2) For the Single-Machine Infinite Bus (SMIB) operation, it is assumed that the rotational speed is constant at the rated value or other given values, yielding

,( )c g cn n f f . (3.2)

(3) Under the off-grid operation, the values of Mg and eg are 0, and the cor-responding equation is

J30 t

dnM

dt

, (3.3)

which can be considered as a special case of (3.1). The generator frequency, fg, is transferred from the speed, n.

3.1.1.2 Waterway system Models of waterway systems in HPPs are only shortly presented, more details can be found in [50]. Considering the elasticity of water and pipe wall, equa-tions for one-dimensional compressible flow in draw water tunnel and pen-stock are described by the continuity equation and the momentum equation.

The continuity equation is

2 2

sin θ 0w wa a VH H V AV V

x t g x gA x

. (3.4)

The momentum equation is

02D

p

V VH V Vg V f

x x t D

. (3.5)

The details of all the symbols in this thesis can be found in Abbreviations and Symbols. The set of hyperbolic partial differential equations are solved by a common and widely used approach, the method of characteristics [133]. Moreover, different forms of pipelines, channels and surge tanks are included in the model, and more information can be found in Paper 1.

40

3.1.1.3 Turbine The Francis turbine model is discussed in this part. Figure 3.3 shows the model and illustrates some of the notations. The equations of the model are presented below.

Figure 3.3. Illustration of the Francis turbine model in TOPSYS

The continuity equation is

S PQ Q . (3.6)

The equations of the method of characteristics are

+C :

C :

P P P P

S M M S

Q C B H

Q C B H

. (3.7)

The turbine flow equation is

211 1 P P SQ Q D H H H . (3.8)

The equations of unitary parameters are

11 1 / P Sn nD H H H , (3.9)

311 1 t P SM M D H H H . (3.10)

The equations of turbine characteristic curve are 11 1 11,Q f n Y , (3.11)

11 2 11,M f n Y . (3.12)

Here, functions f1 and f2 mean the interpolation of the turbine characteristic curves. The equation

30g t

np M

(3.13)

shows the transform from the torque to the power output.

3.1.1.4 Governor system The governor system is a feature of Model 1, with various equations for dif-ferent operating conditions and control modes. Figure 3.4 demonstrates the complete control block diagram of the proportional-integral-derivative (PID) governor system. The main non-linear factors (dead-zone, saturation, rate lim-iting and backlash) are included. All the variables in the governor system are per unit values. The S1, S2 and S3 blocks are selectors between different sig-nals, and the zero input to the selector means no input signal.

C

CP

S

41

Figure 3.4. Block diagram of the governor system in TOPSYS

For the normal operation, which means the isolated and the grid-connected operation with load, there are three control modes: frequency control, opening control and power control. This study establishes a governor model with a switchover function of control mode.

(1) Frequency control mode (PFC) In the frequency control mode, as shown in Figure 3.4, the feedback signal

contains not only the frequency value, but also the opening or power, which forms the frequency control under opening feedback (OF) and power feedback (PF), as respectively described by

2

2

2

2

(1 )

PID PIDp d p p p i PID c

f fd p i f

d y dyb K b K b K y y

dt dt

d x dxK K K x

dt dt

, (3.14)

2

2

2

2

( )

g g PIDp d p p p i g c

f fd p i f

d p dp dye K e K e K p p

dt dt dt

d x dxK K K x

dt dt

. (3.15)

The symbols in governor equations are illustrated in Figure 3.4. (2) Opening control mode (secondary frequency control) In the opening control mode, the governor controls the opening according

to the given value (yc). As demonstrated by Figure 3.4, the opening control is equivalent to the frequency control under the OF without the frequency devi-ation input (xf). The equation of the opening control is

42

2

2(1 )

0

PID c PID cp d p p

p i PID c

d y y d y yb K b K

dt dtb K y y

. (3.16)

Moreover, the modelling of the opening control process can be simplified by ignoring the engagement of the PID controller, i.e. setting the opening di-rectly equal to the given value, as shown in

PID cy y . (3.17)

(3) Power control mode (secondary frequency control). In the power control mode, the governor controls the opening according to

power signals, leading the power output to achieve the given value. The equa-tion is

2

2

( ) ( )

( ) 0

g c g cp d p p

c PIDp i g c

d p p d p pe K e K

dt dtdp dy

e K p pdt dt

. (3.18)

It is worth noting that a simpler controller, without the proportional (P) and derivative (D) terms, is applied in many real HPPs, as shown in

( ) 0c PIDp i g c

dp dye K p p

dt dt . (3.19)

In the governor equations (3.14) – (3.19), only the value of yPID is solved for. For the servo part, the output opening (yservo) is obtained by solving

servoPID y servo

dyy T y

dt . (3.20)

Then, the value of final opening (y) is the value after the non-linear functions i.e. saturation, rate limiting and backlash. Table 3.2. States of selectors in different control modes

Control mode Equation S1 state S2 state S3 state

Frequency control (3.14) 1 1 1

(3.15) 1 3 3

Opening control (3.16) 2 1 1

(3.17) 2 2 1

Power control (3.18) 2 3 3

(3.19) 2 2 3

The selectors (S1, S2 and S3) in the governor system are related to each other. Table 3.2 shows various states of selectors in different control modes. Table

43

3.3 concludes the equation set, in this study, of hydropower units under dif-ferent operating conditions. Discussion of equations for start-up, no-load op-eration, emergency stop and load rejection can be found in Paper I. Table 3.3. Equation set of the hydropower unit under different operating conditions

Operating condition Equation set

Governor Generator Turbine

Normal operation

Frequency control (3.14) or (3.15) (3.1) or (3.2)

(3.6)-(3.13)

Opening control (3.16) or (3.17) (3.1) or (3.2)

Power control (3.18) or (3.19) (3.1) or (3.2)

Start-up Open-loop (3.18) (3.3)

Closed-loop (3.15) (3.3)

No-load operation (3.15) (3.3)

Emergency stop (3.18) (3.3)

Load rejection (3.15) (3.3)

3.1.2 Model 4 and 4-S This subsection is based on Paper X. Comparing to Model 1 with a first-order generator model, Model 4 and 4-S applies a fifth-order model for investigating the rotor angle stability, as demonstrated in Table 3.1 and Figure 3.1. The models of the whole hydraulic-mechanical subsystem (i.e. the waterway sys-tem, the turbine and the governor system) in Model 4 are the same as the ones in Model 1. Hence, only the model of electrical subsystem, including the au-tomatic voltage regulator (AVR) and the PSS, is presented here in a per-unit system.

Model 4-S (“S” is short for “simplified”), a simplified version of Model 4, is designed specifically for studying the hydraulic turbine damping of rotor angle oscillations. Hence Model 4-S is introduced in section 5.2.

3.1.2.1 Generator and power grid The generator is described by a classical fifth-order model [19] that consists of five differential equations, as shown in

44

0

0

0

0

1

( )

( )

( )

j m e

qd fd q d d d

qd q q d d d

dq d q q q

d

dtd

T P PdtdE

T E E I X Xdt

dET E E I X X

dtdE

T E I X Xdt

. (3.21)

The d- and q-axis components of the voltage at the generator terminal are

0 -

= -0

gd d dq

q qdgq

V E IX

E IXV

. (3.22)

The electric active power and the reactive power are described in

=-

dgd gqe

gq gd qe

IV VP

V V IQ

. (3.23)

The network model for the SMIB system is shown in

sin

= = +cos -

gdsd qss

sq s gq d

VV IVX

V V V I

. (3.24)

All the resistances in the system are ignored, and the transformer and trans-mission line are simplified as a reactance (Xs).

3.1.2.2 AVR and PSS The AVR is described by a standard first-order model [15], as presented in Figure 3.5 and the following equation

fdr fd a g PSS

d ET E K V V

dt

. (3.25)

Figure 3.5. Block diagram of the AVR in Model 4 and 4-S

The PSS is described by a second-order model [15], as shown in Figure 3.6 and

45

1 1

0

1 1 1

2 2 2

s

PSS PSS

d V VdK

dt dt T

d V VT d V V

dt T dt T T

. (3.26)

The output limit (saturation) of the AVR and the PSS are included.

Figure 3.6. Block diagram of the PSS in Model 4 and 4-S

3.2 Numerical models in MATLAB In this section, models in MATLAB/Simulink and MATLAB/SPS are pre-sented.

3.2.1 Model 2-L (in Simulink) The main purpose of Model 2-L (“L” is short for “lumped”) is to study con-troller filters for the efficient operation and the grid frequency quality in sec-tion 6.2, which is based on Paper XII.

Model 2-L is for simulating the Nordic power system [129, 134], and it is shown in Figure 3.7. In the model, all the power plants are lumped into one.

Figure 3.7. Mode 2-L: a Simulink model of the Nordic power system, for computing the power system frequency. The governor system is shown in Figure 3.8.

The governor system is the key part, as shown in Figure 3.8. The dead zone [135, 136], floating dead zone [137], and first-order linear filter [138] are in-cluded, as the frequency filter and guide vane opening (GVO) filter. In the

46

numerical model, the floating dead zone is equivalent to the backlash [135], which has an impact during every GVO direction change.

Figure 3.8. The Simulink model of a general turbine governor with different filters, for computing GV movements. The blocks of the filters are highlighted in green.

The transfer functions of the first-order linear filters and the proportional-integral (PI) controller are described by:

Linear frequency filter: 11

1( )

1ff

F sT s

(3.27)

Linear GVO filter: 22

1( )

1ff

F sT s

(3.28)

PI Controller: ( )1

p i

p p p i

K s KC s

b K s b K

(3.29)

The equations of the plant and the grid are as follows.

Plant: 1

( )0.5 1

w

w

T sP s K

T s

(3.30)

Grid: 1

( )G sMs D

(3.31)

The parameters of the plant and the grid are shown in Table 14.4 in Appendix B.

3.2.2 Model 5 and 5-S (in SPS) The aim of Model 5 and 5-S is to conduct simulations for for investigating the rotor angle stability in section 5, as shown in Table 3.1 and Figure 3.1. More exactly, both Model 5 and 5-S are developed for verifications, by applying the widely-used software SPS. Model 5 is built to verify the theoretical eigen-analysis (Model 6) in section 5.1; Model 5-S, a simplified version of Model 5, is utilized specifically for verifying the electrical subsystem in TOPSYS (Model 4) in section 5.1.

Model 5 is introduced in this section based on Paper IX, and Model 5-S is presented in section 5.2 together with Model 4-S. The overall model that con-tains several subsystems is in per-unit system, as illustrated in Figure 3.9, it is a SMIB system with the extended HPP model.

47

Figure 3.9. Block diagram of Model 5: the SMIB system with the extended HPP model. The inputs of reference values are not shown. The signal of rotational speed (ω) is in blue, and the signal of electromagnetic power (Pe) is in red.

3.2.2.1 Generator, transformer, AVR and PSS The generator and the transformer adopt the existing blocks “Synchronous Machine”5 and “Three-Phase Transformer (two windings)”6 from the SPS li-brary. Compared to a standard example “Synchronous Machine”7 existing in SPS, Model 5 is an extended version with the PSS added and with more de-tailed governor, turbine and waterway models.

It is worth noting that the generator model here is seventh-order: the stator transients are included, while they are ignored in the fifth-order generator model in Model 4 and 4-S.

The numerical model of the AVR has the same structure as the one in Model 4 and 4-S, as shown in Figure 3.5. In terms of the PSS, in Model 4 and 4-S, only the input of speed deviation (Δω) is considered; while in Model 5 and 5-S, the input of the deviation of the electromagnetic power (ΔPe) is also included, for the practical case in Swedish HPPs, e.g. HPP 5 in Table 2.1. The PSS is shown in Figure 3.10 and the following equation

1 1

0

1 1 1

2 2 2

, ( )PSSs PSS Pe e

PSS PSS

d Id V VK I K K P

dt dt T

d V VT d V V

dt T dt T T

. (3.32)

5 http://www.mathworks.com/help/physmod/sps/powersys/ref/synchro-nousmachine.html?searchHighlight=Synchronous%20Machine (accessed on March 14th, 2017) 6 http://www.mathworks.com/help/physmod/sps/powersys/ref/threephasetransform-ertwowindings.html?searchHighlight=Three-Phase%20Transformer (accessed on March 14th, 2017) 7 https://www.mathworks.com/help/physmod/sps/examples/synchronous-machine.html (ac-cessed on March 14th, 2017)

48

Figure 3.10. Block diagram of the PSS in Model 5 and 5-S

3.2.2.2 Governor system, turbine and waterway system The numerical model of the governor is presented in Figure 3.11, including the nonlinear components that are important due to their decent influence on the response of GVO. It is worth noting that the PF signal here is the electro-magnetic power, not the mechanical power.

The turbine and waterway system is described by a linearized model, through applying the standard method with six coefficients [15], as shown in Figure 3.12. The elasticity of the water column is considered and the frictional loss is ignored here.

Figure 3.11. Block diagram of the governor system in Model 5. The nonlinear com-ponents are with dashed outline.

Figure 3.12. Block diagram of the linear model of turbine and waterway system in Model 5

3.3 Models for theoretical derivation In this section, various models derived for theoretical analysis are presented.

49

3.3.1 Model 3-F Model 3-F (“F” is short for “Francis turbine”) is mainly for analysing the fre-quency stability, and the results are comparted with simulations by Model 4 in section 4.3, based on Paper III.

Comparing to Model 4, Model 3-F is linearized on the basis of the follow-ing assumptions. (1) Rigid water column equations are adopted in the draw water tunnel, neglecting the elasticity of water and pipe walls. (2) The pen-stock modelling is ignored. (3) Head loss at the bottom of the surge tank is not considered. (4) Steady-state turbine characteristic is linearized by using trans-mission coefficients. (5) Nonlinear characteristics of the governor (e.g. satu-ration, rate limiting and dead zone) are ignored.

3.3.1.1 Details of the model The basic equations are described in the schematic diagram shown in Figure 3.13.

Figure 3.13. Schematic diagram for Model 3-F: a HPP with a surge tank

The equation describing the draw water tunnel is

0

0

2 y yy wy

h dqz q T

H dt . (3.33)

The continuity equation of the surge tank is

y t F

dzq q T

dt . (3.34)

The equations of the turbine torque and the turbine discharge are

t h x y

t qh qx qy

m e z e x e y

q e z e x e y

. (3.35)

The first-order generator equation is

( )a t g g

dxT m m e x

dt . (3.36)

The governor equation for the frequency control is

50

2

2

d d( ) (1 )

d dd

( )d

y p P y p P p i y

p i P i

y yT b K T b K b K T

t tx

b K y K K xt

. (3.37)

Comparing with Equation (3.14), the servo (Ty) is included and Kd is ignored. The governor equation for the power control is

2

2

dd d( )

d d dc

y p i c g

py yT e K p p

t t t . (3.38)

Comparing with Equation (3.19), the servo (Ty) is included.

3.3.1.2 Characteristic equations for Routh-Hurwitz stability criterion The characteristic equations, introduced in section 2, of the system for Routh-Hurwitz stability criterion is presented here. Two characteristic equations are deduced for the frequency control and the power control respectively.

(1) Frequency control By applying the Laplace transform to Equations (3.33) through (3.37), a

fifth-order characteristic equation for frequency control is derived as:

5 4 3 2

5 4 4 2 1 05 4 3 20

d x d x d x d x dxa a a a a a x

dt dt dt dt dt . (3.39)

The coefficients a0 through a5 are explained in Appendix A. (2) Power control By applying the similar treatment as above on Equations (3.33) through

(3.36) and Equation (3.38), a fifth-order characteristic equation for power con-trol is obtained:

5 4 3 2

5 4 3 2 1 05 4 3 20

d x d x d x d x dxa a a a a a x

dt dt dt dt dt . (3.40)

The coefficients through are explained in Appendix A.

3.3.2 Model 3-L Model 3-L (“L” is short for “lumped”) is developed for investigating the sta-bility of the Nordic power system in frequency domain, in section 6.2 based on Paper XII. The corresponding model for time domain simulation is Model 2-L.

3.3.2.1 Details of the model The block diagram of the power system, for frequency domain analysis, is shown in Figure 3.14. Comparing to the time domain simulation Model 2-L, the transfer functions of the PI controller, the plant and the grid in Model 3-L are the same, as shown in Equations (3.29), (3.30) and (3.31). However, the non-linear components of the actuator are ignored here.

51

Figure 3.14. Block diagram of the power system model for frequency domain analysis

The term F(s) in Figure 3.14 is a general transfer function for the filter. It is replaced by specific transfer functions of different filters according to the cor-responding case. The transfer functions of the linear filters are the same as the ones in Model 2-L, as shown in Equations (3.27) and (3.28).

The describing functions, as introduced in section 2.1.3.4, are applied to analyse the nonlinear filters in frequency domain. The describing function of a dead zone of size Edz [130] is

2

1( ) 2 arcsin 12dz dz dz

dzdzN AA A A

E E EA E

. (3.41)

The describing function of a floating dead zone of size Efdz (backlash) [130] is

2

2

21( ) arcsin 1 1

2

21

fdz fdz fdf

z fdz

fdz f

dz

dzfdz

E E E E

E EA

N AA A A A

jA A

E

(3.42)

Here, the parameter A means the amplitude of the periodical input signal.

3.3.2.2 Transfer functions for Nyquist stability criterion In order to investigate the system stability, the open-loop systems with differ-ent filters are examined by the Nyquist stability criterion, as introduced in sec-tion 2.1.3.3.

For analysing the system with the dead zone and floating dead zone, the transfer function of the open-loop system is

1

3 201 11 21 31

4 3 201 11 21 31 41

(s) ( ) P( ) ( )

C s s G s

b s b s b s b

a s a s a s a s a

. (3.43)

For the open-loop system with the linear filter, the transfer function is

2 1

3 202 12 22 32

4 3 202 12 22 32 42

(s) ( ) ( ) P( ) ( )

fF s C s s G s

b s b s b s b

a s a s a s a s a

. (3.44)

All the transfer function coefficients here are shown in Appendix A.

52

3.3.3 Model 6 Model 6 is built for eigen-analysis on the rotor angle stability in section 5.1 based on Paper IX. The corresponding model for time domain simulation is Model 5.

3.3.3.1 Details of the model The overall structure of Model 6 is the same with Model 5, as shown in Figure 3.9. It is a SMIB system with an extended model of a HPP described by a state matrix, ignoring all the nonlinear factors.

The generator and the power grid are modelled with the same approach as Model 4, as shown in Equations (3.21) through (3.24).

The AVR and the PSS (with the speed input and the power input) are the same as the ones in Model 5 and 5-S, with the output saturation removed, as described in Equations (3.25) and (3.32).

The turbine and waterway system is described by a linear model, the same as the one in Model 5. The corresponding equation of the turbine is

qy q qh

m y h

q e y e e h

P e y e e h

. (3.45)

The corresponding equation of the elastic water column is

2 21w

e

T sh

q T s

. (3.46)

From Equations (3.45) and (3.46), two differential equations are deduced:

1

12

1w qy q qh

r

d hh

dt

d h d y d d hh T e e e

dt T dt dt dt

(3.47)

The governor system is a linearized version of the one in Model 5, as illus-trated in Figure 3.11, excluding the nonlinear components with dashed-out-line. The equation of frequency control with the OF is

1 PIp p p i PI p i

d y db K b K y K K

dt dt

. (3.48)

The equation of the PF is

ePIp p p i e p i

d Pd y db K b K P K K

dt dt dt

. (3.49)

Note that the PF signal here is the electromagnetic power, not the mechanical power. The servo is described by

y PI

d yT y y

dt

. (3.50)

53

3.3.3.2 State matrix for eigen-analysis As introduced in section 2.1.3.5, the small signal stability of the system can be analysed by investigating the eigenvalues of the state matrix. For the whole system of Model 6, there are twelve differential equations, i.e. five equations for the generator, one equation for the AVR, two equations for the PSS, two equations for the turbine with the waterway system, one equation for the servo and one equation for the PI controller. Hence, a 12×12 state matrix with twelve corresponding state variables is derived, as shown in

1,2

2,1 2,2 2,4 2,5 2,8 2,10

3,1 3,3 3,4 3,6

4,1 4,3 4,4

5,1 5,5

1

1

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

q

q

d

fd

PI

PSS

a

a a a a a aE

a a a aE a a a

E a a

E

y

y

h

h

V

V

6,1 6,4 6,5 6,6 6,12

7,1 7,2 7,3 7,4 7,5 7,7 7,8 7,10

8,7 8,8

9,1 9,2 9,4 9,5 9,7 9,8 9,9 9,10

10,9

11,1 11,2 11,3 11,4 11,5 11,8 11,10 11,11

12,1 12,2 12

0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

a a a a a

a a a a a a a a

a a

a a a a a a a a

a

a a a a a a a a

a a a

1

1

,3 12,4 12,5 12,8 12,10 12,11 12,120 0 0

q

q

d

fd

PI

PSS

E

E

E

E

y

y

h

h

Va a a a a a V

(3.51) All the non-zero elements, , , of the state matrix are given in Appendix A. Analyses are conducted based on damping ratios corresponding to different eigenvalues in the system for each case. The smallest damping ratio is selected as the main indicator of the system stability.

3.4 Numerical models in MATLAB for HPPs with Kaplan turbines (Model 2-K)

Model 2-K (“K” is short for “Kaplan turbine”) is a numerical HPP model with a Kaplan turbine implemented in applying Simulink, calibrated with measure-ments from two Swedish HPPs (HPP 6 and HPP 7). It is established for quan-tifying relative values of regulation burden and performance of PFC in section 6.3 based on Paper XIV.

Model 2-K is divided into several sub-models, i.e. Model 2-K-1, Model 2-K-2 and Model 2-K-3 for different purposes that are introduced in section 6.3. In this section, Model 2-K-1 and Model 2-K-2 are presented, and Model 2-K-3 is introduced additionally in section 6.3. The per-unit (pu) system is adopted for describing all the models.

The overall structure of the model is shown in Figure 3.15. The open loop “hydropower plant” model with the red dashed outline is Model 2-K-1, for

54

simulating the transient processes within a HPP. In this thesis, it is mainly applied for computing the efficiency (η), power output (pm), GVO and RBA. The closed loop model with the blue dashed outline is Model 2-K-2, for sim-ulating the frequency quality of the whole power system.

Figure 3.15. Model 6: Overall model structure of a hydropower system with a Kaplan turbine. Some detailed set points and feedback signals are omitted here, but included in the more detailed block scheme shown in the following content.

