optimized tuning of parameters for hvdc dynamic...

UPTEC F12038

Examensarbete 30 hpJanuari 2013

Optimized Tuning of Parameters for HVDC Dynamic Performance Studies

Axel Andersson

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Optimized Tuning of Parameters for HVDC DynamicPerformance Studies

Axel Andersson

HVDC (High Voltage Direct Current) is used all over the world for transmission ofelectric power due to lower losses compared to traditional HVAC (High VoltageAlternating Current). However, the procedure of converting AC into DC puts greatdemand on the control system of the converter stations. These control systems needto be tuned properly to give the HVDC system the correct dynamics to handlevariations in the network load and other disturbances.

In this thesis, it was investigated if optimization algorithms can be used for tuning ofthe control parameters. Focus was on three parts of the control system, the CurrentControl Amplifier, Voltage Dependent Current Order Limiter and the Rectifier AlphaMinimum Limiter.

The Nelder & Mead Simplex method was used and several different objectivefunctions were tested, including combinations of integral square error, integralabsolute error, rise time and overshoot. Several different fault cases and scenarioswere tested and results of the optimization were compared to the manually tunedcontrol system.

It was found that the results of the optimization were comparable with the manuallytuned parameters for many of the cases tested. The biggest issue encountered wasthat the optimization algorithm often finds a local minimum in the objective function,leading to a suboptimal solution. This issue could be solved by running theoptimization several times, using different initial values.

It is concluded that using optimization algorithms could be a useful tool for tuning ofthe HVDC control system.

ISSN: 1401-5757, UPTEC F12038Examinator: Tomas NybergÄmnesgranskare: Alexander MedvedevHandledare: Hector Avila, Prerna Bihani

Sammanfattning

HVDC (hogspand likstrom) anvands idag som alternativ till den hogspandavaxelstrommen for att transportera elektrisk energi. Fordelen med attanvanda likstrom istallet for vaxelstrom ar framst att forlusterna blir lagre.Dessvarre kraver HVDC-transmissioner stora och dyra stationer som om-vandlar vaxelstrommen till likstrom innan den transporteras. En annannackdel ar att denna omvandling inte ar helt simpel, tekniskt sett. Detkravs komplicerade reglersystem for att sakerstalla att man levererar rattspanning, strom och effekt ut pa natet.

Innan en HVDC-anlaggning byggs i verkligheten byggs en datormodellav den. Denna modell anvands for diverse tester och simuleringar vars syftear att sakerstalla systemets funktionalitet. En del av dessa tester kallasdynamic performance studies, DPS. I en DPS testas framst hur systemetbeter sig vid vissa felfall och storningar som skulle kunna intraffa vid drift.De olika parametrarna i styrsystemet stalls in for att sakerstalla att sys-temet aterhamtar sig tillrackligt snabbt vid dessa fel. Parametrarna stallsin manuellt enligt ”trial and error”-princip. Antal fall som testas, samt antalparametrar som maste stallas in, gor att DPS:en kan ta valdigt lang tid.

Ett alternativt satt att stalla in dessa parametrar ar att anvanda op-timeringsmetoder. Da stalls parametrarna in automatiskt genom att endator beraknar fram vilka parametrar som ar bast. Detta gors genom attoptimeringsmetoden minimerar en funktion, som kallas malfunktion. Vilkaparametrar som datorn kommer fram till beror pa val av optimeringsmetod,samt hur man definierar malfunktionen.

I detta arbete testas nagra kanda malfunktioner, samt nagra egna ideerpa malfunktioner, for att optimera tre delar av HVDC-reglersystemet. Op-timeringsmetoden som anvandes var Nelder & Mead Simplex-metod ochprogramvaran som anvandes var PSCAD/EMTDC.

Det visades att for tva av de tre delarna av reglersystemet, fann op-timeringsmetoden losningar som var jamforbara med de manuellt funnalosningarna. Slutsatsen blir saledes att optimeringsmetoder kan vara ettbra hjalpmedel vid HVDC-systemstudier.

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Acknowledgements

I would like to thank Hector Avila and Prerna Bihani at ABB HVDC fortheir support and feedback during the course of writing this thesis.

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Contents

1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Purpose and goal . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 22.1 The HVDC system . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 AC conversion . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Control system . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Current Control Amplifier . . . . . . . . . . . . . . . . 52.1.4 Voltage Dependent Current Order Limiter . . . . . . . 62.1.5 Rectifier Alpha Minimum Limiter . . . . . . . . . . . 6

2.2 Dynamic Performance Studies . . . . . . . . . . . . . . . . . . 72.2.1 CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 VDCOL . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 RAML . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Nelder-Mead Simplex Algorithm . . . . . . . . . . . . . . . . 92.3.1 Ordering of vertices . . . . . . . . . . . . . . . . . . . 102.3.2 Calculation of centroid . . . . . . . . . . . . . . . . . . 102.3.3 Simplex transformation . . . . . . . . . . . . . . . . . 112.3.4 Termination test . . . . . . . . . . . . . . . . . . . . . 13

3 Method 133.1 Finding an objective function . . . . . . . . . . . . . . . . . . 13

3.1.1 CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 VDCOL . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 RAML . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Testing the objective function . . . . . . . . . . . . . . . . . . 19