3.4.1 System components

3.4.1.1 Kaplan turbine and waterway system The Kaplan turbine and waterway system model is illustrated in Figure 3.16. The active power from the turbine is described by the classical simplified non-linear model [15, 32]:

0 0q q q G h G h h , (3.52)

3/2mp qh Gh . (3.53)

The above two equations are for single-regulated turbines, e.g. Francis tur-bines. While for the double-regulated turbine, G is a comprehensive gate opening that is identified from the values of GVO (y) and RBA (a), as shown in a fitting function

,FG G y a . (3.54)

The efficiency value is from an interpolation function ,I y a . (3.55)

These two functions are achieved from on-site measurement data that is pre-sented separately in section 3.4.2.

55

Figure 3.16. Block diagram of a model of Kaplan turbine and waterway system in Model 2-K. The signal of RBA is presented in blue, for distinguishing it from the GVO signal.

The head is affected by hydraulic dynamics from the elastic penstock, draw water tunnel and surge tank [32], as shown in

0

p s

h h h

h h h

. (3.56)

In terms of frequency domain, the transfer function describing the head vari-ations due to discharge deviations in the penstock is

2

2 21p wp p e p e p

pe p e

h T T f s T s fG s

q T T s

. (3.57)

For the head variations due to discharge deviations in the surge tank, the trans-fer function is

2 1s wt t

swt s s t

h T s fG s

q T T s T f s

. (3.58)

When the turbine damping (D) [32] is included, the equation of the active power becomes

3/2mp Gh DG f . (3.59)

3.4.1.2 Governor system with filters The model of a governor system for a Kaplan turbine is demonstrated in Fig-ure 3.17.

56

Figure 3.17. Block diagram of governor system of the Kaplan turbine in Model 2-K.

A standard PID (proportional–integral–derivative) controller with droop, common mechanical components and a 2-D (two-dimensional) lookup table, and the artificial filter for the signal of RBA are included. The filter for RBA is a floating dead zone (or floating dead band) that is the same as the one presented in section 3.2.1, which is equivalent to the backlash in the Simulink model.

3.4.1.3 Power grid For investigating the frequency quality under different operation strategies, a simplified model [129, 139] representing the Nordic power grid is applied, as shown in Figure 3.15. The transfer function is the same as the one in Model 2-L, as described in Equation (3.31).

Figure 3.18. Block diagram of a model of lumped HPP in Model 2-K. The model con-tains a governor and a simplified representation of Francis turbine.

3.4.1.4 Lumped HPP The lumped HPP represents the rest of regulating units in the power grid, by assuming that all the regulation in the grid is provided by hydropower. The principle of modelling the lumped HPP is the same as it is for Model 2-L, expect for the disposition of the corresponding scaling factor.

As shown in Figure 3.18, the model contains a governor and a simplified representation of a Francis turbine and waterway system, as described in

57

1

( )0.5 1

wt

w

T sG s

T s

. (3.60)

3.4.2 Turbine characteristic from measurements Model development regarding the turbine characteristics is a key point here. As shown in Table 3.1, a specific model for describing turbine characteristic is developed for Model 2-K, comparing to the common approach by applying turbine characteristics curves. More exactly, the comprehensive gate opening and efficiency, as shown in Equations (3.54) and (3.55), are modelled from specific on-site index tests [140, 141] data from HPP 6 and HPP 7.

Here, the principle of the approach is introduced and more details can be found in Paper XIV.

The measured scatter data of comprehensive gate opening and efficiency can be calculated for a limited operating region and applied for the fitting or the interpolation. By applying the surface fittings, the turbine characteristics for a larger operation range can be obtained. A relatively higher efficiency accuracy is demanded by the economic analysis, and it can be achieved by interpolation of the efficiency data. However the measurement data are only available in a certain range, hence extra data for larger operating points are added from the fitting data for the extrapolation.

3.4.2.1 Fittings The comprehensive gate opening and efficiency for each HPP are fitted to a quadratic polynomial surface using

2 200 10 01 20 11 02,F G G G G G GG G y a p p a p y p a p ay p y , (3.61)

2 200 10 01 20 11 02,F y a p p a p y p a p ay p y . (3.62)

where, pGij and pηij are the coefficients of the fitting. The fittings are demon-strated in Figure 3.19, taking HPP 7 as an example. The fitting of an operating point far from the on-cam range might not be accurate, however, this will not affect the results of this study because the fitting is only applied for small disturbance simulations.

3.4.2.2 Interpolations and extrapolation of efficiency Piecewise cubic interpolation is applied to obtain the final efficiency data, as shown in Figure 3.20. By adopting the added points (in red) from the fitting data, the small operation range covered by the index tests is extended, then the efficiency data can support all the small disturbance simulations (in section 6.3).

58

(a)

(b)

Figure 3.19. Fitting of data of HPP 7: (a) comprehensive gate opening GF and (b) turbine efficiency data. The efficiency value is normalized with respect to the maxi-mum efficiency value.

Figure 3.20. Interpolation of turbine efficiency data of HPP 7. The red scatters are extracted from the fitting for extrapolation.

59

4 Stable operation regarding frequency stability

The stable operation of HPPs regarding frequency stability of power systems is analysed in this chapter. In section 4.1 (based on Paper I), case studies on different operating conditions of HPPs are conducted, and the simulation per-formance of Model 1 based on TOPSYS is presented. In section 4.2 (based on Paper II), response time of PFC in HPPs is investigated under grid-connected operation. In section 4.3 (based on Paper III), frequency stability of HPPs in isolated operation is studied.

All the content in these three sections mainly focus on active power control, of which the ultimate goal is achieving a better frequency stability of power systems.

4.1 Case studies on different operating conditions The application of Model 1 based on TOPSYS is presented in this section by comparing simulations with on-site measurements, based on four engineering cases: a Swedish HPP (HPP 1 shown in Figure 3.2) and three Chinese HPPs (HPP 2 – HPP 4 shown in Figure 4.1).

Figure 4.1. TOPSYS models of three Chinese HPPs: HPP 2 – HPP 4 in Table 2.1.

The aim of Model 1 is to achieve accurate simulation and analysis of different operation cases, e.g. small disturbance, large disturbance, start-up and no-load operation, etc. A good simulation in Model 1 is a crucial basis of the thesis,

60

such as for the studies in section 4.2 (grid-connected operation), section 4.3 (isolated operation) and section 6.1 (grid-connected operation).

4.1.1 Comparison of simulations and measurements For normal operation (small disturbances in grid-connected operation and iso-lated operation), the comparison between simulations with Model 1 and meas-urements are shown in Figure 4.2 through Figure 4.4.

Figure 4.2. Grid-connected operation: power output and opening from simulation and measurement under sinusoidal frequency input (HPP 1). In the figures of this thesis, the “M” refers to measurements and the “S” means simulation.

Figure 4.3. Grid-connected operation: power from simulation (“S-”) and measure-ment (“M-”) under step frequency input (HPP 2).

Overall, the simulation has a good agreement with the measurements. As shown in Figure 4.2, the effect of backlash is reflected: the GVO keeps stable for a short period during the direction change process (e.g. around t = 28 s). In Figure 4.3, after the frequency step change, the phenomenon of power re-verse regulation caused by water inertia is simulated accurately, as well as the

61

gradual power increase or decrease due to the surge (after 20 seconds). How-ever, the simulation of the power decrease has a lower value than the meas-urement. This deviation could be ascribed to the characteristic curve, to some extent, the on-site measurements inevitably deviate from the simulation that is based on the data from the model tests. The oscillation after a load step change is examined by simulation and compared with the measurements, as shown in Figure 4.4. The simulation reflects the real operating condition well: under the power control mode in HPP 3, the power oscillates with the surge oscillation under certain governor parameter settings due to a relatively small cross section of the surge tank.

Figure 4.4. Isolated operation: Simulation (“S-”) and measurement (“M-”) of the power oscillation under power control mode (HPP 3).

Figure 4.5. Frequency and opening from simulation and measurement during a start-up process (HPP 2). The “S” means the simulation with the original characteristic curve of turbine, and the “S2” means the simulation with the modified characteristic curve.

For the start-up process, a case study of HPP 2 is simulated and compared with measurements, as shown in Figure 4.5. In the simulation with the original characteristic curve of the turbine (S), the simulated frequency increase pro-cess is approximately 30% shorter than the measured one, hence the opening

62

from simulation decrease to the no-load opening slightly earlier than the meas-ured opening. Therefore, the curve was modified by decreasing the efficiency. With this revised characteristic curve, the new simulation (S2) fits the meas-urement well. It demonstrates that the inaccurate simulation mainly hinges on errors in the characteristic curve, which is especially error-prone in the small-opening operation range. Due to that for the small-opening operation range, the original input data achieved from the characteristic curve is not accurate enough and very sparse, it is hard to obtain a good predictive simulation.

Figure 4.6. Simulation and measurement of the (a) GVO and pressure in the volute; and (b) pressure in the draft tube, during a load rejection process (HPP 4).

For the load rejection process, the pressures at the inlet of the volute and in the draft tube are simulated and compared with measurements in HPP 4, as demonstrated in Figure 4.6. For the simulated pressure, there is a small static deviation from the measurement after load rejection. It might be due to the water head error caused by the characteristic curve and imprecise parameters of the waterway system.

Moreover, the pulsating pressure at volute and draft tube in the measure-ment cannot be reproduced by the simulations because of the limitation of the one-dimensional characteristic method. The pressure measurement in the draft tube might also be difficult to compare to modelled values, due the swirl not being modelled in the 1-D modelling approach, making the actual water ve-locity past the pressure transducer deviate from the mean velocity in an un-known way.

4.1.2 Discussion The results above show that Model 1 can yield trustworthy simulation results for different physical quantities of the unit under various operating conditions. The main error sources of the simulation are the characteristic curves of the turbine, provided by manufacturers, which directly causes small deviations of power output and affects the rotation speed and pressure values. The reason

63

might be that the characteristic curves do not really describe the on-site dy-namic process accurately, and the error is especially obvious in the small-opening operation range. Furthermore, waterway system parameters might also have errors that impact the simulation.

4.2 Response time for primary frequency control For evaluating the regulation quality of hydro units in PFC, a key is the power response time. How do the regulation and water way system affect the re-sponse time? How should governor parameters be set to control the power response time? These problems are the focuses of this section.

Figure 4.7. Illustration of different times under frequency step disturbance. The open-ing means GVO.

The aim of this section is to investigate general rules for controlling the power response time of PFC. Firstly, specifications of the response of PFC in differ-ent regions are introduced. Then, from the analytical aspect, a time domain solution for GVO response and a response time formula are deduced. Case studies of HPP 2 are conducted by simulations based on Model 1, to investi-gate various influencing factors.

The response time (deployment time) Tresponse and the delay time Tdelay of power response process are the key indicators in this section, as shown in Fig-ure 4.7. The difference between the power response and the GVO response is the focus.

64

4.2.1 Specifications of response of PFC Strictly speaking, the parameters need to be tuned and tested for PFC in every HPP, for meeting the requirement of specifications that varies in different re-gions.

(1) China Electricity Council Based on specifications of China Electricity Council [142], if the units are

operating on 80 % of the rated load, the power response for a frequency step should meet a series of requirements. The most crucial requirements are: the power adjustment quantity should reach 90 % of the static characteristic value within 15 seconds. If the rated head of the unit is larger than 50 m, the power delay time should be less than 4 seconds.

(2) ENTSO-E According to the specifications of ENTSO-E [143], the time for starting the

action of primary control is a few seconds starting from the incident, the de-ployment time of 50 % of the total primary control reserve is at most 15 sec-onds, and the maximum deployment time rises linearly to 30 seconds for the reserve from 50 % to 100 %.

(3) The Nordic power grid Currently, the Norwegian TSO Statnett has no specific requirements on the

response time, but prescribes limits on certain quantities, such as on the delay between frequency deviation and incipient GV motion, on the resolution in frequency measurement, on the permanent droop, and on how to measure these parameters [144].

In Sweden, the TSO Svenska Kraftnät (SvK) has demands on response time, but no requirements on details [145]. The requirements depend on the magnitude of the frequency deviation, and if it exceeds 0.1 Hz, 50 % should be delivered within 5 s, and 100 % within 30 s.

4.2.2 Formula and simulation of response time Based on the theory in section 2.1.3.1, a formula for the GVO response time of PFC under opening control is deduced, for a PI controller with droop and servo. The main variables of the formula are governor parameters, as de-scribed in

1

1ln (1 ) 1p p

p p p i yp i

b KT b K b K T

b K

. (4.1)

Here, Δ is the target value, for example, it is set to 90% according to specifi-cations of China Electricity Council [142].

Simulations based on HPP 2 under different conditions are conducted to analyse the sensitivity of response time with respect to the main parameters. The default settings of the simulation are given in Appendix B. In order to

65

investigate the influence of surge in upstream surge tank, a HPP model with-out surge tank is built, still by applying Model 1. More exactly, in the simpli-fied model, the surge tank and the upstream pipeline before the tank are re-placed by a reservoir. The simulation results are shown in Table 4.1.

The power response time, T4, can be expressed as 4 1 1 1 2 3T T T T T T T . (4.2)

The time difference ΔT (as shown in Figure 4.7), between the power response time (T4) and the analytical response time of GVO (T1), is mainly affected by the rate limiting and numerical algorithm (ΔT1), the water inertia (ΔT2) and the surge (ΔT3).

More detailed results can be found in Paper II.

Table 4.1. The response time of frequency step under different conditions. T1 and T2 are calculated by Equation (4.1) and simulation respectively, and T3 is simulated with the simplified model; Response time of opening or power means the time when the opening or power reaches the target value Δ. All the simulations are conducted with rate limiting which is 12.5%/s. The green bar in each cell indicates the relative mag-nitude of the values in the corresponding cells, except for the cells highlighted by yellow (the values in the yellow cells are much larger).

1 2.0 4.0 0.04 0.020 90% 15.0 14.6 16.0 21.2 -0.4 1.4 5.2

2 2.0 6.0 0.04 0.020 90% 10.0 9.8 11.4 12.4 -0.2 1.6 1.0

3 2.0 2.0 0.04 0.020 90% 30.1 29.2 30.0 232.2 -0.9 0.8 202.2

4 0.2 4.0 0.04 0.020 90% 14.5 15.0 16.4 21.6 0.5 1.4 5.2

5 10.0 4.0 0.04 0.020 90% 17.2 14.6 16.0 21.0 -2.6 1.4 5.0

6 2.0 4.0 0.02 0.020 90% 29.4 29.0 29.2 229.4 -0.4 0.2 200.2

7 2.0 4.0 0.06 0.020 90% 10.2 9.8 12.0 13.2 -0.4 2.2 1.2

8 2.0 4.0 0.04 0.005 90% 15.0 14.8 16.2 21.2 -0.2 1.4 5.0

9 2.0 4.0 0.04 0.500 90% 15.5 15.0 16.4 21.6 -0.5 1.4 5.2

10 2.0 4.0 0.04 0.020 80% 10.4 10.2 11.6 12.4 -0.2 1.4 0.8

11 2.0 4.0 0.04 0.020 70% 7.6 7.6 9.2 9.6 0.0 1.6 0.4

No.

ParametersResponse time of

opening (s)Response time of

power (s)Time difference ΔT (s)

K p K i b p T y ΔΔT3=

T4 -T3

Formula

T1

Simulation

T2

Without surge

tank, T3

With surge

tank, T4

ΔT1=

T2 -T1

ΔT2=

T3-T2

4.3 Frequency stability of isolated operation For HPPs with surge tank, the Thoma criterion [146, 147] is often violated to diminish the cross section of surge tank with the scale getting larger nowa-days. Therefore, the surge fluctuation is aggravated and frequency stability becomes more deteriorative [148]. Recently, some huge Chinese HPPs en-countered this instability problem during the commissioning, measurements

66

under a load step disturbance are shown in Figure 1.3. Hence, the focus of this section is on stabilizing the very low frequency oscillation (see section 1.2.2) of an isolated HPP caused by surge fluctuation.

In this section, by means of theoretical derivation based on Model 3-F, sta-bility conditions under two control modes are contrasted through adopting the Hurwitz criterion. Then, the frequency oscillations are simulated and investi-gated with different governor parameters and operation cases, by applying Model 1. The engineering case here is HPP 2.

4.3.1 Theoretical derivation with the Hurwitz criterion Based on Model 3-F in section 3.3.1 and the theory in section 2.1.3.2, a sta-bility condition of frequency oscillation under frequency control and power control is obtained.

The stability region is the region which satisfies the stability condition in Ki-n coordinates by substituting the system parameters of different states into the stability condition. Here, n (n = F/Fth) stands for the coefficient of cross section area of the surge tank, where F and Fth are the real area and Thoma critical area, respectively.

Figure 4.8. Stability region in Ki-n coordinates of two control modes

A set of curves of stability region boundaries is achieved under two control modes based on the stability condition, as shown in Figure 4.8. The stability region of power control is much larger than which of frequency control. There is no proportional gain (Kp can be regarded as 0) in power control, and it is conducive to the stability. However under frequency control, when Kp is set to near 0, the stability region is still smaller than for power control.

67

4.3.2 Numerical simulation Based on Model 1 in TOPSYS, numerical simulations are conducted to vali-date the result of the theoretical derivation, as shown in Figure 4.9 and Figure 4.10. Through the simulation, the conclusion drawn in the theoretical deriva-tion is verified: the power control produces a better effect on stability than the frequency control. More exactly, under the frequency control, it is hard to sta-bilize the frequency by adopting any of the three sets of parameters. Even when Kp is set to nearly 0, to compare with the power controller that is without proportional component (Kp = 0), the frequency instability still occurs. While under the power control, frequency stability is well ensured, and the contra-diction between rapidity and stability is also indicated.

Figure 4.9. Frequency oscillation under frequency control with different governor parameters

Figure 4.10. Frequency oscillation under power control with different governor pa-rameters

Besides, it is necessary to have an additional discussion on the power control. The applying of the power control in the isolated operation condition is an ideal case, which cannot be implemented in the practical HPP operation. It is because that the load is unknown in reality, therefore the given power cannot be set properly. However, the conclusion based on the idealized case can sup-ply the understanding and guidance for the stability in an islanded operation, which means the operation of a generating unit that is interconnected with a relatively small number of other generating units [136]. In the islanded sys-tem, some units operate in the frequency control mode to balance the changing peak load, and other units adopt the power control to maintain the stability. This issue is also a suggested topic for future work.

More results and discussions are in Paper III.

68

5 Stable operation regarding rotor angle stability

The stable operation of HPPs regarding rotor angle stability of power systems is studied in this chapter. In section 5.1 (based on Paper IX), a fundamental study on hydraulic-mechanical-electrical coupling mechanism for small signal stability of HPPs is conducted by eigen-analysis. Considerable influence from hydraulic-mechanical factors is shown, and it is further quantified in section 5.2 (based on Paper X): An equivalent hydraulic turbine damping coefficient and the corresponding methodology are proposed to quantify the contribution on damping of rotor angle oscillations from hydraulic turbines based on re-fined simulations. In section 5.3, the quick hydraulic – mechanical response is discussed to support the results in this chapter. The engineering case of all sections is HPP 5.

5.1 Hydraulic – mechanical – electrical coupling mechanism: eigen-analysis

This section aims to conduct a fundamental study on hydraulic-mechanical-electrical coupling mechanism for small signal stability of HPPs, focusing on the influence from hydraulic-mechanical factors.

For the local mode oscillation [15] in a SMIB system, the theoretical eigen-analysis (section 2.1.3.5) is the core approach based on Model 6 (section 3.3.3) that is a twelfth-order state matrix. Numerical simulation by applying Model 5 (section 3.2.2) is also conducted for validation. As shown in Figure 1.2, three principal time constants for water column elasticity (Te), water inertia (Tw), and servo (Ty) in the hydraulic-mechanical subsystem are the main study ob-jects. They are analysed under two modes of frequency control (OF and PF) without the PSS. Then, the influence from the hydraulic-mechanical subsys-tem on tuning of the PSS is investigated. Detailed parameter values and oper-ating settings are given in Appendix B.

5.1.1 Influence of water column elasticity (Te) Through Model 6, for each case (one combination of the value of Kp and Te), the smallest damping ratio (ξ) of all oscillation modes is plotted. As shown in

69

Figure 5.1 (a), when the value of Te is small (short penstock), the increased response rapidity of the frequency control (indicated by an increase of the Kp value) with OF leads to a smaller damping ratio of the system. On the contrary, when the value of Te is larger than a certain value, the system becomes more stable with stronger frequency control. Moreover, the trend is inverted when the governor applies PF, as shown in Figure 5.1 (b); the increased strength of the frequency control stabilizes the system with the small value of Te. Also, PF generally leads to higher damping ratios than OF.

(a)

(b)

Figure 5.1. The smallest damping ratio (ξ) of all oscillation modes under different values of governor parameters (Kp) and time constant of water column elasticity (Te). (a) The feedback mode is OF; (b) The feedback mode is PF.

The observations above are validated by time domain simulations. As pre-sented in Figure 5.2, the black line, the blue line and the red line correspond to cases with a high, medium and low damping ratio respectively. The simu-lation results of these three sets of parameters fit the damping ratio well. In short, the impact of water column elasticity is important and it differs from the feedback mode of the frequency control.

70

Figure 5.2. Time domain simulation of the process after the three phase fault: Rota-tional speed. The result validates the cases in Figure 5.1.

5.1.2 Influence of mechanical components of governor (Ty) The rapidity of the GVO response is highly affected by the mechanical com-ponents in the governor system, e.g. servo, backlash, rate limiter, etc. In the state matrix, these components are simplified and represented by the servo time constant (Ty). As shown in Figure 5.3, a small value of Ty leads to a quicker response of GVO, and brings clearer influence on system stability. The influence of Ty is more obvious when Kp is larger. A time domain simu-lation in Figure 5.4 (a) illustrates the influence and it corresponds to the result in Figure 5.3 (b).

Figure 5.3. The smallest damping ratio (ξ) of all the oscillation modes under different values of servo time constant (Ty) and governor parameters (Kp). (a) The feedback mode is OF, Te = 1.0 s; (b) The feedback mode is OF, Te = 0.01 s; (c) The feedback mode is PF, Te = 1.0 s; (d) The feedback mode is PF, Te = 0.01 s.

71

(a)

(b)

Figure 5.4. Simulation of rotational speed after the three phase fault. The governor adopts the OF; (a) cases under different values of Ty: the result validates the cases in Figure 5.3 (b); (b) cases with and without rate limiter

Moreover, the linear theoretical model can result in negative damping ratios (Figure 5.1 and Figure 5.3). However the simulated oscillations are not diver-gent, because of the added damping by the nonlinear components (mainly from the rate limiter) in the numerical model, as shown in Figure 5.4 (b).

5.1.3 Influence of water inertia (Tw) The water inertia, represented by the water starting time constant (Tw), is nor-mally regarded as adverse to system stability, especially in islanded operating conditions, as presented in section 4.3. By contrast, for the SMIB system, the influence of water inertia is not monotonic, as demonstrated in Figure 5.5 and Figure 5.6.

It is shown that the effect of water inertia differs from that of water column elasticity. Figure 5.5 (a) and Figure 5.6 present that a larger value of Tw leads to smaller damping ratio when the value of water column elasticity (Te) is around 0.4. However, the increase of Tw results in slightly more stable cases when the value of Te is large or small, as demonstrated in Figure 5.6 and Figure 5.5 (b). When the governor adopts the PF, the system is more stable under

72

larger water inertia in this case, and this is validated by time domain simula-tions (Figure 5.7).

Figure 5.5. The smallest damping ratio (ξ) of all oscillation modes under different values of water starting time constant (Tw) and governor parameters (Kp). (a) The governor adopts the OF, Te = 0.4 s; (b) The governor adopts the OF, Te = 0.01 s; (c) The governor adopts the PF, Te = 0.4 s; (d) The governor adopts the PF, Te = 0.01 s.

Figure 5.6. The smallest damping ratio (ξ) of all the oscillation modes under different values of Tw and Te. The governor adopts OF.

73

Figure 5.7. Simulation of rotational speed after the three phase fault under different values of Tw. The governor adopts PF. The result validates the cases in Figure 5.5 (d).

5.1.4 Influence on tuning of PSS Here, the influence of hydraulic-mechanical factors on tuning of the PSS is investigated. The damping effects under implementation of the PSS with dif-ferent settings of gain Ks under various conditions are shown in Figure 5.8.

Figure 5.8. The smallest damping ratio (ξ) of all the oscillation modes under different gains of PSS (Ks): (a) OF in governor and speed input in PSS; (b) OF in governor and power input in PSS; (c) PF in governor and speed input in PSS; (d) PF in gover-nor and power input in PSS.

Two main insights are obtained here. (1) The optimal parameters values vary with different types of feedback. More exactly, for achieving the largest damp-ing ratio, the Ks value differs in four cases, shown in Figure 5.8 (a) - (d). The optimal value of Ks changes with different types of PSS, meanwhile, the tun-ing is also affected by the water column elasticity. (2) The stability margin changes considerably under various conditions. The frequency control with

74

the PF generally results in a higher damping ratio, and this is validated by the time domain simulations in Figure 5.9.

In short, it can be observed that there is still room for optimizing the pa-rameters and performance of the PSS by considering the effect of the hydrau-lic-mechanical factors.

Figure 5.9. Simulation of rotational speed after the three phase fault under different modes (OF and PF) of governor. The PSS adopts speed input and the gain (Ks) is set to 4.0. The result validates the cases in Figure 5.8 (a) and (c).

5.2 Quantification of hydraulic damping: numerical simulation

Damping coefficient is a common term (D) used in power system stability analysis, and its general form is described in a linearization of the swing equa-tion

j m eT P P D (5.1)

However, the variation range of D in the hydropower field is still unclear; it is normally assumed to be positive and often set to zero to obtain a conservative result. Therefore the swing equation is rewritten as

j m eT P P . (5.2)

The aim of this section is to quantify the contribution from a hydraulic tur-bine to the damping of local mode electromechanical oscillations [15]. An equivalent hydraulic turbine damping coefficient (Dt, simplified as “the damp-ing coefficient” in the following context) is introduced here, as described in

,j m e m const t eT P P P D P . (5.3)

In this study, the focus is on the mechanical power (Pm), instead of the elec-tromagnetic power (Pe) that is the main analysis object in previous studies.

The purpose of introducing the damping coefficient is as follows. (1) Quan-tifying the value of Dt can clarify an long-standing issue: how large is the damping contribution from the hydraulic system? (2) For analysis of large

75

power systems, the mechanical power simulation in HPPs is inevitably sim-plified and less accurate, misleading the analysis of power system oscillations. The quantified damping coefficient can be easily implemented in models of complex multiple-machine systems, hence the mechanical power in the sys-tem can be set to constant without losing the influence from the hydraulic sys-tem on the system stability (shown in section 5.2.3). (3) Considering the damping coefficient can affect the system parameter tuning, including the PSS tuning (shown in section 5.2.3).

In this section, firstly, the corresponding methodology is introduced. Then, the quantitative results of the damping coefficient are presented in different cases with and without the application of PSS. Lastly, the influence and sig-nificance of the damping coefficient are demonstrated in case studies.

5.2.1 Method and model

5.2.1.1 Method of quantifying the damping coefficient The method of quantifying the damping coefficient is based on simulations by Model 4 and Model 4-S (“S” is short for “simplified”), as shown in Table 5.1. The only difference between these two models is that the swing equation in Model 4 and Model 4-S is (5.2) and (5.3) respectively; it means that the me-chanical power is simplified as constant and the whole model of the hydraulic subsystem is ignored in Model 4-S.

Table 5.1. Different numerical models in this section

Model Description Purpose

4 TOPSYS model with refined hydrau-lic-mechanical subsystem introduced in section 3.1.2

Refined simulation

4-S Simplified version of Model 4, with the swing equation in (5.3)

Quantifying the damping co-efficient by the comparison with Model 4

5-S A MATLAB/SPS model mentioned in section 3.2.2

Verifying the TOPSYS model (Model 4 and 4-S)

The detailed steps are as follows:

(1) Step 1: Apply Model 4 to simulate transient processes of a HPP after a three phase fault;

(2) Step 2: Adopt Model 4-S to simulate transient processes with different values of the damping coefficient Dt. Other conditions remain the same as the case in Step 1.

(3) Step 3: Compare the damping performance (reflected by the rotational speed) from the two simulation models; among results from Model 4-S with

76

different values of Dt, one of the curves has the best agreement with the sim-ulation from Model 4; thus the corresponding value of Dt is the quantified damping coefficient for this case.

The detailed method for determining the agreement between results from two models in Step 3 is by applying the root mean square error (RMSE), as described in

2

4, 4 ,1

1 N

RMSE i S ii

P P PN

(5.4)

Here, Pi means a value of a local maximum or local minimum (peaks of a curve) of speed deviation (Δω); the subscript 4 and 4-S stand for Model 4 and Model 4-S respectively; N is the total number of local maxima and minima during a certain time period (10 seconds in this paper). The minimum value of the PRMSE indicates the best agreement of two curves of rotational speed simulated by Model 4 and Model 4-S.

The engineering case of this section is HPP 5, as shown in Figure 5.10. Detailed parameter values and operating settings are given in Appendix B.

Figure 5.10. TOPSYS model of HPP 5, of which a single unit is connected to an infi-nite bus

5.2.1.2 Model verification Here, the verification of the TOPSYS Model is presented. The hydraulic-me-chanical subsystem of the model has been verified by measurements in differ-ent cases, as shown in section 4.1. Therefore, the main focus here is verifying the electrical subsystem, by comparing the electrical transients simulated from Model 4-S (TOPSYS) and Model 5-S (SPS) in Table 5.1.

Model 5-S is a standard SPS model, as shown in Figure 5.11, and the swing equation of it is described by (5.3).

77

Figure 5.11. Model 5-S in SPS. The block with dashed outline shows the implementa-tion of the simplified mechanical power by applying the damping coefficient (Dt).

(a)

(b)

(c)

(d)

Figure 5.12. Comparison between Model 4 (TOPSYS) and Model 4-S (SPS), without the application of PSS. In both models, the mechanical power Pm is constant. (a) Ro-tational speed; (b) Excitation voltage; (c) Generator terminal voltage; (d) Electro-magnetic active power.

Comparisons between simulations by Model 4-S and Model 5-S are shown in Figure 5.12. In both models, the mechanical power is constant. Generally, the simulations by Model 4-S have a good agreement with the results from Model

78

5-S. The main difference occurs in the damping performance, which is ex-pected. The reason is the stator transients, which contribute to the damping [15], are included in the SPS model but ignored in the equivalent circuit model in TOPSYS. Therefore, the TOPSYS model leads to slightly more conserva-tive results; however it does not affect much the quantification of the damping coefficient, since both Step 1 and 2 are conducted by the TOPSYS model.

5.2.2 Quantification of the damping coefficient Here, the quantification results of the damping coefficient are presented for different cases with and without the application of the PSS. The simulation cases are listed in Table 5.2.

Table 5.2. Different simulation cases under the three phase fault. Other settings of all the cases are the same, apart from the descriptions. The “delay” means the delay time in the turbine governor.

Case Description Purpose

1 Delay = 0.30 s; No PSS Demonstrating a positive and a negative damping coefficient under cases without the PSS 2 Delay = 0.50 s; No PSS

3 Delay = 0.25 s; with PSS Demonstrating a positive and a negative damping coefficient under cases with the PSS 4 Delay = 0.50 s; with PSS

Without the application of the PSS, two examples of quantifying the damping coefficients are shown in Figure 5.13. The values of the damping coefficient are quantified as 2.0 and -1.1 respectively. The main reason for the difference in the damping performance is the phase shift in the mechanical power re-sponse with respect to the rotational speed deviation, and the delay time is the most influential fact affecting the phase shift.

A crucial point here is that the damping coefficient can vary over a consid-erable range and can even be negative, while previously the contribution is unclear and normally assumed to be positive.

The change processes of the GVO and mechanical power under case 1 and case 2 are shown in Figure 5.14. The phase shift between the mechanical power and GVO is approximately 180º, clearly demonstrating the non-mini-mum-phase response of the mechanical power. Also, it is shown clearly that the phase shift between the power and the speed is changed due to different delay times.

79

(a)

(b)

Figure 5.13. Quantification of the damping coefficients for cases without the PSS

(a)

(b)

Figure 5.14. Simulation of GVO, mechanical power and speed. (a) Case 1; (b) Case 2. The GVO and mechanical power are deviations from initial values. The curves of speed are exactly the same as the ones in Figure 5.13.

For the cases with application of the PSS, the influence from the hydraulic turbine is still obvious. As shown in Figure 5.15, the damping coefficient is quantified as 1.5 and -2.1 respectively for case 3 and case 4. The quantifying method is basically the same as above, the only difference is that the PSS is activated in both Model 4 and Model 4-S.

Considering the influence from mechanical power can contribute to a better tuning of PSS, as further discussed below. The quantified damping coefficient

80

is convenient to add in cases of PSS tuning for complex multiple-machine systems in which the detailed hydraulic modelling needs to be ignored.

(a)

(b)

Figure 5.15. Quantification of the damping coefficients under the application of the PSS

Figure 5.16. Simulations of three cases with the implementation of PSS

5.2.3 Influence and significance of the damping coefficient In this part, firstly, the influence of the damping coefficient on the PSS tuning is presented for a SMIB system. Secondly, the effect of the damping coeffi-cient on multi-machine system stability is shown, based on the WSCC 3-ma-chine 9-bus system [149] without the implementation of PSS.

5.2.3.1 Influence on the PSS tuning Three cases after the three phase fault are simulated by Model 4-S, as shown in Figure 5.16. In order to neutralize the effect of a negative damping (-2.0), the gain of PSS (Ks) needs to be increased from 2.0 to 9.0.

81

5.2.3.2 Influence on multi-machine system stability For showing the influence of the damping coefficient on multi-machine sys-tem stability, a simple case study is conducted based on the WSCC 3-machine 9-bus system [149], by applying the PSAT8.

Figure 5.17. Model of the WSCC 3-machine 9-bus system in PSAT

Figure 5.18. Rotor angle difference (δ21) between machine 1 and machine 2 under two conditions

The system model is shown in Figure 5.17, a fault occurs on bus 7 at 1.0 s and the clearing time is 1.083 s. The PSS is not applied and the AVR type is the same as the one above. The setting of the AVR is: Ka = 400 pu, Tr = 0.01 s. Two cases are compared: the first is the original system in which the damping coefficients of all the machines are 0. For the second, a positive damping co-efficient (Dt = 2.0 pu) is applied in machine 2.

8 Power System Analysis Toolbox (PSAT): http://faraday1.ucd.ie/psat.html (accessed on March 14th, 2017)

82

The results are shown in Figure 5.18. The rotor angle difference (δ21) be-tween machine 1 and machine 2 is taken as an indicator of the system stability [149]. The original system (Dt = 0) is unstable (red dashed line); while under the second condition, the contribution from the damping (Dt = 2.0 pu) in ma-chine 2 leads to a stable system.

5.3 Discussion on quick response of hydraulic – mechanical subsystem

A key point of this chapter is whether the responses of GVO and mechanical power of turbines are quick enough to trigger an obvious coupling effect be-tween the hydraulic-mechanical subsystem and the electrical subsystem. Pre-viously the effect of turbine governor has often been ignored in the small sig-nal stability analysis [15]; however in recent years, the rapidity of PFC has been demanded in order to ensure quick response, as shown in section 4.2.

Figure 5.19 shows the on-site measurements in a Swedish HPP, supporting the simulations in this chapter in the following aspects. (1) The fast GVO re-sponse is demonstrated clearly, and the largest rate can reach the rate limit (0.1pu/s). (2) A delay in the governor system, around 0.25 s between two GVO signals, is shown. (3) In terms of the measured active power, a non-minimum phase response is clearly presented.

Furthermore, the rapid GVO response after a three phase fault is also shown in simulations in previous studies [81, 82, 85]. Also, in this thesis, practical nonlinear components (servo, backlash, rate limit) are included and all these factors tend to slow the governor response.

In short, the concern on the response rapidity of hydraulic – mechanical subsystem is fully considered in this study, and the cases are realistic.

Figure 5.19. Measurements in a Swedish HPP after a step change of the GVO set-point: active power and deviation of GVO control signal and feedback signal

83

6 Efficient operation and balancing renewable power systems

In this chapter, the efficient operation of HPPs during balancing actions for renewable power systems is studied, focusing on PFC that acts on a time scale from seconds to minutes. In section 6.1 (based on Paper XI, Paper XII and Paper XIII), the problem description, cause and initial analysis of wear and tear of turbines are presented. Based on the analysis results, a controller filter is proposed in section 6.2 (based on Paper XII) as a solution for reducing the wear of turbines and maintaining the regulation performance, reflected by the frequency quality of power systems. Then in section 6.3 (based on Paper XIV), the study is further extended by proposing a framework that combines technical plant operation with economic indicators, to obtain relative values of regulation burden and performance of PFC.

6.1 Wear and tear due to frequency control 6.1.1 Description and definition In terms of wear and tear of hydropower turbines, there are different views and corresponding indicators to evaluate. From a point of view of control, this study focuses on the movements of the GVs in Francis turbines. The GV movements are expressed by the variations of GVO. Two core indicators are discussed, as shown in Figure 6.1:

(1) The first is the movement distance which is the accumulated distance of GV movements;

(2) The second is the movement amount which means the total number of movement direction changes. One movement corresponds to one direction change.

Some distance and amount of GV movement for a regulating unit are al-ways expected. Hence, blindly decreasing the movement is obviously not ad-visable. However, excessive values of these two indicators bring three types of wear and tear as follows.

(1) In the perspective of tribology, there is a linear positive correlation be-tween movement distance and material deterioration on the bearing [100].

(2) From the standpoint of hydraulics, direction changes of the actuator leads to dynamic loads on the turbine runner [13, 96].

84

(3) A huge amount of actuator movement implies a multitude of load cy-cles, which might increase the structure fatigue.

Figure 6.1. Illustration of two very important indices of wear and tear: distance and amount of GV movement.

Figure 6.2. Histogram of simulated GVO movements for real frequency record of a week in March, 2012. 19942 is the number of movements with the distance from 0 to 0.2%.

In [97], the measurements show that when Kaplan turbines operate in fre-quency control mode instead of in discharge control mode, the movement dis-tance of blade angle range will be increased up to ten times, and the amount of load cycles increases from 3 – 136 to 172 – 700 for the same period of time.

It is worth noting that there is a great amount of GV movements with small amplitudes. Reference [150] found that during 4 months of observations in HPPs, between 75 and 90% of all GVO movements are less than 0.2% of full stroke. In this thesis, a simulation is conducted and it demonstrates a similar result: 93.9% of all GVO movements are less than 0.2% of full stroke, as shown in Figure 6.2. More importantly, from the engineering experience, the wear and tear on the materials from small movements is believed to be more serious than from large movements. On the other hand, the regulation value from the small movements is not very obvious. Therefore, decreasing the number of small movements should be a priority.

85

6.1.2 Cause Here, a crucial reason for small GV movements is revealed: fluctuations of power system frequency. In order to exemplify the characteristics, measured frequency data of the Nordic power grid frequency is discussed, as shown in Figure 6.3 and Figure 6.4.

Figure 6.3. Time-domain illustration of small frequency fluctuations. Ts is the sam-pling time. The frequency change process from point A to B is a “frequency fluctua-tion”, as used in this thesis.

From intuitive observations of Figure 6.3, the frequency oscillation can be roughly divided into two “components”: (1) very low frequency “fundamen-tal” (with long period, larger than 10~20 s); (2) high frequency “harmonic” (with short period, less than 1~2 s depending on the sampling time).

The “fundamental” is generally a random signal, while in recent years, the grid frequency oscillations with some specific long periods appear in different power systems, e.g. Nordic power grid (with the period around 60 s) [129, 151], Colombian power grid (with the period around 20 s) [18]. Besides, long period oscillation of grid frequency in Great Britain and Turkey are presented in [57] and [59] respectively. For the short-period “harmonic”, this work de-fines a “frequency fluctuation” as a monotonic frequency changing process between a local maximum (or minimum) and a neighbouring local minimum (or maximum), as shown in Figure 6.3.

From Figure 6.3, one could have an intuition that, for the frequency chang-ing process, direction variations happen every one or two sampling periods. The intuition is further verified by Figure 6.4. The green columns are only for the frequency fluctuations with small values which are within ±2.5 mHz. Fig-ure 6.4 demonstrates two important features: (1) the values of frequency fluc-tuations are mostly very small, within ±2.5 mHz which equals to 5×10-5 pu; (2) the time lengths of small fluctuations are also extremely small.

In short, the results indicate that the power system frequency experiences both long period “fundamental” oscillations and “harmonic” fluctuations with small amplitude and high frequency. The frequency input would lead to the unfavourable amount of small GV movements.

More data and discussions can be found in Paper XI and Paper XII.

86

(a)

(b)

Figure 6.4. For the frequency data with 1 s sampling time in the month (March 2012): (a) Histogram of values of frequency fluctuations; the total amount of frequency fluc-tuations is 899308. The total amount of small frequency fluctuations is 666241. (b) Histogram of time lengths of the 666241 small frequency fluctuations.

6.1.3 Analysis on influencing factors In this part based on Paper XI, the GV movement is analysed by theoretical analysis based on ideal sinusoidal frequency input and simulations with real frequency records. The influences on wear and tear of different factors, e.g. governor parameters, PF mode and nonlinear governor factors, are explored.

6.1.3.1 Method and model Numerical simulations of PFC are conducted under both OF and PF, by ap-plying Model 1. The engineering case here is HPP 1. Detailed settings are given in Appendix B.

In terms of theoretical analysis, basic analytical formulas based on ideal-ized frequency deviation signals are deduced. For idealized frequency devia-tion signals as described in

sin 2 sin( )f f ff A t T A t , (6.1)

the following formula is proposed to estimate the accumulated movement dis-tance (Dy):

4 totaly PI m f

f

TD G G A

T . (6.2)

Here, Af is the amplitude of the sinusoidal input frequency signal, Ttotal and Tf represent the total time and a period respectively. GPI is the gain of the PI controller and Gm is the product of the gains of mechanical components (back-lash and lag), as shown below:

87

2 2

2 2

21

1

121

p

f i

PIp p p

f p i

K

T KG

b b K

T b K

, [pu/ pu], (6.3)

1 gvbacklash

in

BLG

A , [pu/ pu], (6.4)

2

2

1

21

lag

yf

G

T T

(6.5)

Here, Ain is the input amplitude, while BLgv represents the value of backlash. Ty stands for the lag in the main servomotor. Here, a simpler representation of the backlash is applied, comparing to Equation (3.42) that is based on the de-scribing function method.

Additionally, the response time (section 4.2), i.e. the time it takes for the opening to reach 63.2 % (≈1-e-1) of its final value after a step disturbance is

1

1 ln(1 )p pr p p

p i

b KT b K

b K

, [s]. (6.6)

The response time indicates the rapidity of PFC.

6.1.3.2 Results: influencing factors on wear and tear Here, the influence of different factors is discussed, based on Equation (6.2) and simulations under sinusoidal input signal, Δf with amplitude 0.025 Hz (0.0005 pu), and with a real record of frequency deviation in the Nordic power system.

As shown in Table 6.1, the governor parameters have essential influence on the GVO movements. The theoretical formulae for ideal input reflects the trend for real movements well, as can be seen from the comparison between the formula calculation and simulation results of movement distance. In Table 6.1, under different parameter settings, the change tendencies of movement distance under ideal and real frequency are in good agreement, for both OF and PF modes. Therefore the formulae are effective to achieve a good ten-dency estimation. Note that in the formula, the gain and movement distance are directly determined by the period. However, the “period” of real frequency is changing all the time and hard to get an approximate value. This brings a difficulty of applying the formula to estimate the real condition, but it will not influence the tendency prediction.

88

Table 6.1. Influence of governor parameters under OF and PF. For the movement distance, the total calculation and simulation time is two hours. Movement distance under OF and the response time are calculated by formulas; other values are from simulation. PF – “75%” and “100%” mean that the GVO set point is at 75% and 100% of rated value, respectively. 4. Groups 1 to 4 are Vattenfall standard sets of parameters.

T f = 60s

T f = 120s

T f = 60s

T f = 120s

OFPF -75%

PF -100%

1 0.10 1.00 0.167 59.71 0.17 0.23 0.19 0.26 0.23 0.27 0.282 0.04 1.00 0.417 59.95 0.70 0.76 0.77 0.82 0.71 0.75 0.783 0.02 1.00 0.833 59.99 1.62 1.67 1.70 1.75 1.54 1.57 1.644 0.01 2.00 1.667 59.99 3.49 3.46 3.65 3.62 3.25 3.24 3.415 0 1 10 5.00 10.35 5.68 11.83 6.24 7.69 7.22 8.906 0 10 0.83 58.90 2.29 1.70 2.88 2.09 1.99 2.61 2.677 0 2 2 24.98 3.92 3.44 4.32 3.76 3.09 3.20 3.43

Movement distance under real frequency

OF PF-100%No.

b P

[pu]K P

[pu]

K I

[s-1]

Response time (s)

Movement distance (Dy) under Sinusoidal frequency (with

backlash)

In terms of the application of PF, as shown in Table 6.1, the influence is

normally not too large; however the difference could be relatively substantial under large gain conditions. Besides, PF may lead to either increase or de-crease of movement, comparing with OF.

More discussions on nonlinear factors and two main influence factors un-der PF, operation set point and surge (water level fluctuation in surge tank), can be found in Paper XI.

6.2 Controller filters for wear reduction considering frequency quality of power systems

Aiming at the aforementioned problem in section 6.1 and the initial results, in this section, applying a suitable filter in the turbine controller is proposed as a solution for wear reduction. However, the controller filters impact the active power output and then affects the power system frequency. Therefore, the pur-pose of this section is the trade-off between two objectives: (1) reducing the wear and tear of the turbines; (2) maintaining the regulation performance, re-flected by frequency quality of power systems.

The widely-used dead zone is compared with a floating dead zone and a linear filter, by time domain simulation and frequency domain analysis. The filters are introduced in section 3.2.1.

89

6.2.1 Method and model The governor system (Figure 3.8) in Model 2-L (section 3.2.1) is applied to simulate the GV movements, based on a measured one-day grid frequency data. Then, the frequency under the influence of the different filters can be simulated and compared by using Model 2-L, based on a certain load disturb-ance. The mean value and standard deviation (SD) are chosen as the indicators to evaluate the frequency quality [152, 153]. However, load disturbances in power systems are unknown. Therefore a “grid inverse” [129] model is built, as shown in Figure 6.5, to compute a load disturbance from the existing meas-ured frequency. The transfer function of the grid inverse model is shown in

( )1r

p

Ms DG s

t s

. (6.7)

To avoid high amplification of high frequency noise in the grid frequency sig-nal, a pole with time constant tp (set to 0.1) is added [135].

Figure 6.5. Simulink model of a “grid inverse” for computing the load disturbance, as highlighted by red

In terms of the theoretical analysis, the describing functions (section 2.1.3.4) and Nyquist criterion (section 2.1.3.3) are adopted to examine the frequency response and stability of the system with different filters. Model 3-L is ap-plied, as described in section 3.3.2.

6.2.2 On-site measurements Here, on-site measurements and its comparison with simulations are pre-sented. The measurement was conducted in HPP 8. The measurement length is 6800 s, and Figure 6.6 shows a 3000 s period of comparison between the simulation and the measurement. The detailed information is shown in Ap-pendix B. As shown in Figure 6.6, the simulation matches the measurement well in time domain.

90

Figure 6.6. Comparison of the simulated GVO and the GVO measured in HPP 8. The measurement noise is demonstrated in the small plot inside the right figure.

Table 6.2. Statistics of the one-day GVO movements under different conditions of the servo (Ty) and backlash

Ty = 0 Ty = 0.1 Ty = 0.2 Ty = 0 Ty = 0.1 Ty = 0.20 128.13 48.83 45.50 2519586 577804 330166

0.01% 56.67 43.64 43.47 510284 8122 34720.02% 45.20 43.27 43.16 90310 3078 29460.03% 43.56 42.98 42.87 15306 2850 28060.04% 43.12 42.70 42.59 4260 2744 27040.05% 42.82 42.43 42.33 2992 2656 26280.10% 41.53 41.18 41.09 2462 2366 2342

Number of movementsBacklash

Movement distance (full strokes)

6.2.3 Time domain simulation Further results of time domain simulation are presented here. In the following simulations, the governor parameter setting adopts Ep3 (see Table 14.2 in Ap-pendix B), which leads to relatively large gain and GV movements. Other set-tings are also given in Appendix B.

Firstly, the cases without any filters are presented in Table 6.3, showing that without any filters, the governor system (especially the actuator) inher-ently filters the majority of the frequency fluctuations. This can also be ob-served from the measurements above: The time length of a GV movement is relatively long, the average value is 35.1 s (6800 s divided by 194 move-ments). The period value is 60~70 s, corresponding to the period of “funda-mental” frequency fluctuation in Nordic power grid. This further shows that the GV movement is mainly determined by the “fundamental” component of the input frequency, and the influence of the “harmonic” component is not significant.

The performances of different filters are presented in Table 6.3. It is shown that the traditional filter, dead zone, indeed reduces the movement distance, but not the movement amount. In contrast, the floating dead zone has a good

91

performance for the distance and especially for the movement amount. The linear filter could also decrease both two indicators, however the effect is not obvious.

Table 6.3. Statistics of the one-day GVO movements and the frequency quality with different filter types. The GVO movement is simulated by the governor model in Model 2-L, under two frequency input data with different sampling times. The frequency quality is computed by Model 2-L. “Ts” is the sampling time of the data. In the column “50-Mean”, the values refer to the mean frequency deviations; for example, the value 0.00104 Hz means the mean value of the frequency without any filter is 49.99896 Hz. “SD” stands for the standard deviation.

distance amount distance amount 50 - Mean SD

No filter(abs) \42.25

strokes2614

42.32 strokes

26280.00104

Hz0.0414

HzNo filter (pu) \ 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

±0.01% 89.9% 99.5% 89.9% 99.1% 121.0% 108.9%±0.02% 80.4% 96.8% 80.4% 96.6% 142.7% 118.1%±0.05% 54.6% 83.9% 54.5% 84.6% 221.1% 146.2%±0.1% 22.4% 43.8% 22.3% 43.1% 354.1% 193.8%

2×0.01% 96.5% 87.7% 99.4% 97.0% 105.4% 103.7%2×0.02% 88.5% 71.7% 93.6% 81.1% 103.5% 112.9%2×0.05% 56.1% 34.5% 62.5% 40.9% 114.9% 149.1%2×0.1% 17.2% 4.7% 19.7% 6.5% 104.2% 229.7%

1.0 99.1% 96.4% 99.0% 97.0% 100.0% 102.0%2.0 97.5% 93.5% 97.4% 92.2% 100.1% 106.3%3.0 95.4% 89.8% 95.3% 88.9% 100.1% 125.1%

±0.1% 98.7% 99.7% 98.7% 99.7% 102.6% 101.0%±1.0% 87.6% 92.8% 87.6% 93.3% 124.8% 110.7%±2.0% 77.2% 85.7% 77.1% 85.7% 149.8% 121.5%±5.0% 46.7% 55.0% 46.7% 55.1% 232.4% 153.9%0.1% 94.4% 83.6% 94.4% 83.5% 100.0% 101.0%0.5% 77.6% 56.8% 77.6% 56.5% 99.6% 107.1%1.0% 62.5% 40.9% 62.5% 40.8% 101.2% 117.3%2.0% 43.0% 23.5% 43.0% 23.4% 97.6% 143.7%1.0 99.1% 96.4% 99.0% 97.0% 100.0% 102.0%2.0 97.5% 93.5% 97.4% 92.2% 100.1% 106.3%3.0 95.4% 89.8% 95.3% 88.9% 100.1% 125.1%

GVO filter -

Linear (Tf2)

Filter typeValue setting (pu)

Guide vane openingFrequency quality

Ts = 1 s Ts = 0.02 s

Frequency filter - Dead

zone (Edz)

Frequency filter -

Floating dead zone Frequency

filter -

Linear (Tf1)

GVO filter - Dead zone

(Edz)

GVO filter - Floating

dead zone

(Efdz)

The frequency quality, the crucial trade-off factor, is analysed under different filters here. As demonstrated in Table 6.3, the dead zone leads to a poor fre-quency quality. By contrast, the linear filter only slightly increases the stand-ard deviation. The floating dead zone results in a medium performance of the frequency quality.

92

The effects of these three types of filters are obvious from the histogram in Figure 6.7. The floating dead zone and the linear filter increase the standard deviation, but the distribution is still single-peaked even with a large parame-ter setting. However, the dead zone leads to a very unfavourable double-peaked distribution.

Figure 6.7. Histogram of the one-day frequency simulated by the Nordic power system model with three types of filters

6.2.4 Frequency domain analysis: stability of the system Here, the Nyquist stability criterion is applied to test the stability of the system with different filters. Firstly, the nonlinear filters are tested, as shown in Figure 6.8.

The system is stable with the dead zone under these parameter settings. In contrast, the floating dead zone leads to a limit cycle oscillation in the system with the Ep3 parameter setting; while when the system adopts the Ep0 setting, the oscillation is avoided. However, the describing function framework only gives indication of the system stability, therefore the result is verified by a time-domain simulation of a load step change (+ 1×10-2 pu), as shown in Fig-ure 6.8 (b). Nevertheless, in section 6.2.3, even with the limit cycle oscillation under the Ep3 setting, the frequency quality under the floating dead zone is still acceptable.

Then, the influence of the linear filter is discussed, as described in Figure 6.9 (a). For the system with the Ep3 parameter setting, the critical value of the filter constant, Tf1 or Tf2, is approximately 4.0 s. It is validated clearly by time-domain simulation, as shown in Figure 6.9 (b).

93

(a) (b)

Figure 6.8. (a) Frequency-domain result: the Nyquist curve of the open-loop system under different governor parameters and the negative reciprocal of the describing functions. (b) Time-domain result: the system frequency after a load step change (+ 1×10-2 pu), simulated by Model 2-L.

(a) (b)

Figure 6.9. (a) Frequency-domain result: gain margin and phase margin of the system with the linear filter. (b) Time-domain result: the system frequency after a load step change (+ 1×10-2 pu), simulated by Model 2-L.

6.2.5 Concluding comparison between different filters The main conclusion of this section is shown in Table 6.4. It suggests that the floating dead zone, especially the GVO filter after the controller, outperforms the widely-used dead zone on the trade-off between the wear reduction and frequency quality.

94

Table 6.4. Comparison between different filters

Filter Advantages Disadvantages

Dead zone

1. Deceasing the GV movement distance effectively 2. No obvious influence on system stability

Two fatal points: 1. Worst frequency quality: the dou-ble-peaked distribution 2. Not very effective in reducing the GV movement amount, which is the main goal

Floating dead zone

1. Well reducing the GV move-ment amount 2. Deceasing the GV movement distance effectively 3. Good frequency quality, espe-cially with the GVO filter

Might cause limit cycle oscillation (however it can be avoided, i.e. by tuning the governor parameters; even with the limit cycle oscillation, the frequency quality is still acceptable)

Linear filter

1. Best frequency quality 2. Can decrease both the move-ment distance and amount to some extent

1. Cannot obviously decrease both the movement distance and amount 2. Might cause system instability (however it can be avoided, i.e. by tuning the governor parameters)

6.3 Framework for evaluating the regulation of hydropower units

In this section, a framework is proposed as shown in Figure 6.10, combining technical operation strategies with economic indicators, to obtain relative val-ues of regulation burden and performance of PFC.

Figure 6.10. Framework for quantifying and evaluating the regulation of hydropower units. Efficiency loss as well as wear and fatigue are adopted to represent the burden; regulation mileage and frequency quality are applied to evaluate the regulation per-formance.

For the quantification, Model 2-K is applied and calibrated with measure-ments from HPP 6 and HPP 7. Kaplan turbines are studied here, since they are more complicated in terms of control. Hence, the methodology and results can

95

easily be simplified and extended to other turbine types. Burden relief strate-gies and their consequences are discussed, under two idealized remuneration schemes for PFC, inspired by the ones used in Sweden and in parts of the USA. They differ in the underlying pricing philosophy mainly in that the Swe-dish one does not take actual delivery into account, but rather compensates for the reserved capacity.

6.3.1 The framework In the framework, burden is represented by efficiency loss and wear/fatigue, and regulation performance is evaluated using regulation mileage and fre-quency quality. The technique to quantify burden and regulation performance and the corresponding indicators that serve as the main outputs of the numer-ical simulations are introduced in subsection 6.3.2.

Optimizing the regulation conditions of hydro units is the key to easing their incurred burden. Various regulation conditions are comprehensively compared by varying the turbine governor parameters (Ep1 – Ep3 in Table 14.2 in Appendix B), operating set-points (seven points in Figure 6.11) and regulation strategies (subsection 6.3.2).

(a) (b)

Figure 6.11. Illustration of the combinator table for the turbine in the HPP 6 (a) and HPP 7 (b). In each figure, seven on-cam operating points are highlighted by blue scatters, they are within the maximum efficiency range.

Under different conditions, the following two idealized pricing schemes of regulation payments are concisely analysed: strength payment and mileage payment that are inspired respectively by the ones used by the TSO SvK in Sweden and by PJM Interconnection LLC (PJM), a regional transmission or-ganization in the USA [154]. The schemes are detailed in subsection 6.3.2.

Various simulations are conducted to test the above-mentioned indicators based on HPP 6 and HPP 7, as illustrated in Figure 6.12.

96

Figure 6.12. Illustration of the simulation structure. The models 2-K-1 through 2-K-3 are introduced in subsection 6.3.2. The blocks with dashed outline represent the selections in simulation setting based on different conditions: i varies from 1 to 3, for three sets of governor parameters Ep1 – Ep3; j varies from 1 to 4, for three regulation strategies (S1 – S3) and an ideal on-cam case (S0); k varies from 1 to 2 for two HPPs; n varies from 1 to 7 for seven operating set-points. The set in the parenthesis with simulation presents different cases conducted in the model. In total, there are 168 (3×4×2×7) and 24 (3×4×2×1) simulation cases conducted in Model 1 and Model 2 respectively. The terms in the parenthesis with the output variables show the actual needed set of results for analysing the corresponding indicator in this paper.

6.3.2 Methods Here, the detailed methods are introduced. It is worth noting that the burden (efficiency loss, wear and fatigue) is discussed from a physical perspective. Further economic modelling to obtain the gain or loss of profit from regulation is not included, while it is necessary in the future to fully characterize the ef-fects of PFC on system economics.

6.3.2.1 Numerical models As shown in Figure 6.12, three models are applied. Model 2-K-1 and Model 2-K-2 are presented in section 3.4, and Model 2-K-3 is introduced here. In subsection 6.2.1, a method of simulating and evaluating the frequency quality for units with Francis turbines is introduced. Here, the method is improved and extended for Kaplan turbines.

Model 2-K-3 is for computing the unknown sequence of one-day load dis-turbance from the original measured frequency, as shown in Figure 6.13. The “grid inverse” model is shown in Equation (6.7) in subsection 6.2.1, and the detailed parameters are given in Table 14.5 in Appendix B. The value of time constant tp is set to 0.1 s in this study.

97

Figure 6.13. Block diagram of Model 2-K-3 with the “grid inverse” model for com-puting the load disturbance; the lumped HPP block is described in the part with dashed outline in Figure 3.18.

(a)

(b)

Figure 6.14. Comparison of measurement and simulation during a period of normal PFC. (a) GVO (y) and RBA (α), the deviation value is shown. (b) Active power.

In Model 2-K-3, all the regulating HPPs in the Nordic power grid are lumped into one scaled HPP with the scaling factor (K3). The model of the lumped HPP is introduced in subsection 3.4.1.

HPP 6 and HPP 7 are applied as engineering cases. One unit of each HPP is taken as the study case. The detailed parameter values of the two HPPs are shown in Table 14.6 in Appendix B. For HPP 6, the dynamic processes under normal PFC is simulated by Model 2-K-1 and compared to the measurements under the governor parameters Ep1 (Table 14.2), as shown in Figure 6.14. The simulation of the GVO, the RBA and the power output has a good agreement

98

with the measurements, showing that the model can yield trustworthy simula-tion results.

6.3.2.2 Regulation strategies The following operation strategies S1 – S3 and an ideal case S0 are analysed. (1) S1: normal PFC in which GV and RB regulate without any artificial filter (widely implemented); (2) S2: PFC with a floating dead zone filter for reduc-ing the movement of runner blades; (3) S3: PFC with the runner blades being totally fixed (no RB movements); (4) S0: Normal PFC in an ideal on-cam condition, and it is unrealistic and only implemented to identify the off-cam loss in normal PFC.

6.3.2.3 Method of quantification Here, the method of quantification of the burden and quality of regulation is introduced.

(1) Efficiency loss The losses are classified into the following compositions, as illustrated in

Figure 6.15.

Figure 6.15. Compositions of efficiency losses analysed in this work

The loss in steady state operation, -Δηst, is 1st st , [pu]. (6.8)

A negative value of the efficiency change Δη indicates an efficiency loss. ηst is the on-cam steady state efficiency that is a constant value taken from the interpolation, and it varies for different operating points. In this section, the main object is the extra efficiency loss due to regulation, which is given as Sj st Sj , [pu]. (6.9)

Here, ηSj is the average value of the instantaneous efficiency during the oper-ation period (one day) under a specific strategy (Sj). More specifically, the efficiency loss in transient due to deviation from the set-point (on-cam) can be obtained as

0 0S st S , [pu]. (6.10)

99

The loss due to off-cam in normal PFC is achieved by the difference between ΔηS0 and ΔηS1. For example, the extra loss due to off-cam condition under strategy S2 and S3 is ΔηS0 - ΔηS2 and ΔηS0 - ΔηS3, respectively.

(2) Wear and fatigue For quantifying the wear and fatigue of turbines, we use the two indicators

introduced in subsection 6.1.1 for both GV and RB: the movement distance and the movement amount. The movement distance can be described in

, 1

1

RB, 11

N

GV dist is isis

N

dist is isis

Y y y

Y a a

. (6.11)

Here, N is the total amount of samples and is means the sample number. y and a represent GVO and RBA respectively.

(3) Regulation mileage The regulation mileage is introduced to quantify the amount of work hy-

dropower units expend to follow a regulation signal, as described in

, , 11

N

R m ism r m isis

atedPM p p

, [MW]. (6.12)

Here, pm is the active power in per unit; Pm-rated is rated power of the Kaplan turbine, and its unit is MW.

(4) Frequency quality The frequency quality is evaluated to comprehensively reflect the regula-

tion performance of the hydropower unit. As presented in subsection 6.2.1 and in the lower part of Figure 6.12, the core idea is comparing the new frequency sequence of the power system under different regulation conditions, to exam-ine whether the frequency quality is deteriorated. The frequency quality is mainly evaluated through the root mean square error (RMSE with respect to the rated frequency 50 Hz) of the frequency sequence.

In section 6.2, the frequency quality reflects the influence of a lumped unit that represents all the generating units in the grid. By contrast, the method here examines the influence of the regulation from a single Kaplan unit on the fre-quency of the whole grid.

6.3.2.4 Regulation payment In this section, only relative values of payment are considered. Clearing prices are not considered in the quantification. The strength payment Paystrength, in-spired by SvK, is computed as

, [MW/Hz]

0.1

, [pu]

stepR

Rstrength

R base

PS

SPay

S

. (6.13)

100

Here SR is the regulation strength; Pstep is the increase in output power caused by a frequency step change that is selected as from 50 Hz to 49.9 Hz by SvK. SR-base is a base value for normalizing the payment, and here it is set to the regulation strength (41.08 MW/Hz) of the unit in HPP 6 under Ep1 and S1. The payment will depend on governor parameterization, turbine characteris-tics, burden relief strategy, and operating point.

The mileage payment Paymile, inspired by PJM, is based on the mileage in power output

Rmile

R base

MPay

M

, [pu]. (6.14)

MR-base is a base value for normalizing the payment, and here it is set to the regulation mileage (449.5 MW in Table 6.5 below) of the unit in HPP 6 under Ep1 and S1.

6.3.2.5 Setting of simulations For all the simulations, the length is 24 hours and time-step is 0.02 s. The input signal of Model 2-K-1 and Model 2-K-3 is a sequence of measured one-day (24-hour) Nordic grid frequency, and its sampling time is 1.0 s. The imple-mentation of the strategy S2 and S3 is by setting the value of the filter that is the floating dead zone for RBA. For the S2 and S3, the value of the floating dead zone is set to 0.03 pu and 1.0 pu respective; while for the S1, this filter value is set to 0. For the ideal case S0, all the mechanical components in runner blade part (after the combinator in Figure 3.17) are removed to achieve a purely on-cam operation. The initial steady-state value of water head is set to the rated value (1.0 pu).

(1) Settings for simulating the GVO, RBA and the efficiency loss The adopted model is Model 2-K-1, without the engagement of the power

grid components. Hence the scaling factor (K) is set to 1.0. As presented in Figure 6.12, there are 168 (3×4×2×7) simulation cases conducted.

(2) Simulating and evaluating the frequency quality Firstly, Model 2-K-3 is applied for computing the load disturbance. Cur-

rently, the Nordic TSOs require a fixed amount of regulating power of PFC in the whole grid: 753MW for 0.1 Hz frequency deviation, hence the total regu-lation strength SRT is 7530 MW/Hz, and it is normalized as SRT-pu that is 10.0 pu [129]. Consequently, in Model 2-K-3, the product of the governor static gain 1/bp and K3 is 10.0 pu [129]. Here, the governor parameter set is selected as Ep1 for computing the load, and the droop in Model 2-K-3 is noted as bp3 (bp0 = 0.04 pu). Therefore, the value of K3 is set to 0.4 pu (10×bp3 pu).

Then, with the simulated load as input, Model 2-K-2 is applied to simulate new frequency sequences. In Model 2-K-2, the regulation power in the grid is provided by the Kaplan unit (Model 2-K-1) that is the examining object as well as the rest of units (Francis unit in the lumped HPP). In Model 2-K-2, the values of two scaling factors (K1 and K2) are described in

101

1 2

1 2 1 1 2 2

R R RT

R pu R pu RT pu

S S S

S S G K G K S

(6.15)

Here the SR1 and SR2 are the regulation strength of the Kaplan unit and the lumped HPP respectively; their corresponding values in per unit are repre-sented as SR1-pu and SR2-pu. G1 and G2 are the gains from frequency deviation to power deviation for the Kaplan unit and the lumped HPP, as presented in

11

22

10

1

R

step m rated

SG

f P

Gbp

, [pu]. (6.16)

Here, Δfstep is the step change value of frequency 0.1Hz (i.e. 0.002 pu); Pm-rated is the rated power of the Kaplan unit, bp2 means the droop in the lumped HPP.

In this section, the regulation strength SR2 is kept as constant. The SR1 of the Kaplan turbine changes with various conditions and influences the frequency quality. For each HPP, the new frequency sequences are compared to the fre-quency sequence from Ep1 and S1. The operating point has little influence on the frequency quality, hence the operating point stays the same (selected as point 5) in all simulations, as shown in Figure 6.12.

6.3.3 Burden quantification In this section, the efficiency loss as well as wear and fatigue due to regulation are discussed. The overall results under various operation conditions are shown in Table 6.5, and detailed results of efficiency loss are presented in Table 6.6. It is shown that the governor parameter directly influences the effi-ciency loss: the lower the droop (the higher the static gain), the more effi-ciency loss. However, the loss is not linearly related to the droop due to the complexity and nonlinearity in the system.

The compositions of efficiency loss in regulation are studied by comparing the results in different strategies. The difference between efficiency losses un-der S1 and S0 is very small, indicating that the loss due to normal PFC is mainly caused by the trajectory deviation from the set-point. The extra loss due to off-cam operation is negligible for normal PFC, but not for wear reduc-tion strategies. The operation strategy S3 leads to a considerable efficiency loss that is larger than 1.0 % under the high gain setting Ep3 and mainly caused by off-cam operation, showing the economic drawback of the strategy. While the strategy S2 only causes a slight increase (approximately 0.02 %) of effi-ciency loss compared to the S1.

102

Table 6.5. Overall results of different operation conditions from one-day simulation. The bar in each cell indicates the relative magnitude of the values with the same color. The results of efficiency loss are condensed from Table 6.6. The results of GV move-ment are based on HPP 6, because there is little difference on the indicators between two HPPs. The operating point does not influence much on movement of GV and RB, hence only the results from point 5 are shown. For frequency quality, the values of the change of root mean square error are shown; negative and positive values are shown with green and grey bars respectively, and positive values indicate better frequency quality. The regulation mileages are shown with purple bar, and the detailed results are in Figure 6.16.

S1 S2 S3 S1 S2 S3 S1 S2 S3

Avg-HPP 6 -0.023% -0.034% -0.069% -0.085% -0.105% -0.281% -0.340% -0.375% -1.167%

Min-HPP 6 -0.196% -0.207% -0.217% -0.370% -0.394% -0.476% -0.680% -0.712% -1.441%

Avg-HPP 7 -0.016% -0.046% -0.091% -0.083% -0.108% -0.409% -0.365% -0.397% -1.652%

Min-HPP 7 -0.403% -0.422% -0.461% -0.585% -0.609% -1.004% -1.100% -1.127% -2.242%

Distance 7.177 7.177 7.177 14.782 14.782 14.782 30.094 30.094 30.094

Amount 2039 2039 2039 2257 2257 2257 2550 2550 2550

Dist.-HPP 6 4.027 0.245 0 10.731 1.515 0 24.663 5.831 0

Dist.-HPP 7 5.843 0.509 0 13.649 2.131 0 28.186 6.866 0

Amount-HPP 6 1237 19 0 1567 48 0 1877 174 0

Amount-HPP 7 1329 28 0 1621 58 0 1887 184 0

HPP 6 0 -0.09% -0.28% 0.52% 0.45% -0.07% 1.56% 1.48% 0.38%

HPP 7 0 -0.06% -0.24% 0.43% 0.38% -0.04% 1.31% 1.25% 0.38%

HPP 6 449.5 229.3 207.9 1094.8 573.2 482.0 2412.3 1324.2 996.5

HPP 7 404.1 207.5 187.0 954.7 506.1 442.3 2026.6 1156.8 958.9

HPP 6 100.0% 79.7% 33.5% 204.6% 191.1% 54.6% 421.9% 422.3% 43.6%

HPP 7 81.3% 67.5% 25.6% 162.9% 150.2% 32.1% 317.3% 308.4% < 0

HPP 6 100.0% 51.0% 46.3% 243.6% 127.5% 107.2% 536.7% 294.6% 221.7%

HPP 7 89.9% 46.2% 41.6% 212.4% 112.6% 98.4% 450.9% 257.4% 213.3%

Bur

den

Efficiency change

[pu]

GV movement

[/]

RB movement

[pu]

Parameter Ep1 (b p = 0.04) Ep2 (b p = 0.02) Ep3 (b p = 0.01)

Strategy

Reg

ulat

ion

pe

rfor

man

ce

Frequency quality [pu]

Mileage [MW]

Pay

men

t Strength [pu]

Mileage [pu]

It is found that the operating set-point has a considerable influence not only on the steady state efficiency but also on the efficiency loss in transients, as shown in Table 6.6. The differences between the maximum and minimum of efficiency changes under seven points are considerable (e.g. 0.774%, 1.087% and 1.610% under Ep1 through Ep3 respectively for HPP 7). This causes com-plexity in the quantifying and evaluating process. In some cases, the efficiency even increases compared to the steady state value, as highlighted by blue in the table cell. The main explanation is that the on-cam efficiency of a set-point (e.g. point 1) is not the global maximum.

103

Table 6.6. Detailed simulation results of efficiency change (ΔηSj). The seven operating points of each HPP are shown in Figure 6.11. Negative values mean efficiency loss. The bar in each cell is a graphical representation of the value, and the negative and positive values are highlighted by red and blue respectively. The average values of seven points are selected as the representative results listed in Table 6.5. The value of “Max-Min” is the difference between the maximum and minimum of the efficiency changes of seven points.

Δη S0

(S0)

Δη S1

(S1)

Δη S2

(S2)

Δη S3

(S3)

Δη S0

(S0)

Δη S1

(S1)

Δη S2

(S2)

Δη S3

(S3)1 -0.130% -0.130% -0.123% -0.136% -0.370% -0.370% -0.383% -0.332%2 0.019% 0.019% 0.014% -0.028% 0.301% 0.301% 0.269% 0.205%3 0.112% 0.112% 0.092% 0.023% -0.300% -0.300% -0.353% -0.461%4 0.045% 0.045% 0.039% 0.003% 0.235% 0.235% 0.212% 0.159%5 -0.085% -0.085% -0.083% -0.097% 0.371% 0.371% 0.312% 0.123%6 -0.196% -0.196% -0.207% -0.217% -0.403% -0.403% -0.422% -0.334%7 0.075% 0.075% 0.030% -0.031% 0.057% 0.056% 0.041% 0.006%

Average -0.023% -0.023% -0.034% -0.069% -0.015% -0.016% -0.046% -0.091%Max - Min 0.309% 0.309% 0.299% 0.241% 0.774% 0.774% 0.734% 0.666%

1 -0.229% -0.229% -0.241% -0.437% -0.391% -0.391% -0.398% -0.604%2 0.031% 0.031% 0.022% -0.204% 0.502% 0.502% 0.483% 0.040%3 0.186% 0.185% 0.171% -0.073% -0.518% -0.518% -0.550% -1.004%4 0.042% 0.042% 0.022% -0.175% 0.215% 0.215% 0.183% -0.238%5 -0.224% -0.224% -0.248% -0.375% 0.305% 0.304% 0.259% -0.073%6 -0.369% -0.370% -0.394% -0.476% -0.585% -0.585% -0.609% -0.758%7 -0.031% -0.031% -0.069% -0.230% -0.103% -0.103% -0.127% -0.229%

Average -0.085% -0.085% -0.105% -0.281% -0.082% -0.083% -0.108% -0.409%Max - Min 0.555% 0.555% 0.565% 0.403% 1.087% 1.087% 1.092% 1.043%

1 -0.427% -0.427% -0.460% -1.373% -0.399% -0.399% -0.424% -1.763%2 -0.163% -0.163% -0.197% -1.016% 0.511% 0.511% 0.482% -1.189%3 -0.036% -0.037% -0.075% -0.860% -0.687% -0.688% -0.725% -2.242%4 -0.211% -0.211% -0.247% -1.019% -0.048% -0.048% -0.087% -1.554%5 -0.507% -0.507% -0.541% -1.308% -0.077% -0.077% -0.112% -1.320%6 -0.679% -0.680% -0.712% -1.441% -1.099% -1.100% -1.127% -2.025%7 -0.357% -0.357% -0.390% -1.154% -0.752% -0.753% -0.783% -1.473%

Average -0.340% -0.340% -0.375% -1.167% -0.364% -0.365% -0.397% -1.652%Max - Min 0.643% 0.643% 0.638% 0.582% 1.610% 1.610% 1.608% 1.053%

Ep3

Para. Operation

point

HPP 6 HPP 7

Ep1

Ep2

In terms of wear and fatigue, the governor parameters has a direct influence, as shown in Table 6.5. A lower droop leads to longer distance and larger amount of movements. For the GV movements, results do not differ from S1 to S3, because different strategies only affect the runner blade side. The bur-den relief strategy S2 leads to a significant decrease of both the distance and amount of RB movements, and strategy S3 totally diminishes the RB move-ment. Besides, the influence from different operating points is small, and this decreases the complexity in analysis.

104

6.3.4 Regulation performance Here, the regulation mileage and frequency quality are discussed as crucial trade-off perspectives of the burden. As shown in Table 6.5 and Figure 6.16, there is a high correlation between regulation mileage and the static gain (1/bp). As to the operating strategy, the wear reduction strategy S2 and S3, especially S3, lead to decreases in mileage. The results are not influenced much by the operating set-points, and this is conducive to simplify the evalu-ation.

(a) (b)

Figure 6.16. Regulation mileage of two HPPs for 24 hours. For each case, results from all seven operating points are close, hence the scatters overlap each other; the average values of seven points in various conditions are presented in Table 6.5.

In terms of frequency quality, the results are presented in Table 6.5. A larger value indicates a better frequency quality. Overall, lower droop that corre-sponds to higher regulation strength of the unit results in better frequency quality, based on our setting that the regulation strength of the rest of the units in the grid is unchanged. Besides, the influence of the off-cam operation is shown clearly: when the strategy S2 is applied, the frequency quality worsens compared to the performance of S1 under the same Ep setting. The deteriora-tion of the frequency quality is more obvious under S3.

6.3.5 Regulation payment Evaluating consequences of regulation payment schemes is of great im-portance for both power producers and TSOs. Here, the two idealized schemes, inspired by the ones used by SvK in Sweden and PJM in the USA, are analysed through their relative payment values. Relative values of pay-ments under different conditions are presented in Table 6.5.

Both strength payment and mileage payment lead to compensation overall aligned with performance, as it correlates well with RB movement and fre-quency quality. Compared to strategy S1, strategy S3 leads to a considerable compensation decrease while causing an obvious drop in efficiency and poor frequency quality. Therefore, S3 may not be a promising choice for producers.

105

Alternatively, strategy S2 can be suitable for producers, because it does not result in significant compensation decline (especially for HPP 7) and can largely decrease the RB movements without deteriorating the efficiency and frequency quality too much. An important finding is a large difference in two payments under burden relief strategies S2 and S3. This illustrates the influ-ence that payment schemes might have on operating strategies chosen by pro-ducers.

(a) (b)

Figure 6.17. Strength payments based on the strength obtained from the -0.1 Hz fre-quency step change and a -0.05 Hz frequency step change. (a) HPP 6; (b) HPP 7 .For each frequency change case, the relative values are normalized with the respect to the strength under Ep1 and S1 for each HPPs.

For HPP 7, the strength under S3 with Ep3 is negative, because the turbine efficiency sharply decreases in the off-cam operation and the power output after the frequency change is even smaller than the original value. However, the regulation performance of the case is positive (with +0.38% in frequency quality and 958.6 MW in mileage) since the average frequency deviation is considerably smaller than the 0.1 Hz used to determine strength. This demon-strates a limitation of the strength payment: it ignores the performance of ac-tual delivered regulation. As shown in Figure 6.17, the negative strength is avoided by varying the frequency change to -0.05 Hz from the original value 0.1 Hz: the payment increases to 63.5 % from -9.9% under S3 with Ep3 for HPP 7. This reflects the complexity of achieving an appropriate implementa-tion of strength payment.

106

7 Summary of results

The main results of the thesis are summarized in the following three aspects. Some of the conclusions are based on results that are only displayed in Papers I – III, and Papers IX – XIV.

1. Stable operation regarding frequency stability

(1) For the TOPSYS model (Model 1) and different operating conditions of hydropower units: The TOPSYS model (Model 1) is reliable for computing different phys-

ical quantities of the unit by comparing the simulation with on-site meas-urements of different operating conditions, e.g. start-up, no-load opera-tion, normal operation and load rejection in different control modes.

The main error source of the simulation might be the characteristic curves of the turbine, and the error is especially obvious in the small-opening operation range.

(2) For the response time for PFC of hydropower units: The time difference ΔT, between the power response time and the analyt-

ical response time of opening, is mainly affected by rate limiting and numerical algorithm (ΔT1), water inertia (ΔT2) and surge (ΔT3).

The most direct and effective method to control the response time is still adjusting the governor parameters. Especially for a HPP without surge tank, the formula of opening response time can also help to predict the power response and supply a flexible guidance of parameter tuning.

(3) For the frequency stability of an isolated HPP with surge tank: The power control has a better performance on frequency stability, com-

paring with the frequency control. By contrast, power control leads to poorer rapidity.

The water inertia of the final state is a key factor of frequency stabil-ity. The worst operation case for stability is with large load, large load disturbance amplitude and low water head of the final state.

107

2. Stable operation regarding rotor angle stability

(1) For eigen-analysis on hydraulic – mechanical – electrical coupling mech-anism: A core contribution is the state matrix. It could be an efficient approach

for further research and engineering applications. The effect of turbine governor actions is no longer ignorable due to

the increasing rapidity of frequency control nowadays. Opening feedback and power feedback mode of frequency control lead to different perfor-mances, and their impacts mainly differ according to the water column elasticity.

Influences from single factors are not monotonic and depend on the interactions with other factors. The consideration of water column elas-ticity (Te) is of great importance. The mechanical component in the gov-ernor (Ty) also has a non-ignorable effect. The influence of water inertia (Tw), which is normally regarded as adverse to the system stability, varies in different conditions, and it can contribute to the system stability in cer-tain cases.

There is still room for parameter tuning of controllers by considering the influence of the hydraulic-mechanical factors.

(2) For quantification of the hydraulic damping: The equivalent hydraulic turbine damping coefficient and the corre-

sponding methodology are proposed to quantify the contribution on damping of rotor angle oscillations from hydraulic turbines.

The damping effect from hydraulic turbines can be considerable. For the cases in this Swedish HPP, the damping coefficient can vary from +2.0 to -2.1. The damping effect can hence also be negative, while previ-ously the contribution is unclear and normally assumed to be positive.

The damping performance under the application of the PSS is also obviously affected. An important application of the damping coefficient is to add in models of complex multiple-machine systems in which the detailed hydraulic modelling needs to be ignored.

The TOPSYS model (Model 5) could be a solid tool to identify the char-acteristics of complex hydropower systems and predict the dynamic pro-cesses.

3. Efficient operation and balancing renewable power systems

(1) For the influence from PFC on wear and tear: The problem of wear and tear of hydropower units is defined from

the control perspective.

108

Parameter tuning is a key approach to affect the movements. The the-oretical formulas reflect the trend for real movements well, and they are effective to achieve a good estimate.

The influence of the power feedback mode is normally not too large. The operation point affects the movements by changing of the ratio Δy/Δp. The surge normally does not cause much opening movement change, but it could lead to a “resonance”.

(2) For the controller filters for wear reduction considering frequency quality of power systems: The characteristics of frequency fluctuations in the Nordic power grid

are presented, as an important cause of small GV movements. The floating dead zone leads to a best performance on the trade-off

between the wear reduction and frequency quality. Without any artificial filters, the majority of the frequency fluctuations are

inherently filtered by the governor system, especially the actuator. The GV movement is mainly determined by the “fundamental” compo-nent of the input frequency.

(3) For the framework for evaluating the regulation of hydropower units: A framework is proposed, combining technical operation strategies

with economic indicators, to obtain relative values of regulation burden and performance of PFC.

The efficiency loss, wear and fatigue, regulation mileage, and fre-quency quality are quantified and characterized with respect to various plant conditions. They differ across operating conditions and turbines, in-dicating that site-specific analysis may be necessary to quantify the bur-den of regulation at each HPP.

For producers, the model can be a solid tool for optimizing the oper-ating strategy to obtain more compensation, smaller efficiency losses and less wear and fatigue. Another benefit is the possibility to evaluate remu-neration schemes.

For TSOs, remuneration analyses using this framework can be used to formulate detailed operating guidelines or best practices for HPPs. The two payment approaches represent different pricing philoso-phies: strength payment mainly compensating for reserved capacity by scaling the clearing price with the regulation strength, and mileage pay-ment evaluating the actual delivery with respect to regulation mileage. Neither of the schemes is affected directly by variations in efficiency loss, which varies with turbine characteristics and operating points.

109

8 Conclusion and discussion

In this section, more general conclusions are condensed and discussed, com-paring to the relatively detailed summary of results in section 7. A main focus here is the contribution of the thesis in the following aspects.

(1) Regarding physical characteristics: Understanding of hydraulic – me-chanical – electrical coupling mechanism is approached; the oscillations, with various periods from less than one to hundreds of seconds, of multiple physi-cal quantities are analysed. The coupling between the hydraulic – mechanical subsystem and the electrical subsystem can be considerable and should be considered with higher attention.

(2) Regarding mathematical method: Multiple models, for numerical sim-ulations and theoretical analysis, with various degrees of complexity are built for different purposes, and on-site measurements are applied for validations. Effectiveness and applicability of different models are shown, supplying sug-gestions for further model optimization.

(3) Regarding engineering measures: Control strategies for various operat-ing conditions are investigated and recommended for a more stable and effi-cient operation of HPPs. It is demonstrated that conducting individual case studies is necessary, since the dynamic processes highly depend on the system characteristics. This also reflects the importance of an effective and accurate modelling and analysis approach.

Interaction between HPPs and power systems is a core emphasis through-out the thesis. It is also underlined in the title and can be categorised in the following two perspectives.

(1) Influence from power systems on HPPs: The dynamic processes and corresponding control strategies of HPPs under diverse disturbances and re-quirements from power systems are addressed in the thesis, e.g. the response time under frequency disturbance, the frequency oscillation under load dis-turbance, the rotor angle oscillation under voltage disturbance, and the wear and efficiency loss due to the delivery of grid services, etc.

(2) Influence from HPPs on power systems: Quantifications of the influ-ences from HPPs on power systems are conducted and the corresponding methodologies are proposed, e.g. the grid frequency quality from hydropower regulation, and the equivalent hydraulic damping coefficient on low frequency oscillations of power systems, etc.

110

9 Future work

The future work is suggested from the aspects of research content, research object and research method.

Regarding the research content: Further study on the hydraulic – mechani-cal – electrical coupling mechanisms is important, and features of and reme-dies to oscillations of multiple physical quantities should be in-depth investi-gated. The mechanism and influencing factors of efficient and economic op-eration need to be explored, a more comprehensive economic evaluation of regulation of units is of importance.

Regarding the research object: More work on pumped storage plants, espe-cially on plants with variable speed units, are very important. Further research on Kaplan turbines and other types of turbines, e.g. Pelton turbines and bulb turbines is meaningful. The work also can be developed in terms of the object scale: extension towards studying on larger scale, e.g. power systems with multiple hydropower units, is a crucial direction; while more sophisticated in-vestigations on details of system components, e.g. characteristics of turbines and generators, are also essential.

Regarding the research method: Theoretical analysis could be enhanced with respect to better description and analysis of nonlinear features in the sys-tem. The sophistication of numerical modelling can be further improved, more details of the system should be considered and the modelling and numerical accuracy needs to be thoroughly discussed. More model tests and on-site measurements are needed, in order to obtain a better observation and under-standing of the physical dynamic behaviours and to validate the reliability of theoretical analysis and numerical simulation.

111

10 Summary of papers

Paper I A mathematical model and its application for hydro power units under different operating conditions

This paper presents a mathematical model of hydropower units based on the basic version of the software TOPSYS. Based on one Swedish hydropower plant and three Chinese plants, the simulation and on-site measurements are compared for start-up, no-load operation, normal operation, and load rejection in different control modes.

The author performed programming work, simulations and discussions, and wrote the manuscript.

Published in Energies, 2015, 8(9), 10260-10275.

Paper II Response time for primary frequency control of hydroelectric generating unit

This paper builds a suitable model for conducting reliable simulation and to investigate the general rules for controlling the power response time. This paper deduces a time domain solution for guide vane opening response and a response time formula. Based on two hydropower plants, the factors which cause the time difference, between the power response time and the analytical response time of opening, are investigated from aspects of both regulation and water way system.

The author built the numerical model, performed simulations and analytical deviations, and wrote the manuscript.

Published in International Journal of Electrical Power and Energy Sys-tems, 74(2016):16–24.

Paper III Frequency stability of isolated hydropower plant with surge tank under different turbine control modes

The focus of this paper is on stabilizing the low frequency oscillation of an isolated hydropower plant (HPP) caused by surge fluctuation. In a theoretical derivation, the governor equations of frequency control and power control are introduced to the mathematical model. For numerical simulation, a governor

112

model with control mode switch-over function is built. The frequency oscilla-tions under frequency control, power control and control mode switch-over are simulated and investigated respectively.

The author built the numerical model, performed simulations and discus-sion, and wrote the manuscript.

Published in Electric Power Components and Systems, 43(15): 1707-1716.

Paper IV Regulation quality for frequency response of turbine regulating system of isolated hydroelectric power plant with surge tank

Aiming at an isolated hydropower plant (HPP) with surge tank, this paper studies the regulation quality of the turbine regulating system under load dis-turbances. A fifth order transfer function for frequency response under step load disturbance is derived. The complete fifth order system is solved and the regulation quality for frequency response is studied.

The author participated in the modelling and analysis, commented on the manuscript.

Published in International Journal of Electrical Power and Energy Sys-tems, 2015, 73: 528-538.

Paper V Time response of the frequency of hydroelectric generator unit with surge tank under isolated operation based on turbine regulating modes

For isolated turbine regulating systems, a complete mathematical model under three regulation modes is established. Based on dominant poles and ze-ros, the method of order reduction for a high-order system of time response of the frequency is proposed. The regulation quality for time response of the fre-quency is studied, by solving the complete high-order systems.

The author participated in the modelling and analysis, and commented on the manuscript.

Published in Electric Power Components and Systems, 2015, 43(20), 2341-2355.

Paper VI Instability analysis of pumped-storage stations at no-load conditions us-ing a parameter-varying model

This study set out to theoretically explore the root cause of the instability of pumped-storage stations at no-load conditions and the dominant factors influencing it. A detailed study of the two key factors affecting system stability was carried out based on the system model. The Laplace transform and inverse transform decomposition were used to obtain a mathematical expression for the no-load oscillation. Simulations were performed for further validation.

113

The author engaged in the modelling and simulations, and commented on the manuscript.

Published in Renewable Energy, 90 (2016): 420-429.

Paper VII Extreme water-hammer pressure during one-after-another load shed-ding in pumped-storage stations

This study theoretically analysed the hydraulic connections between the turbines and revealed the mechanism of the rapid rise in the water-hammer pressure under the one-after-another load-shedding conditions. Numerical simulations and model experiments were performed for validation. An analy-sis was conducted on the distribution of the water inertia in branch pipes, and corresponding engineering measures were proposed.

The author engaged in the numerical modelling and experiments, and pol-ished the manuscript.

Published in Renewable Energy, 99 (2016): 35-44.

Paper VIII Simulation of wind speed in the ventilation tunnel for surge tank in tran-sient process

To obtain the wind speed in a ventilation tunnel for a surge tank during transient processes, this article adopts a one-dimensional numerical simulation method and establishes a mathematical model of a wind speed. The effective mechanism of water-level fluctuation in a surge tank and the shape of the ven-tilation tunnel for the wind speed distribution and the change process are dis-covered. A comparison between the simulation results of 1D and 3D compu-tational fluid dynamics were also conduced.

The author conducted part of case studies and discussions. Published in Energies, 9.2 (2016): 95.

Paper IX Eigen-analysis of hydraulic-mechanical-electrical coupling mechanism for small signal stability of hydropower plant

This work aims to conduct a fundamental study on hydraulic-mechanical-electrical coupling mechanism for small signal stability of hydropower plants. A twelfth-order state matrix is established for theoretical eigen-analysis. A simulation model based on SimPowerSystems is built for validation. The in-fluencing mechanisms of water column elasticity, governor mechanical com-ponent and water inertia are studied under different control modes of the tur-bine governor.

The author conducted the theoretical eigen-analysis, built the numerical model and did the simulations, and wrote the manuscript.

Submitted to Renewable Energy, 2017.

114

Paper X Hydraulic damping on rotor angle oscillations: quantification using a nu-merical hydropower plant model

This paper aims to quantify the hydraulic turbine damping of rotor angle oscillations. An equivalent hydraulic turbine damping coefficient is intro-duced and the corresponding methodology is proposed. A sophisticated HPP model that combines electrical subsystems with a refined hydraulic-mechani-cal subsystem is established and verified. Then, the quantitative results of the damping coefficient are analysed under different cases. Observations and dis-cussions of on-site measurements are included to support the analysis.

The author proposed the methodology, built the numerical model and did the simulations, and wrote the manuscript.

Submitted to IEEE Transactions on Energy Conversion, 2017.

Paper XI Wear and tear on hydro power turbines – influence from primary fre-quency control

This paper studies wear and tear on hydropower turbines by applying nu-merical simulation and theoretical derivation, from the point of view of regu-lation and control. Governor models are built and validated by measurement data. The core indicator, guide vane movement, is analysed based on ideal sinusoidal frequency input and real frequency records. Influences of different factors, e.g. governor parameters, power feedback mode and nonlinear gover-nor factors are analysed.

The author deduced the theory formula, performed the simulations and analysis, and wrote the manuscript.

Published in Renewable Energy, 87(2015) 88-95.

Paper XII Wear reduction for hydro power turbines considering frequency quality of power systems: a study on controller filters

In this paper, a controller filter is proposed as a solution to the trade-off between reducing the wear of turbines and maintaining the frequency quality of the power systems. The dead zone is compared with a floating dead zone and a linear filter. Time domain simulation is used to investigate the guide vane movement, the load disturbance and the power system frequency. The describing functions method and the Nyquist criterion are adopted to examine the stability of the system.

The author engaged in the measurements, performed the theoretical analy-sis and simulations, and wrote the manuscript.

Published in IEEE Transactions on Power Systems, 2016.

115

Paper XIII Analysis on regulation strategies for extending service life of hydro power turbines

The aim of this paper is to extend the service life of hydropower turbines, by reasonably decreasing the guide vane (GV) movements with appropriate regulation strategies, e.g. settings of PI (proportional-integral) governor pa-rameters and controller filters. Basic theoretical formulas are discussed and compared to the simulation results, showing reasonable correspondence.

The author performed the theoretical analysis and simulations, and wrote the manuscript.

Presented at 28th IAHR Symposium on Hydraulic Machinery and Systems, Grenoble, France, July, 2016; Published in IOP Conference Series: Earth and Environmental Science, 2016.

Paper XIV Burden on hydropower units for balancing renewable power systems

In this paper, a framework combining technical plant operation with eco-nomic indicators is proposed. Primary frequency control (PFC), acting on a time scale from seconds to minutes, are analysed. A model integrating hydrau-lic, mechanical, and electrical subsystems is developed to characterize effi-ciency loss, wear and fatigue, regulation mileage, and frequency quality. Bur-den relief strategies and their consequences are evaluated under two idealized remuneration schemes for PFC, inspired by those used in Sweden and in the USA. The study can be beneficial to both producers and transmission system operators: a favourable alternative strategy of burden relief is shown for pro-ducers, and insights for transmission system operators are provided on the re-lation between incentive structures and the quality of the delivered service.

The author participated in formulating the research question, built the nu-merical model, conducted the simulations, analysed the results and drafted the manuscript.

Submitted to Nature Energy, under formal peer review, 2017.

Paper XV Allocation of frequency control reserves and its impact on wear on a hy-dropower fleet

This paper investigates how the allocation of a sold volume of frequency control reserves within a large hydropower production fleet can affect the costs of providing primary and secondary reserves, in terms of its impact on wear and fatigue, production losses and the quality of the delivered frequency control.

The author participated in the numerical modelling and discussions, and commented on the manuscript.

Revised and resubmitted to IEEE Transactions on Power Systems, 2016.

116

Paper XVI Linear synthetic inertia for improved frequency quality and reduced hy-dropower wear and tear

This paper investigates how the frequency quality and frequency control-ling hydropower units are affected by decreasing inertia and damping, using the Nordic power system as a case study. A new type of synthetic inertia (SI), which is linear and continuously active, is suggested as a means to mitigate the impacts on these units.

The author engaged in the numerical modelling and discussions, and com-mented on the manuscript.

Submitted to IEEE Transactions on Power Systems, 2016.

117

11 Acknowledgements

Many people said that an introduction section might be the most difficult part to write in a thesis, and I also agreed until I came to this very special section, just because I have so much gratitude to express, and I will feel guilty if I use any casual word for any one of them… Here I really would like to add the “Prof.” or “Dr.” title when I mention someone, as a Chinese tradition; but please allow me to neglect the title as a Swedish style, also because there will be too many titles in the text otherwise…

The thesis was carried out as a part of "Swedish Hydropower Centre - SVC". I thank the China Scholarship Council (CSC), StandUp for Energy, and I appreciate the scholarships and travel grants from the Anna Maria Lundin’s scholarship committee (twice), the Wallenberg, the ÅForsk, the Liljewalch and the Sederholm foundations!

For my supervisors Per Norrlund and Urban Lundin, I think I can write a whole acknowledgement section just for you! To my main supervisor Per, I would like to express my sincere gratitude for your great guidance, your super nice character and your rigorous working attitude as a doctor in the field of numerical analysis! To my second supervisor Urban, thank you so much for your guidance, support and influence! In my mind, what you do is far more beyond a standard duty of a second supervisor. In many Chinese universities, there is a common prize similar to “Top 10 of the nicest supervisors”; I think that both of you will be winners if Uppsala University initiates this kind of award!

To the head of our Division of Electricity, Mats Leijon, thank you for ac-cepting me as a PhD student and helping me to get my PhD scholarship!

Dear hydropower group members in Uppsala: Niklas Dahlbäck, Johan Abrahamsson, Birger Marcusson, Jose Perez, Jonas Noland, Fredrik Evestedt, thank you for your support and inspirations to my research. Espe-cially to my co-authors and office-mates Linn Saarinen and Johan Bladh: thank you a lot for your comments and discussion on my works, and teaching me much knowledge and experience; I really appreciate and learned a lot from your professional and efficient way of working! To all the members of SVC reference group, thank you for your comments and supports to my project.

I would like to thank my collaborators in Wuhan University in China. Thank you very much Jiandong Yang, you are like my “third supervisor” and I really appreciate your support and guidance! To all my co-authors and mem-bers in the hydropower group in Wuhan, especially Wencheng Guo, Chao

118

Wang, Wei Zeng and Jiebin Yang, thank you for your discussion and help on my research, and the happy time we spent together! To all my friends in Wuhan and Changsha, thank you for companying, supporting and the happy hours we had together in my vacations!

To the Energy-Water Resource Systems Team in the Oak Ridge National Lab in the USA: thank you Brennan Smith, Boualem Hadjerioua, Adam Witt and Stephen Signore for your wonderful cooperation and discussion. I will always keep in mind the hot but exciting days in Oak Ridge!

To the SMart grid And Renewable energy Technology (SMART) Lab in the University of Saskatchewan in Canada: thank you C.Y. Chung for your great cooperation and insightful guidance! I also thank all the lab members, I indeed appreciate and enjoyed the atmosphere of your lab.

My sincere thanks also go to all my teachers of the courses in Uppsala, KTH and Luleå: I not only gained the knowledge, but also had deep feelings on the high-quality Swedish education. I heartily thank the organizers, speak-ers and all fellow attendants of the UK Energy Research Centre Energy Summer School and the Energy Summer School in University of Gro-ningen: the two summer schools really had profound influence on my mind, much more than I thought before I attended!

To Thomas Götschl, thank you for your support on my computer. Thank you, Maria Nordengren, Gunnel Ivarsson, Ingrid Ringård, Anna Wiström and Emma Holmberg, for the really helpful administrative works.

Thank you Rafael Waters, Liguo Wang, Liselotte Ulvgård, Nat-takarn Suntornwipat and all the student assistants, my teaching experi-ence with you was great and smooth! Thank you Nicole Carpman for your help on ordering circuit components for the teaching!

Thank all my friends, lunch-mates and colleagues in the Division of Elec-tricity: André L., Anke B., Arvind P., Aya A., Eduard D., Flore R., Fran-cisco F., Johan F., Juan de S., Kaspars S., Maria A. C., Minh Thao N., Pauline E., Per R., Tatiana P., Tobias K. and Victor M., thank you for the great time we had together in Angstrom, in nations and parties! To Anders G., Dalina J., Dana S., Irina D., Jennifer L., Jon O., Linnea S., Magnus H., Malin G., Marianna G., Markus G., Morgan R., Muzafar I., Petter E., Saman M., Simon T., Valeria C. and all other colleagues, thank you for sharing your interesting life and diverse knowledge from all over the world! To my new office-mates Anna F. and Jonathan S., thank you for company-ing for the last months of my PhD study!

Especially thank all my Mandarin-speaking colleges in the division, Wei L., Ling H., Yue H., Liguo W., Xiao Z., Wenchuang C., Jinming W., Qiulin L., Wooi Chin L., Lai Mun O., it’s my great fortune to meet you!

I give my great gratitude to my landlords Shuxi Z. and Guihua L., you are like my family in Uppsala! To my previous and current flat-mates, Shunguo W., Yongmei G., Keqiang G., Xiaoyang S. and Huiying Q., thank you for

119

all the wonderful days we had and all the ups and downs we went through together!

To my pals, Changqing R., Feiyan L., Fengzhen S., Jiajie Y., Lichuan W., Meiyuan G. and Yi R., thank you for being by my side from the begin-ning till the end of my PhD period! Thank you Hao C., Xiaoting Z., and Yan-ran Z., being together with you is always so nice and relaxing!

To my dear Chinese friends, Chenjuan L., Chunling S., Cong S. , Dan W., Fei H., Fengjiao Z., Haoyu L., Huimin Z., Jingyi H., Le F., Le G., Lei T., Lei Z., Lin S., Ling X. , Liyang S., Mingzhi J., Na X., Ping Y., Qifan X., Rui S., Shihuai W., Song C., Teng Z., Tianyi S., Wen H., Wenxing Y., Xiaoliang L., Xiaowen L., Yuan T., Yuanyuan H., Zhibing Y., and Zhen Q., for all the happiness we shared and all the delicious Chinese food we had together! Really thank you Weiwei S., especially for your help and driving in Oak Ridge!

I give unique thanks to unique groups: my football-mates from all over the world in Campus 1477, Flogsta fotboll, our division team, the Angstrom group and Barbafarsorna United, as well as Jiangtao C. and other Chinese teammates! Now I am proud to say that I achieved two goals of coming to Sweden: having a doctoral study and improving my football skill!

Finally, to my family, especially to my parents and my love: any words

here are pale and I will express my deepest gratitude with action! 最后，致我的家人，致我的父母和我的挚爱：深情不及久伴，厚爱

无需多言，我将用行动表达对你们最衷心的爱！

Weijia Yang (杨威嘉) March 27th, 2017 Uppsala

120

12 Svensk sammanfattning

Globalt sett utgör vattenkraft den största förnybara källan till elektrisk energi, och axlar en stor del av uppdraget att reglera många kraftsystem. Stabil och effektiv drift av vattenkraftverk och deras samspel med kraftsystemet har stor betydelse. Vattenkraft utgör ett relativt moget teknikområde, men nya utma-ningar uppkommer i samband med utbyggnad av andra förnybara energikällor. En större andel av exempelvis sol- och vindkraft, vars intermittens medför variabilitet och oförutsägbarhet, ställer nya krav på vattenkraftens funktion som leverantör av reglertjänster och energilagring. Dessutom ger dessa källor inte automatiskt något tillskott till kraftsystemets tröghet, vilket kan leda till försämring av kraftsystemets elektriska frekvens. För vattenkraften, som i många kraftsystem står för det mesta av frekvensregleringen, innebär det en ökning av slitagedrivande reglerarbete.

En annan faktor som ger förändrade förutsättningar är den växande kom-plexiteten och storleken hos vattenkraftverk och elnät, exempelvis i Kina där det finns dussintals planerade, designade, konstruerade eller idrifttagna vat-tenkraftverk med effekter över 1000 MW.

I den här avhandlingen används teoretisk analys, numerisk simulering och fältmätningar som huvudsakliga metoder. Flera numeriska kraftverksmodeller utformas, med varierande komplexitetsgrad beroende på deras syfte. Huvud-parten av analysen och resultaten baseras på åtta vattenkraftverk i Sverige och Kina.

Stabil drift (frekvensstabilitet och rotorvinkelstabilitet) och effektiv drift utgör två viktiga mål, vilket motiverar avhandlingens undertitel. Uttrycket ”ef-fektiv drift” (”efficient operation”) används här i en generell betydelse och syftar på ekonomiskt sund drift i ett större perspektiv. Det inkluderar både produktion vid hög turbinverkningsgrad och faktorer som relaterar till slitage och utmattning. Uttrycket ”ekonomisk drift” undviks eftersom problemen stu-deras med ett tekniskt perspektiv och konkreta ekonomiska mått saknas.

Flera driftförutsättningar analyseras avseende stabil drift; svarstid vid fre-kvensreglering och systemstabilitet vid önätsdrift undersöks. Med frekvensre-glering avses här i första hand primärreglering. Även sekundärreglering är re-levant och berörs i artiklar, men behandlas inte explicit i avhandlingen. Fre-kvensreglering hanteras på olika vis i olika länder. I exempelvis Sverige är det en tjänst som systemoperatören köper från kraftproducenterna medan det i andra länder, som Norge och Kina, också finns en plikt att leverera utan er-sättning. Även system som bygger på ersättning kan variera i sin utformning.

121

I det svenska systemet baseras ersättningen huvudsakligen på volymen av den reserverade reserven, alltså beredskapen att leverera tjänsten. I ett av USA:s delsystem tas istället hänsyn till volym och kvalitet hos den faktiska leveran-sen. Följder av skillnader i ersättningsprincip berörs i avhandlingen.

En grundläggande studie genomförs beträffande kopplingar mellan hydrau-liska, mekaniska och elektriska system och hur de påverkar småsignalstabili-teten. En metodik föreslås för kvantifiering vattenturbiners bidrag till dämp-ning av lågfrekventa oscillationer. Oscillationer med periodtider från mindre än en och upp till hundratals sekunder, analyseras.

Vad beträffar effektiv drift presenteras en beskrivning och en inledande analys av slitage och utmattning av turbiner; ett regulatorfilter föreslås som en slitagereducerande åtgärd som bibehåller kraftsystemets frekvenskvalitet. Stu-dien utökas med ett lovande ramverk som kombinerar tekniska driftaspekter med ekonomiska indikatorer, för att bestämma relativa färden på reglerbörda och prestation kopplad till frekvensreglering.

Ändamålsenlighet och tillämpbarhet betraktas för olika numeriska mo-deller och förslag på optimering av modeller lämnas. De använda modellerna har varierande komplexitetsgrad. Högst komplexitet har modeller baserade på TOPSYS, en programvara utvecklad mestadels vid universitetet i Wuhan, Kina. Därefter kommer modeller implementerade i MATLAB. Lägst kom-plexitet har analytiska modeller som härletts exempelvis för studier av stabi-litetskriterier.

Resultaten visar att kopplingen mellan det hydrauliskt-mekaniska och det elektriska delsystemen kan vara av betydelse och bör betraktas noggrannare. Kraftsystemets påverkan på vattenkraftverk behandlas genom studier av dy-namiska processer och motsvarande reglerstrategier som motiveras av olika typer av störningar och av kravställningar kopplade till kraftsystemet. Exem-pel på påverkan i denna riktning, som studeras här, är svarstid vid frekvensre-glering, svallning vid laststörning, rotorvinkelpendling vid spänningsstörning och slitage och verkningsgradsförlust vid leverans av reglertjänster.

Den andra riktningen, alltså vattenkraftverkens påverkan på kraftsystemet behandlas genom kvantifiering av den aktiva regleringens betydelse för fre-kvenskvaliteten och hur den hydrauliska dämpningen i vattenturbiner bidrar, med hjälp av föreslagna metodiker.

122

13 中文概要 (Summary in Chinese)

水力发电作为目前世界上规模最大的可再生能源，在许多电力系统中亦承担着大量而关键的调节任务。如今，这一相对成熟的技术仍面临着许多新的挑战：（1）全球间歇性可再生能源发电比重的不断提升，水电站有功功率调峰、调频及无功功率调节性能需要进一步提升；（2）水电站自身规模越来越大、水力-机械-电气耦合系统的结构越来越复杂，水电站运行控制策略需要相应地提升；（3）全球许多地区（如北欧、中国西南地区等）形成了以水电为主要电源成分的电力系统，且大型水电站群远离负荷中心，故水电站与电力系统相互作用特性及网源协调机制亟需研究；（4）在各国电力市场体系的形成及变革趋势下，水电机组在可再生能源电力系统中运行调节的高效性、经济性问题研究意义重大。因此，本文针对水电站与电力系统的相互作用机理，以稳定性（频率稳定与功角稳定）和高效性为核心研究目的，开展了水电站动态过程与控制的研究。本文的主要研究内容和思路可见图 1.7。 本文采用理论分析、数值模拟和现场实测三种主要研究方法。针对不同的研究目的，本文分别基于水电站过渡过程计算软件 TOPSYS、MATLAB（Simulink 与 SimPowerSystems）及理论分析方法建立了复杂度与精度各不相同的 10 余个水电站系统数学模型。本文主要采用一维的数学分析计算方法，如图 1.6 所示，为系统部件（水轮机、发电机等）的精细分析及大型系统（电力系统、梯级电站群等）的整体研究搭建桥梁是本文及后续工作的长远目标之一。本文各项具体的研究工作都是基于真实的工程实例，总共采用了 8 个瑞典或中国的水电站作为分析对象。 关于频率稳定性问题，本文针对水电站多种运行条件以及不同调节模式开展分析，并通过 4个大型水电站的实测数据验证了 TOPSYS软件拓展功能的可靠性。针对欧洲输电运营联盟 (ENTSO-E)、北欧输电运营商 Statnett、中国国家电网的规范需求，推导了频率阶跃下机组导叶开度响应理论公式，揭示了机组一次调频功率响应时间的变化规律及关键因素影响机理。针对水电站孤网运行下由调压室水位波动引起的机组频率振荡问题，从理论分析层面阐明了不同水轮机控制模式的不同作用机制，并通过数值模型以进行验证，分析了各项关键因素的影响规律。 关于功角稳定性问题，本文首先针对水电站系统的小信号稳定性，开展了水力-机械-电气耦合机制的机理性研究；采用特征根分析和数值模拟的方法，揭示了水力-机械中各因素（水体弹性、水力阻尼、调速系统接力器等）对电力系统本地振荡模式的影响机制。而后，本文进

123

一步开展量化水力阻尼对功角振荡的研究工作，提出了等效水轮机阻尼系数以及基于精细数值模拟（TOPSYS 软件拓展功能）的相应量化方法；分别对投入、切除电力系统稳定器的三相短路工况开展探讨，并利用现场实测资料对研究结论进行了补充分析。 关于水电站在可再生能源系统中运行与调节的高效性问题，本文首次基于机组调节控制层面阐述了一次调频过程中水电机组磨损问题机理，定义了描述机组磨损的调节量化指标，并对水力-机械系统中各项影响因素规律进行了探索。而后，本文进一步提出使用一种频率信号滤波器，在维持电力系统频率质量的前提下减小水电机组磨损，研究表明该滤波器的综合性能明显优于电力系统现广泛使用的人工死区。最后，本文针对水电机组在电网调频过程中的“负担”与调节品质这双方面互相权衡的问题，提出了一套结合电站技术特性和经济性指标的量化评价体系；以效率损失、机组磨损、调节“里程”、频率质量四项指标为落脚点，对机组“减负”运行策略及其在两种经济补偿机制（分别基于瑞典与美国的相关现行机制）下的综合性能进行了评价。 通过上述的系列研究工作，本文得出的各项具体结论可参见第 7 章，在此主要对本文的核心结论与观点进行凝练与阐述： （1）物理特性层面：本文加深了对水电站水力-机械-电气耦合机制的认识；分析了水力发电系统中具有不同周期的各种振荡过程及作用机理，如周期可低于 1 秒的功角振荡、周期约 60 秒的北欧电力系统频率振荡及周期可达数百秒的调压室水位波动等。本文得出的一个核心结论为，水力-机械子系统与电气子系统之间可产生较强的耦合作用，并值得更多关注及后续研究。 （2）数学方法层面：本文建立了针对理论分析与数值模拟的多种数学模型，并通过现场实测进行了验证；不同模型的适用性、高效性得以展现，对后续研究中复杂水电系统的数学描述及模型优化有一定启发。 （3）工程措施层面：本文针对水电站多种运行条件下的稳定性与高效性进行了分析，并提出与优化了一系列针对性的控制策略。研究也表明，水电站动态过程受系统各部分特性的影响明显，针对各个电站的个例分析非常必要，亦体现出准确、高效的数学建模及分析方法的重要性。 正如本文标题所强调，水电站与电力系统的相互作用为本文的研究重点并贯穿于全文，具体可归纳为以下两方面。 （1）电力系统对水电站的影响：本文研究了电力系统多种扰动及不同需求下的水电站动态响应规律及控制策略，体现在频率扰动下的机组响应时间、负荷扰动下的系统频率振荡、电压扰动下的功角振荡、电网调频服务下的机组磨损及效率损失等具体研究内容。 （2）水电站对电力系统的影响：本文开展了水电站对电力系统影响的评价工作并提出了相应的量化分析方法，例如水电机组调节下的电网频率质量（频率稳定性）、针对电力系统低频振荡的等效水轮机阻尼系数（功角稳定性）等。

124

14 Appendices

14.1 Appendix A (1) Symbols in section 3.3.1.2 for frequency control

0

0

2 ywy

hm T s

H , 5 4 (1 )y P Pa f T b K ,

4 5 4(1 ) (1 )y P P P P P I ya f T b K f b K b K T

3 6 5 4 1(1 ) (1 )y P P P P P I y P I Pa f T b K f b K b K T f b K f K ,

2 7 6 5 2 1(1 ) (1 )y P P P P P I y P I Pa f T b K f b K b K T f b K f K f ,

1 7 6 3 2(1 )P P P I y P I P Ia f b K b K T f b K f K f K ,

0 7 3P I Ia f b K f K 1 y F wyf e T T ,0

20

2( )y

y qh wy F h qy wy

hf e e T T e e T

H ,

0 03

0 0

2 2(1 )y y

y qh h qy

h hf e e e e

H H , 4 a F w yf T T T ,

05

0

2( ) ( )y

F wy g x a F wy qh

hf T T e e T T T e

H ,

0 06

0 0

2 2( )( ) (1 )y y

g x F wy qh a qh h qx wy

h hf e e T T e T e e e T

H H ,

0 07

0 0

2 2( )(1 )y y

g x qh h qx

h hf e e e e e

H H .

(2) Symbols in section 3.3.1.2 for power control

5 wy F y aa T T T T , 4 2 0wy F y y a wy F aa T T T b T T b T T T ,

3 0 2 1 2 0 4y y a wy y h qx wy F a y wy Fa T b b T T b T T e e T T b T b e T T b ,

2 1 2 3 0 2 1

0 4 5 ( )

y y qx a

wy h qx y wy h qy y wy F

a T b b T e b b b T b

T e e e b T e e b e T T b

,

1 1 2 3 0 5 1 3 4( ) ( )qx y wy h qy y qya b b e b e b T e e b e b e b b ,

125

0 1 3 5( )y qya e b e b b , 00

0

2 y Fwy qh

h Tb T e

H , 0

10

21y qhh e

bH

, 2 g xb e e ,

03

0

2 y hh eb

H , 4 p I ab e K T , 5 ( 1)p I gb e K e .

(3) Coefficients of transfer functions Φ1 in section 3.3.2.2

0 1 0b , 11 p wb K KT , 21 p i wb K K K KT , 3 1 ib K K , 01 0a ,

11 0.5 1w p pa T M b K , 21 1 0.5p p w p i p pa M b K T b K M D b K D ,

31 0.5p i p p w p ia b K M D b K D T b K D , 4 1 p ia b K D

(4) Coefficients of transfer functions Φ2 in section 3.3.2.2 b02 = b01, b12 = b11, b22 = b21, b32 = b31,

02 0.5 1f w p pa T T M b K ,

12 0.5 1 1 0.5w p p f p p w p i p pa T M b K T M b K T b K M D b K D ,

22 1 0.5

0.5

p p w p i p p

f p i p p w p i

a M b K T b K M D b K D

T b K M D b K D T b K D

,

3 2 0 .5p i p p w p i f p ia b K M D b K D T b K D T b K D ,

42 p ia b K D .

(5) Elements of the state matrix in section 3.3.3.2

1,2 0 02a f , 12,1

j

Ka

T , 2,2

j

ea

T , 2

2,4j

Ka

T , 3

2,5j

Ka

T ,

2,8y

j

ea

T , 2,10

h

j

ea

T ,

43,1

0

d d

d

K X Xa

T

, 3,3

0

1

d

aT

,

53,4

0

d d

d

K X Xa

T

, 3,6

0

1

d

aT

, 4

4,10

d d

d

K X Xa

T

, 4,3

0

1

d

aT

,

5 54,4

0

1d d

d

K X K Xa

T

,

6

5,10

q q

q

K X Xa

T

, 7 7

5,50

1q q

q

K X K Xa

T

,

86,1

a

r

K Ka

T , 9

6,4a

r

K Ka

T , 10

6,5a

r

K Ka

T , 6,6

1

r

aT

, 6,12a

r

Ka

T ,

8,7

1

y

aT

, 8,8

1

y

aT

,2,1

9,1 2

w q

e

T e aa

T

,

2,29,2 2

w q

e

T e aa

T

,

126

2,49,4 2

w q

e

T e aa

T

,

2,59,5 2

w q

e

T e aa

T

,

8,79,7 2

w qy

e

T e aa

T ,

2,8 8,89,8 2

w q w qy

e

T e a T e aa

T

, 9,9 2

w qh

e

T ea

T ,

2,109,10 2

1 w q

e

T e aa

T

, 10,9 1a

, 11,1 2,1 2 4,1 3 5,1s s Pe s Pea K K a K K K a K K K a , 11,2 2,2 1 1,2s s Pea K K a K K K a ,

11,3 2 4,3s Pea K K K a , 11,4 2,4 2 4,4s s Pea K K a K K K a ,

11,5 2,5 3 5,5s s Pea K K a K K K a , 11,8 2,8sa K K a , 11,10 2,10sa K K a , 11,110

1a

T ,

112,1 11,1

2

Ta a

T , 1

12,2 11,22

Ta a

T , 1

12,3 11,32

Ta a

T , 1

12,4 11,42

Ta a

T , 1

12,5 11,52

Ta a

T

, 112,8 11,8

2

Ta a

T , 1

12,10 11,102

Ta a

T , 1

12,11 11,112 2

1Ta a

T T , 12,12

2

1a

T .

For opening feedback:

2,1( )7,1 1

pOF

p p

K aa

b K

, 2,2

( )7,2 1p i

OFp p

K a Ka

b K

, ( )7,3 0OFa ,

2,4( )7,4 1

pOF

p p

K aa

b K

, 2,5

( )7,5 1p

OFp p

K aa

b K

, ( )7,7 1

p iOF

p p

b Ka

b K

,

2,8( )7,8 1

pOF

p p

K aa

b K

, 2,10

( )7,10 1p

OFp p

K aa

b K

.

For power feedback:

( )7,1 1 3 5,1 2 4,1 2,1PF p i p p p p pa b K K b K K a b K K a K a ,

( )7,2 1 1,2 2,2PF p i p ia b K K a K a K , ( )7,3 2 4,3PF p pa b K K a ,

( )7,4 2 2 4,4 2,4PF p i p p pa b K K b K K a K a ,

( )7,5 3 3 5,5 2,5PF p i p p pa b K K b K K a K a , ( )7,7 0PFa , ( )7,8 2,8PF pa K a ,

( )7,10 2,10PF pa K a .

Here, K1 – K10 are:

1 2 3

e d d q q d q q d

e e ee q d q d

q d

P E I E I I I X X

P P PP E E K K E K E

E E

127

4 5

cos;q s d d

d d q qd q

E V I II I E K K E

X E

6 7

sin; q qs d

q q d dq d

I IV EI I E K K E

X E

2 2

8 9 10

sin coss d q s s q d sg

q d

g g gg q d q d

q d

X E X V X E X VV

X X

V V VV E E K K E K E

E E

2

1

cos cos 2 sincossin q d q s s d sq sd s

d q d q

X X E V V E VE VE VK

X X X X

,

2

sinsin q d s dd s d

d q d q

X X V EE V EK

X X X X

,

3

coscos q d s qq s q

d q d q

X X V EE V EK

X X X X

, 4

sins

d

VK

X

,

5

1

d

KX

,6

coss

q

VK

X

,7

1

q

KX

,

8

cos singd s q gq s d

g q g d

V V X V V XK

V X V X

, 9

gq s

g d

V xK

V X

, 10gd s

g q

V xK

V X

.

The values of the state variables in the coefficients K1 – K10 are initial steady-state values.

128

14.2 Appendix B (1) Simulation settings in section 4.2

Table 14.1. The default settings of the simulation of PFC

Upstream level [m]

Down-stream

level [m]

Initial power [MW]

Frequency step [Hz]

bp Kp , Ki , Kd Ey , Ef

1639.3 1332.3 476 -0.2 0.04 9, 8, 0 0, 0.05

(2) Parameters and settings in section 4.3 Governor parameters:

Kd = 0, Ty = 0.02 s, bp = 0.04 pu, ep = 0.04 pu, Ef = Ey =Ep= 0; Characteristic coefficient of power grid load: eg = 0.0; Transmission coefficient of ideal turbine: eh = 1.5 pu, ex = -1 pu ,ey = 1 pu,

eqh = 0.5 pu, eqx = 0, eqy = 1 pu; Cross section area of surge tank (F): 415.64 m2; Thoma critical section area for stability (Fth): 416.08 m2.

(3) Parameter values of HPP 5 in chapter 5

The values of the generator parameters are estimated from field simulations of standard tests in [127, 155]. Generator: the nominal apparent power is 206 MVA, and the line-to-line

voltage is 21 kV. Xd =0.768 pu, =0.249 pu, =0.187 pu, Xq =0.512 pu, =0.189 pu,

=7.880 s, =0.049 s, =0.0283 s, Tj =7.0 s; Transformer and transmission line: Xs =0.30 pu; Turbine characteristic (for a normal operating point): eqy =0.66 pu, eqω

=0.1 pu, eqh =0.47 pu, ey =0.5 pu, eω =-0.96 pu, eh =1.45 pu; Penstock: Tw =1.34 s (calculated under the rated condition: discharge is

275.0 m3/s and water head is 73.0 m), α=0.33 pu, Te =0.115 s (length of the penstock is 115 m).

(4) Operating settings of HPP 5 in section 5.1 Initial steady-state condition: Pe =0.90 pu, cosφ =0.90 (Qg =0.436), Vs

=1.00 pu; Turbine governor (for both two feedback modes): bp =0.04 pu, Kp =9.0 pu,

Ki =5.0 s-1, Ty =0.2 s, Backlash=0.001 pu, Limiting rate=0.1 pu/s; AVR: Tr =0.05, Ka =100, the regulator output limit is ±2.0 pu; PSS (speed input): Kω =1 pu, KPe =0, Ks =9.5 pu, T0 =1.4 s, T1 =0.154 s,

T2 =0.033 s;

129

PSS (power input): Kω =0, KPe =1 pu, Ks =2.5 pu, T0 =1.4 s, T1 =0.154 s, T2 =3.0 s.

(5) Operating settings of HPP 5 in section 5.2 Initial steady-state condition: Pe =0.90 pu, cosφ =0.90 (Qe =0.436), Vs

=1.00 pu; Governor: bp =0.04 pu, Kp =8.0 pu, Ki =1.0 s-1, Ty =0.2 s, the backlash

value is 0.001 pu, the limiting rate is 0.1 pu/s; AVR: Tr =0.05, Ka =100 pu, the output limit is ±4.0 pu; PSS: Kω =1, KPe =0, Ks =9.5 pu, T0 =1.4 s, T1 =0.354 s, T2 =0.033 s; the

output limit is ±0.05 pu.

(6) Default simulation settings in section 6.1 Upstream level and downstream level: 213.1m and 78.3 m; Initial power: 122.0 MW; Amplitude and period of sinusoidal frequency: 0.1 Hz and 60 s; Turbine governor: Kp =1.0 pu, Ki =0.833 s-1, Kd =0, bp =0.02 pu, Edz =0,

Backlash-By =0.001 pu, Ty (lag) =0.02 s.

(7) Detailed information of measurements in section 6.2 The original sampling frequency is 200 Hz, and the sampling time of the

signals is averaged to 0.2 s. The governor parameter in the simulation is the same as the values in the HPP during the measurement, which are the standard parameter settings EP1 in Vattenfall HPPs, see Table 14.2. The values of the lag, the backlash and the delay are set to 0.25 s, 0.00029 pu and 0.097 s re-spectively.

Table 14.2. Standard controller parameters in Vattenfall HPPs

Parameter Ep0 Ep1 Ep2 Ep3

bp (or Ep) 0.1 0.04 0.02 0.01

Kp 1 1 1 2

Ki 1/6 5/12 5/6 5/3

Table 14.3. The default parameter settings of the actuator in section 6.2

Parameter Servo (Ty) Saturation Rate limiting Backlash

Value 0.2 [0,1 pu] ±0.1 pu/s 0.05×10-2 pu

130

Table 14.4. Parameters of the plant and the grid in section 6.2

Symbol Parameter Value

K Scaling factor 10×bp [pu] Tw Water time constant 1.5 [s]

M System inertia 13 [s] D Load damping constant 0.5 [pu]

(8) Detailed information for section 6.3

Table 14.5. Parameters of the plant and the grid in section 6.3. The value of M and D are updated, comparing from the values in Table 14.4.

Symbol Parameter Value

Tw Water time constant 1.5 [s]

M System inertia 14.6 [s] D Load damping constant 0.66 [pu]

Table 14.6. Parameter values of the HPP 6 and HPP 7 for simulation settings. There is no surge tank in HPP 7, hence the values of Twt, Ts and ft are shown as N/A.

HPP 6 HPP 7

Para. Value Para. Value Para. Value Para. Value

Ty 0.25 s Twt 13.2 s Ty 0.25 s Twt N/A

Tya 0.90 s Tr 0.1 s Tya 0.90 s Tr 0.07 s

Tdel-gv 0.097 s Ts 350.0 s Tdel-gv 0.097 s Ts N/A

Tdel-a 0.410 s α 0.33 Tdel-a 0.410 s α 0.33

BLgv 0.00029 pu ft 0.0065×q0

BLgv 0.00029 pu ft N/A

BLa 0.00132 pu fp 0.0120×q0

BLa 0.00132 pu fp 0.010×q0

Twp 1.7 s Dt 0 Twp 1.01 s Dt 0

131

15 References

[1] Q. Schiermeier, J. Tollefson, T. Scully, A. Witze, and O. Morton, "Electricity without carbon," Nature, vol. 454, pp. 816-823, 2008.

[2] C. Mitchell, "Momentum is increasing towards a flexible electricity system based on renewables," Nature Energy, vol. 1, p. 15030, 02/01/online 2016.

[3] T. Rintamäki, A. S. Siddiqui, and A. Salo, "How much is enough? Optimal sup-port payments in a renewable-rich power system," Energy, vol. 117, pp. 300-313, 2016.

[4] A. S. Brouwer, M. van den Broek, A. Seebregts, and A. Faaij, "Operational flex-ibility and economics of power plants in future low-carbon power systems," Ap-plied Energy, vol. 156, pp. 107-128, 2015.

[5] D. Elliott, "A balancing act for renewables," Nature Energy, vol. 1, p. 15003, 2016.

[6] A. S. Brouwer, M. van den Broek, W. Zappa, W. C. Turkenburg, and A. Faaij, "Least-cost options for integrating intermittent renewables in low-carbon power systems," Applied Energy, vol. 161, pp. 48-74, 2016.

[7] J. Olauson, M. N. Ayob, M. Bergkvist, N. Carpman, V. Castellucci, A. Goude, et al., "Net load variability in Nordic countries with a highly or fully renewable power system," Nature Energy, vol. 1, p. 16175, 2016.

[8] E. Ørum, M. Kuivaniemi, M. Laasonen, A. I. Bruseth, E. A. Jansson, A. Danell, et al., "Future system inertia," ENTSO- E2015.

[9] X. Chang, X. Liu, and W. Zhou, "Hydropower in China at present and its further development," Energy, vol. 35, pp. 4400-4406, 2010.

[10] J. Jia, "A Technical Review of Hydro-Project Development in China," Engineer-ing, vol. 2, pp. 302-312, 2016.

[11] J. Shen, C. Cheng, X. Cheng, and J. R. Lund, "Coordinated operations of large-scale UHVDC hydropower and conventional hydro energies about regional power grid," Energy, vol. 95, pp. 433-446, 2016.

[12] H. Zhou, Y. Su, Y. Chen, Q. Ma, and W. Mo, "The China Southern Power Grid: Solutions to Operation Risks and Planning Challenges," IEEE Power and En-ergy Magazine, vol. 14, pp. 72-78, 2016.

[13] P. Storli and T. Nielsen, "Dynamic load on a Francis turbine runner from simu-lations based on measurements," in IOP Conference Series: Earth and Environ-mental Science, 2014, p. 032056.

[14] E. Doujak, "Effects of Increased Solar and Wind Energy on Hydro Plant Opera-tion," Hydro Review Worldwide, vol. 2, pp. 28-31, 2014.

[15] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability and control vol. 7: McGraw-hill New York, 1994.

[16] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, et al., "Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions," IEEE transactions on Power Systems, vol. 19, pp. 1387-1401, 2004.

132

[17] K. Prasertwong, N. Mithulananthan, and D. Thakur, "Understanding low-fre-quency oscillation in power systems," International Journal of Electrical Engi-neering Education, vol. 47, pp. 248-262, 2010.

[18] H. V. Pico, J. D. McCalley, A. Angel, R. Leon, and N. J. Castrillon, "Analysis of Very Low Frequency Oscillations in Hydro-Dominant Power Systems Using Multi-Unit Modeling," IEEE Transactions on Power Systems, vol. 27, pp. 1906-1915, 2012.

[19] J. Machowski, J. Bialek, and J. Bumby, Power system dynamics: stability and control: John Wiley & Sons, 2011.

[20] F. P. Demello and C. Concordia, "Concepts of synchronous machine stability as affected by excitation control," IEEE Transactions on Power Apparatus and Sys-tems, vol. 88, pp. 316-329, 1969.

[21] H. A. M. Moussa and Y.-n. Yu, "Optimal power system stabilization through excitation and/or governor control," IEEE Transactions on Power Apparatus and Systems, pp. 1166-1174, 1972.

[22] R. Grondin, I. Kamwa, L. Soulieres, J. Potvin, and R. Champagne, "An approach to PSS design for transient stability improvement through supplementary damp-ing of the common low-frequency," IEEE Transactions on Power Systems, vol. 8, pp. 954-963, 1993.

[23] G. Robert and D. Hurtado, "Optimal design of reactive power PI regulator for hydro power plants," in 2008 IEEE International Conference on Control Appli-cations, 2008, pp. 775-780.

[24] M. Alizadeh Bidgoli and S. M. T. Bathaee, "Full-state Variables Control of a Grid-connected Pumped Storage Power Plant Using Non-linear Controllers," Electric Power Components and Systems, vol. 43, pp. 260-270, 2015.

[25] J. Dai, D. Xiao, F. Shokooh, C. Schaeffer, and A. Benge, "Emergency generator startup study of a hydro turbine unit for a nuclear generation facility," IEEE Transactions on Industry Applications, vol. 40, pp. 1191-1199, 2004.

[26] D. Rimorov, I. Kamwa, and G. Joós, "Quasi-Steady-State Approach for Analysis of Frequency Oscillations and Damping Controller Design," IEEE Transactions on Power Systems, vol. 31, pp. 3212-3220, 2016.

[27] W. Tan, "Unified tuning of PID load frequency controller for power systems via IMC," IEEE Transactions on power systems, vol. 25, pp. 341-350, 2010.

[28] S. A. N. Sarmadi and V. Venkatasubramanian, "Inter-Area Resonance in Power Systems From Forced Oscillations," IEEE Transactions on Power Systems, vol. 31, pp. 378-386, 2016.

[29] L. Jiang, W. Yao, Q. H. Wu, J. Y. Wen, and S. J. Cheng, "Delay-dependent stability for load frequency control with constant and time-varying delays," IEEE Transactions on Power systems, vol. 27, pp. 932-941, 2012.

[30] C. K. Sanathanan, "Accurate low order model for hydraulic turbine-penstock," IEEE Transactions on Energy Conversion, vol. 2, pp. 196-200, 1987.

[31] C. D. Vournas, "Second order hydraulic turbine models for multimachine stabil-ity studies," IEEE Transactions on Energy Conversion, vol. 5, pp. 239-244, 1990.

[32] F. Demello, R. Koessler, J. Agee, P. Anderson, J. Doudna, J. Fish, et al., "Hy-draulic-turbine and turbine control-models for system dynamic studies," IEEE Transactions on Power Systems, vol. 7, pp. 167-179, 1992.

[33] E. De Jaeger, N. Janssens, B. Malfliet, and F. Van De Meulebroeke, "Hydro turbine model for system dynamic studies," IEEE Transactions on Power Sys-tems, vol. 9, pp. 1709-1715, 1994.

133

[34] L. N. Hannett, J. W. Feltes, and B. Fardanesh, "Field tests to validate hydro tur-bine-governor model structure and parameters," IEEE Transactions on Power Systems, vol. 9, pp. 1744-1751, 1994.

[35] L. N. Hannett, J. W. Feltes, B. Fardanesh, and W. Crean, "Modeling and control tuning of a hydro station with units sharing a common penstock section," IEEE Transactions on Power Systems, vol. 14, pp. 1407-1414, 1999.

[36] O. Souza Jr, N. Barbieri, and A. Santos, "Study of hydraulic transients in hydro-power plants through simulation of nonlinear model of penstock and hydraulic turbine model," IEEE Transactions on Power Systems, vol. 14, pp. 1269-1272, 1999.

[37] H. Fang, L. Chen, N. Dlakavu, and Z. Shen, "Basic modeling and simulation tool for analysis of hydraulic transients in hydroelectric power plants," IEEE Trans-actions on Energy Conversion, vol. 23, pp. 834-841, 2008.

[38] P. Pennacchi, S. Chatterton, and A. Vania, "Modeling of the dynamic response of a Francis turbine," Mechanical Systems and Signal Processing, vol. 29, pp. 107-119, 2012.

[39] D. Chen, C. Ding, Y. Do, X. Ma, H. Zhao, and Y. Wang, "Nonlinear dynamic analysis for a Francis hydro-turbine governing system and its control," Journal of the Franklin Institute, vol. 351, pp. 4596-4618, 2014.

[40] C. Wang, H. Nilsson, J. Yang, and O. Petit, "1D–3D coupling for hydraulic sys-tem transient simulations," Computer Physics Communications, vol. 210, pp. 1-9, 2017.

[41] D. R. Giosio, A. D. Henderson, J. M. Walker, and P. A. Brandner, "Physics Based Hydraulic Turbine Model for System Dynamics Studies," IEEE Transac-tions on Power Systems, 2016.

[42] M. Brezovec, I. Kuzle, and T. Tomisa, "Nonlinear digital simulation model of hydroelectric power unit with Kaplan turbine," IEEE Transactions on Energy Conversion, vol. 21, pp. 235-241, 2006.

[43] D. Kranjcic and G. Štumberger, "Differential evolution-based identification of the nonlinear kaplan turbine model," IEEE Transactions on Energy Conversion, vol. 29, pp. 178-187, 2014.

[44] N. Kishor, R. Saini, and S. Singh, "A review on hydropower plant models and control," Renewable and Sustainable Energy Reviews, vol. 11, pp. 776-796, 2007.

[45] N. Kishor and J. Fraile-Ardanuy, Modeling and dynamic behaviour of hydro-power plants: IET, 2017.

[46] G. A. Munoz-Hernandez, S. a. P. Mansoor, and D. I. Jones, Modelling and Con-trolling Hydropower Plants: Springer, 2013.

[47] C. Nicolet, B. Greiveldinger, J.-J. Hérou, B. Kawkabani, P. Allenbach, J.-J. Si-mond, et al., "High-order modeling of hydraulic power plant in islanded power network," IEEE Transactions on Power Systems, vol. 22, pp. 1870-1880, 2007.

[48] A. Padoan, B. Kawkabani, A. Schwery, C. Ramirez, C. Nicolet, J.-J. Simond, et al., "Dynamical behavior comparison between variable speed and synchronous machines with pss," IEEE Transactions on Power Systems, vol. 25, pp. 1555-1565, 2010.

[49] G. Zhao, "Study on the United Transient Process for Hydraulic, Mechanical and Power System (in Chinese)," Doctor, Department of Water Conservancy and Hydropower Engineering, Wuhan University, 2004.

[50] H. Bao, "Research on setting condition of surge chamber and operation control of the hydropower station (in Chinese)," Doctor, Water Resources and Hydro-power Engineering, Wuhan University, Wuhan, 2010.

134

[51] W. Li, L. Vanfretti, and Y. Chompoobutrgool, "Development and implementa-tion of hydro turbine and governor models in a free and open source software package," Simulation Modelling Practice and Theory, vol. 24, pp. 84-102, 2012.

[52] B. Svingen, "Application of LabVIEW for dynamic simulations of Hydraulic piping systems," in SIMS 2005 46th Conference on Simulation and Modeling, 2005.

[53] J. A. S. Sena, M. C. P. Fonseca, I. F. Di Paolo, W. Barra, J. A. L. Barreiros, C. T. Costa, et al., "An object-oriented framework applied to the study of electro-mechanical oscillations at Tucuruí hydroelectric power plant," Electric Power Systems Research, vol. 81, pp. 2081-2087, 2011.

[54] J. W. Tsotie and R. Wamkeue, "Advanced-model of synchronous generator for hydropower plants numerical simulations," Electric Power Systems Research, 2016.

[55] J. Liang, X. Yuan, Y. Yuan, Z. Chen, and Y. Li, "Nonlinear dynamic analysis and robust controller design for Francis hydraulic turbine regulating system with a straight-tube surge tank," Mechanical Systems and Signal Processing, vol. 85, pp. 927-946, 2017.

[56] S. Wei, Hydraulic turbine regulation (in Chinese) Wuhan: Huazhong University of Science and Technology Press, 2009.

[57] S. Mansoor, D. Jones, D. A. Bradley, F. Aris, and G. Jones, "Reproducing oscil-latory behaviour of a hydroelectric power station by computer simulation," Con-trol Engineering Practice, vol. 8, pp. 1261-1272, 2000.

[58] D. Kosterev, "Hydro turbine-governor model validation in pacific northwest," IEEE Transactions on Power Systems, vol. 19, pp. 1144-1149, 2004.

[59] M. E. Cebeci, U. Karaağaç, O. B. Tör, and A. Ertaş, "The effects of hydro power plants’ governor settings on the stability of Turkish power system frequency," presented at the ELECO Conf, 2007.

[60] H. N. V. Pico, D. C. Aliprantis, J. D. McCalley, N. Elia, and N. J. Castrillon, "Analysis of hydro-coupled power plants and design of robust control to damp oscillatory modes," IEEE Transactions on Power Systems, vol. 30, pp. 632-643, 2015.

[61] B. Strah, O. Kuljaca, and Z. Vukic, "Speed and active power control of hydro turbine unit," IEEE Transactions on Energy Conversion, vol. 20, pp. 424-434, 2005.

[62] J. Aguero, P. Arnera, M. Barbieri, M. Beroqui, R. B. Lastra, J. Mastronardi, et al., "Hydraulic transients in hydropower plant impact on power system dynamic stability," presented at the Power and Energy Society General Meeting-Conver-sion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE, 2008.

[63] J. Zhao, L. Wang, D. Liu, J. Wang, Y. Zhao, T. Liu, et al., "Dynamic Model of Kaplan Turbine Regulating System Suitable for Power System Analysis," Math-ematical Problems in Engineering, vol. 2015, 2015.

[64] J. I. Pérez-Díaz, J. I. Sarasúa, and J. R. Wilhelmi, "Contribution of a hydraulic short-circuit pumped-storage power plant to the load–frequency regulation of an isolated power system," International Journal of Electrical Power & Energy Systems, vol. 62, pp. 199-211, 2014.

[65] G. Martínez-Lucas, J. I. Sarasúa, J. Á. Sánchez-Fernández, and J. R. Wilhelmi, "Power-frequency control of hydropower plants with long penstocks in isolated systems with wind generation," Renewable Energy, vol. 83, pp. 245-255, 2015.

[66] G. Martínez-Lucas, J. I. Sarasúa, J. Á. Sánchez-Fernández, and J. R. Wilhelmi, "Frequency control support of a wind-solar isolated system by a hydropower plant with long tail-race tunnel," Renewable Energy, vol. 90, pp. 362-376, 2016.

135

[67] C. Landry, C. Nicolet, S. Giacomini, and F. Avellan, "Influence of the hydraulic system layout on the stability of a mixed islanded power network," in Advances in Hydroinformatics, ed: Springer, 2014, pp. 307-323.

[68] T. K. Nielsen, "The importance of including elastic property of penstock in the evaluation of stability of hydropowewr plants," in 6th IAHR International Meet-ing of the Workgroup on Cavitation and Dynamic Problems in Hydrauli Ma-chinery and Systems, Ljubljana, Slovenia, 2015.

[69] K. Vereide, B. Svingen, T. Nielsen, and L. Lia, "The Effect of Surge Tank Throt-tling on Governor Stability, Power Control, and Hydraulic Transients in Hydro-power Plants," IEEE Transactions on Energy Conversion, 2016.

[70] X. Yu, J. Zhang, C. Fan, and S. Chen, "Stability analysis of governor-turbine-hydraulic system by state space method and graph theory," Energy, vol. 114, pp. 613-622, 2016.

[71] X. Yuan, Z. Chen, Y. Yuan, and Y. Huang, "Design of fuzzy sliding mode con-troller for hydraulic turbine regulating system via input state feedback lineariza-tion method," Energy, vol. 93, pp. 173-187, 2015.

[72] H. Li, D. Chen, H. Zhang, C. Wu, and X. Wang, "Hamiltonian analysis of a hydro-energy generation system in the transient of sudden load increasing," Ap-plied Energy, vol. 185, pp. 244-253, 2017.

[73] B. Xu, D. Chen, H. Zhang, and R. Zhou, "Dynamic analysis and modeling of a novel fractional-order hydro-turbine-generator unit," Nonlinear Dynamics, vol. 81, pp. 1263-1274, 2015.

[74] H. Zhang, D. Chen, B. Xu, and F. Wang, "Nonlinear modeling and dynamic analysis of hydro-turbine governing system in the process of load rejection tran-sient," Energy Conversion and Management, vol. 90, pp. 128-137, 2015.

[75] B. Xu, F. Wang, D. Chen, and H. Zhang, "Hamiltonian modeling of multi-hydro-turbine governing systems with sharing common penstock and dynamic analyses under shock load," Energy Conversion and Management, vol. 108, pp. 478-487, 2016.

[76] L. Wang, Q. Han, D. Chen, C. Wu, and X. Wang, "Nonlinear modeling and sta-bility analysis of the pump-turbine governing system at pump mode," IET Re-newable Power Generation, 2017.

[77] W. Guo, J. Yang, M. Wang, and X. Lai, "Nonlinear modeling and stability anal-ysis of hydro-turbine governing system with sloping ceiling tailrace tunnel under load disturbance," Energy Conversion and Management, vol. 106, pp. 127-138, 2015.

[78] W. Guo, J. Yang, J. Chen, and M. Wang, "Nonlinear modeling and dynamic control of hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel," Nonlinear Dynamics, vol. 84, pp. 1383-1397, 2016.

[79] B. Wang, W. Guo, and J. Yang, "Analytical solutions for determining extreme water levels in surge tank of hydropower station under combined operating con-ditions," Communications in Nonlinear Science and Numerical Simulation, vol. 47, pp. 394-406, 2017.

[80] W. Guo, J. Yang, and Y. Teng, "Surge wave characteristics for hydropower sta-tion with upstream series double surge tanks in load rejection transient," Renew-able Energy, vol. 108, pp. 488-501, 2017.

[81] O. Akhrif, F. A. Okou, L. A. Dessaint, and R. Champagne, "Application of a multivariable feedback linearization scheme for rotor angle stability and voltage regulation of power systems," IEEE Transactions on Power Systems, vol. 14, pp. 620-628, 1999.

136

[82] D. M. Dobrijevic and M. V. Jankovic, "An approach to the damping of local modes of oscillations resulting from large hydraulic transients," IEEE Transac-tions on Energy Conversion, vol. 14, pp. 754-759, 1999.

[83] Q. Lu, Y. Sun, Y. Sun, F. F. Wu, Y. Ni, A. Yokoyama, et al., "Nonlinear decen-tralized robust governor control for hydroturbine-generator sets in multi-ma-chine power systems," International Journal of Electrical Power & Energy Sys-tems, vol. 26, pp. 333-339, 2004.

[84] M. J. Jin, W. Hu, F. Liu, S. W. Mei, and Q. Lu, "Nonlinear co-ordinated control of excitation and governor for hydraulic power plants," IEE Proceedings-Gen-eration, Transmission and Distribution, vol. 152, pp. 544-548, 2005.

[85] S. Mei, X. Gui, C. Shen, and Q. Lu, "Dynamic extending nonlinear H∞ control and its application to hydraulic turbine governor," Science in China Series E: Technological Sciences, vol. 50, pp. 618-635, 2007.

[86] M. Weixelbraun, H. Renner, O. Kirkeluten, and S. Lovlund, "Damping low fre-quency oscillations with hydro governors," in PowerTech (POWERTECH), 2013 IEEE Grenoble, 2013, pp. 1-6.

[87] C. Gencoglu, O. B. Tor, E. Cebeci, O. Yilmaz, and A. N. Guven, "Assessment of the effect of hydroelectric power plants' governor settings on low frequency inter area oscillations," in Power System Technology (POWERCON), 2010 In-ternational Conference on, 2010, pp. 1-8.

[88] P. C. O. Silva, S. Alligné, P. Allenbach, C. Nicolet, and B. Kawkabani, "A fully modular tool for small-signal stability analysis of hydroelectric systems," in Electrical Machines (ICEM), 2014 International Conference on, 2014, pp. 1697-1703.

[89] P. C. O. Silva, C. Nicolet, P. Grillot, J. L. Drommi, and B. Kawkabani, "Assess-ment of Power Swings in Hydropower Plants through High-Order Modelling and Eigenanalysis," IEEE Transactions on Industry Applications, vol. PP, pp. 1-1, 2017.

[90] X. Liu and C. Liu, "Eigenanalysis of oscillatory instability of a hydropower plant including water conduit dynamics," IEEE Transactions on Power Systems, vol. 22, pp. 675-681, 2007.

[91] H. Gao, X. Xie, J. Zhang, C. Wu, and K. Sun, "Second‐order oscillation mode study of hydropower system based on linear elastic model and modal series method," International Transactions on Electrical Energy Systems, 2016.

[92] U. Dorji and R. Ghomashchi, "Hydro turbine failure mechanisms: An overview," Engineering Failure Analysis, vol. 44, pp. 136-147, Sep 2014.

[93] X. Liu, Y. Luo, and Z. Wang, "A review on fatigue damage mechanism in hydro turbines," Renewable and Sustainable Energy Reviews, vol. 54, pp. 1-14, 2// 2016.

[94] H.-J. Huth, "Fatigue design of hydraulic turbine runners," Doctor, Norwegian University of Science and Technology, 2005.

[95] M. Gagnon, S. Tahan, P. Bocher, and D. Thibault, "Impact of startup scheme on Francis runner life expectancy," in IOP Conference Series: Earth and Environ-mental Science, 2010, p. 012107.

[96] C. Trivedi, B. Gandhi, and C. J. Michel, "Effect of transients on Francis turbine runner life: a review," Journal of Hydraulic Research, vol. 51, pp. 121-132, 2013.

[97] E. Wurm, "Consequences of primary control to the residual service life of Kaplan runners," presented at the Russia Power 2013 & Hydrovision Russia 2013, Moscow, Russia, 2013.

137

[98] U. Seidel, C. Mende, B. Hübner, W. Weber, and A. Otto, "Dynamic loads in Francis runners and their impact on fatigue life," in IOP Conference Series: Earth and Environmental Science, 2014, p. 032054.

[99] J. Ukonsaari, "Wear and friction of synthetic esters in a boundary lubricated journal bearing," Tribology international, vol. 36, pp. 821-826, 2003.

[100] R. Gawarkiewicz and M. Wasilczuk, "Wear measurements of self-lubricating bearing materials in small oscillatory movement," Wear, vol. 263, pp. 458-462, 2007.

[101] G. F. Simmons, "Journal bearing design, lubrication and operation for enhanced performance," Luleå University of Technology, 2013.

[102] G. Liu and S. Daley, "Optimal-tuning nonlinear PID control of hydraulic sys-tems," Control Engineering Practice, vol. 8, pp. 1045-1053, 2000.

[103] E. Çam, "Application of fuzzy logic for load frequency control of hydroelectri-cal power plants," Energy conversion and management, vol. 48, pp. 1281-1288, 2007.

[104] S. Doolla and T. Bhatti, "Load frequency control of an isolated small-hydro power plant with reduced dump load," IEEE Transactions on Power Systems, vol. 21, pp. 1912-1919, 2006.

[105] "Hydro Life Extension ModernizationGuides - Volume 2: Hydromechanical Equipment," EPRI2000.

[106] X. Liu, Y. Luo, B. W. Karney, and W. Wang, "A selected literature review of efficiency improvements in hydraulic turbines," Renewable and Sustainable En-ergy Reviews, vol. 51, pp. 18-28, 2015.

[107] S. Soares and C. T. Salmazo, "Minimum loss predispatch model for hydroelec-tric power systems," IEEE Transactions on Power Systems, vol. 12, pp. 1220-1228, 1997.

[108] A. Arce, T. Ohishi, and S. Soares, "Optimal dispatch of generating units of the Itaipú hydroelectric plant," IEEE Transactions on power systems, vol. 17, pp. 154-158, 2002.

[109] S. Soares, T. Ohishi, M. Cicogna, and A. Arce, "Dynamic dispatch of hydro generating units," in Power Tech Conference Proceedings, 2003 IEEE Bologna, 2003.

[110] C. Ma, H. Wang, and J. Lian, "Short-term electricity dispatch optimization of Ertan hydropower plant based on data by field tests," Journal of Renewable and Sustainable Energy, vol. 3, p. 063109, 2011.

[111] E. C. Bortoni, G. S. Bastos, T. M. Abreu, and B. Kawkabani, "Online optimal power distribution between units of a hydro power plant," Renewable Energy, vol. 75, pp. 30-36, 2015.

[112] T. K. Siu, G. A. Nash, and Z. K. Shawwash, "A practical hydro, dynamic unit commitment and loading model," IEEE transactions on power systems, vol. 16, pp. 301-306, 2001.

[113] T. Dal’Santo and A. S. Costa, "Hydroelectric unit commitment for power plants composed of distinct groups of generating units," Electric Power Systems Re-search, vol. 137, pp. 16-25, 2016.

[114] B. G. Mulu, P. P. Jonsson, and M. J. Cervantes, "Experimental investigation of a Kaplan draft tube–Part I: Best efficiency point," Applied energy, vol. 93, pp. 695-706, 2012.

[115] P. P. Jonsson, B. G. Mulu, and M. J. Cervantes, "Experimental investigation of a Kaplan draft tube–Part II: Off-design conditions," Applied energy, vol. 94, pp. 71-83, 2012.

138

[116] P. Schniter and L. Wozniak, "Efficiency based optimal control of Kaplan hydro-generators," IEEE Transactions on Energy Conversion, vol. 10, pp. 348-353, 1995.

[117] K. T. Atta, A. Johansson, M. J. Cervantes, and T. Gustafsson, "Phasor extremum seeking and its application in kaplan turbine control," in 2014 IEEE Conference on Control Applications (CCA), Part of 2014 IEEE Multi-conference on Systems and Control, Antibes, France, 2014, pp. 298-303.

[118] "Best Practice Catalog Propeller-Kaplan Turbine," Oak Ridge National Labora-tory2011.

[119] O. Nilsson and D. Sjelvgren, "Variable splitting applied to modelling of start-up costs in short term hydro generation scheduling," IEEE Transactions on Power Systems, vol. 12, pp. 770-775, 1997.

[120] O. Nilsson and D. Sjelvgren, "Hydro unit start-up costs and their impact on the short term scheduling strategies of Swedish power producers," IEEE Transac-tions on Power Systems, vol. 12, pp. 38-44, 1997.

[121] B. H. Bakken and T. Bjorkvoll, "Hydropower unit start-up costs," in Power En-gineering Society Summer Meeting, 2002 IEEE, 2002, pp. 1522-1527.

[122] N. G. Bean, M. M. O'Reilly, and J. E. Sargison, "A stochastic fluid flow model of the operation and maintenance of power generation systems," Power Systems, IEEE Transactions on, vol. 25, pp. 1361-1374, 2010.

[123] G. A. Aggidis, E. Luchinskaya, R. Rothschild, and D. Howard, "The costs of small-scale hydro power production: Impact on the development of existing po-tential," Renewable Energy, vol. 35, pp. 2632-2638, 2010.

[124] I. Guisández, J. I. Pérez-Díaz, and J. R. Wilhelmi, "Assessment of the economic impact of environmental constraints on annual hydropower plant operation," En-ergy Policy, vol. 61, pp. 1332-1343, 2013.

[125] S.-R. Huang, P.-L. Chang, Y.-W. Hwang, and Y.-H. Ma, "Evaluating the productivity and financial feasibility of a vertical-axis micro-hydro energy gen-eration project using operation simulations," Renewable Energy, vol. 66, pp. 241-250, 2014.

[126] M. Ranlöf, "Electromagnetic analysis of hydroelectric generators," Doctor, Uppsala University, 2011.

[127] J. Bladh, "Hydropower generator and power system interaction," Doctor, Upp-sala University, 2012.

[128] M. Wallin, "Measurement and modelling of unbalanced magnetic pull in hydro-power generators," Uppsala University, 2013.

[129] L. Saarinen, "The Frequency of the Frequency: On Hydropower and Grid Fre-quency Control," Doctor, Uppsala University, 2017.

[130] W. E. Vander Velde, Multiple-input describing functions and nonlinear system design: McGraw-Hill, New York, 1968.

[131] F. Golnaraghi and B. C. Kuo, Automatic control systems, 2010. [132] H. Bao, J. Yang, and L. Fu, "Study on Nonlinear Dynamical Model and Control

Strategy of Transient Process in Hydropower Station with Francis Turbine," in Power and Energy Engineering Conference, 2009. APPEEC 2009. Asia-Pacific, 2009, pp. 1-6.

[133] V. L. Streeter and E. B. Wylie, Fluid transients: McGraw-Hill New York, 1978. [134] L. Saarinen, P. Norrlund, and U. Lundin, "Tuning primary frequency controllers

using robust control theory in a power system dominated by hydropower," in CIGRE Session 2016, 2016.

[135] L. Ljung and T. Glad, Control theory-multivariable and nonlinear methods: Taylor and Francis, 2000.

139

[136] IEEE Guide for the Application of Turbine Governing Systems for Hydroelec-tric Generating Units, 2011.

[137] W. D. Ghrist III, "Floating deadband for speed feedback in turbine load control," US 4593364 A, 1986.

[138] T. Hägglund, "A unified discussion on signal filtering in PID control," Control Engineering Practice, vol. 21, pp. 994-1006, 2013.

[139] L. Saarinen, P. Norrlund, and U. Lundin, "Tuning primary frequency controllers using robust control theory in a power system dominated by hydropower," pre-sented at the CIGRE Session 2016, Paris, France, 2016.

[140] V. Kercan, V. Djelic, T. Rus, and V. Vujanic, "Experience with Kaplan turbine efficiency measurements–Current meters and/or index test flow measurement," Proc. IGHEM, Montreal, Canada, 1996.

[141] A. Adamkowski, M. Lewandowski, and S. Lewandowski, "Selected Experi-ences with Optimization Tests of the Kaplan-Type Hydraulic Turbines," Journal of Energy and Power Engineering, vol. 8, 2014.

[142] "China Electricity Council: DL/T 1040-2007 -The grid operation code (in Chi-nese)," ed, 2007.

[143] European Network of Transmission System Operators for Electricity, ENTSO-E Operation Handbook, Policy 1 (2009): Load-Frequency Control and Perfor-mance, 2009.

[144] Statnett, Funksjonskrav i kraftsystemet 2012 (in Norwegian), 2012. [145] Svenska Kraftnät, Regler för upphandling och rapportering av primärreglering

FCR-N och FCR-D (från 16 november 2012, in Swedish), 2012. [146] C. Jaeger, Fluid transients in hydro-electric engineering practice: Blackie,

1977. [147] G. I. Krivchenko, "Admissibility of deviating from the Thoma criterion when

designating the cross-sectional area of surge tanks," Hydrotechnical Construc-tion, vol. 22, pp. 403-409, 1988.

[148] F. Liang, Y. Jiandong, B. Haiyan, and L. Jinping, "Effect of Turbine Character-istic on the Response of Hydroturbine Governing System with Surge Tank," in Power and Energy Engineering Conference, 2009. APPEEC 2009. Asia-Pacific, 2009, pp. 1-6.

[149] P. M. Anderson and A. A. Fouad, Power System Control and Stability: John Wiley & Sons, 2003.

[150] J. A. Jones, R. A. Palylyk, P. Willis, and R. A. Weber, "Greaseless Bushings for Hydropower Applications: Program, Testing, and Results," US Army Corps of Engineers, Engineer Research and Development Center1999. (http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA371845).

[151] M. Grøtterud, "Analysis of the Slow Floating in Grid Frequency of the Nordic Power System: Impact of Hydraulic System Characteristics," Master, Depart-ment of Electric Power Engineering, Norwegian University of Science and Technology, 2012.

[152] Z. Xu, J. Ostergaard, M. Togeby, and F. R. Isleifsson, "Evaluating frequency quality of Nordic system using PMU data," in Power and Energy Society Gen-eral Meeting-Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE, 2008, pp. 1-5.

[153] Y. Chen, Z. Xu, and J. Østergaard, "Frequency analysis for planned islanding operation in the Danish distribution system-Bornholm," in Universities Power Engineering Conference, 2008. UPEC 2008. 43rd International, 2008, pp. 1-5.

[154] "PJM Manual 15: Cost Development Guidelines," Cost Development Subcom-mittee of PJM2016.

140

[155] J. Lidenholm and U. Lundin, "Estimation of hydropower generator parameters through field simulations of standard tests," IEEE Transactions on Energy Con-version, vol. 25, pp. 931-939, 2010.

Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1494

Editor: The Dean of the Faculty of Science and Technology

A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally throughthe series Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)

Distribution: publications.uu.seurn:nbn:se:uu:diva-318470

ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2017