4 Simulation setup 194.1 PSCAD’s Optimum Run . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Initial step size . . . . . . . . . . . . . . . . . . . . . . 204.1.2 Normalization of parameters . . . . . . . . . . . . . . 204.1.3 Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 HVDC test system . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Results and discussion 215.1 CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1.1 Initial objective functions . . . . . . . . . . . . . . . . 215.1.2 Objective function modification . . . . . . . . . . . . . 235.1.3 Testing the objective function . . . . . . . . . . . . . . 25

5.2 VDCOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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5.2.1 Initial objective functions . . . . . . . . . . . . . . . . 265.2.2 Modification of the objective function . . . . . . . . . 285.2.3 Testing the objective function . . . . . . . . . . . . . . 32

5.3 RAML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.1 Initial objective function . . . . . . . . . . . . . . . . . 345.3.2 Objective function modification . . . . . . . . . . . . . 365.3.3 Testing the objective function . . . . . . . . . . . . . . 40

6 Conclusions 42

References 44

Appendix 45

A VDCOL Comparisons 45

B RAML Comparisons 63

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1 Introduction

1.1 Introduction

Today, using HVDC (High Voltage Direct Current) is the most efficient wayof transporting large quantities of electric power over long distances. Tworeasons for this are that direct current does not suffer from reactive lossesand that transmission line costs are lower. HVDC systems are also used forconnecting asynchronous AC networks and upholding stability in grids.

The downside to using HVDC are the large converter stations needed toconvert the AC into DC. These stations contain a large number of compo-nents and are very costly. The procedure of converting AC into DC also putsgreat demand on the control system of the stations. These control systemsneed to be tuned properly to give the HVDC system the correct dynamicsto handle variations in the network load and disturbances.

Before implementing the real HVDC system, a software model of thesystem is built. On this model, extensive tests are carried out to ensurethe proper performance and robustness of the system during transient con-ditions. These tests are called dynamic performance studies (DPS) and arean essential part of the development of the HVDC system.

In the DPS, several different cases and configurations are tested to ensurethe system complies to specifications. These tests range from minor voltagedrops to cable breaks. Several variables need to be taken into account whenrating these tests, including recovery time, phase margin and overshoot. Theparameters of the control system are tuned until the system characteristicssatisfy the specification.

The parameters of the control system have up to this time been tunedby hand, using trial and error. The amount of tests and parameters thatneed to be taken into account has made the DPS a very time consumingprocess.

1.2 Purpose and goal

The purpose of this thesis is to find a method to automatize the dynamicperformance studies using optimization algorithms in order to save timeand resources. Using optimization algorithms could also help find solutionswith better performance and robustness than the manually found solutions.The solution the algorithm finds optimal depends on the function which itminimizes. This function is called the objective function.

The goal of this thesis is to find and implement an optimization algo-rithm and an objective function, so that the algorithm is quicker and givesbetter results than manual trial and error. The algorithm and objectivefunction will be implemented in PSCAD/EMTDC, a software for powersystem simulations.

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1.3 Scope of the thesis

The HVDC control system contain a huge number of different controllers andfunctions with corresponding parameters to be tuned. This thesis focuseson the tuning of three core components of the HVDC control system, theCurrent Control Amplifier (CCA), Voltage Dependent Current Order Lim-iter (VDCOL) and Rectifier Alpha Minimum Limiter (RAML), all essentialfor the performance and stability of the system.

2 Background

The goal of this chapter is to explain the necessary concepts needed tounderstand the problems, methods and results of this thesis.

2.1 The HVDC system

This section describes the basic functionality of the HVDC system. The sys-tem described, and used for this thesis, is the conventional line-commutatedcurrent-source converter type.

2.1.1 AC conversion

The principle used to convert AC into DC in an HVDC system is the sameprinciple used in electronics. In electronics, diodes are used for rectifyingthe AC voltage. In HVDC systems, thyristors are used. Thyristors are es-sentially diodes, which conduct in the forward direction but block in thebackward direction. However, they have one important feature diodes lack.Thyristors have an input that controls when the thyristor conducts. How-ever, when the thyristor has switched on and is conducting, it is not possibleto switch it off, it will conduct until the voltage across it crosses zero.

Figure 2.1: Three phase rectifier bridge

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Figure 2.1 shows a three phase rectifier bridge using thyristors. It issimilar to the classic rectifier bridge used in electronics where diodes areused.

Assume the thyristors conduct at all times in the positive region, effec-tively making them behave like diodes. The output will be a direct voltageas shown in figure 2.2.

Figure 2.2: Voltage output from rectifier bridge

Now assume that a delay is introduced, so that the thyristors conduct afraction later than they do in the diode case, see figure 2.3.

Figure 2.3: Voltage output from rectifier bridge using firing angle α

This delay is called α or the firing angle. It can be seen that the DCvoltage is reduced compared to the case where α = 0.

It can be shown [1] that the DC-voltage is given by

Udc =3√2

πUac cos(α) (2.1)

where Uac is the phase-to-phase RMS voltage. It can be seen that the DCvoltage can be controlled by α which can be varied between 0 and 180

degrees. This corresponds to a change in the DC voltage from 3√2

π Uac to

−3√2

π Uac

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Figure 2.4: Voltage output from rectifier bridge using firing angle α, showingthe overlap angle µ

In reality, the thyristors are not ideal. There will be some overlap be-tween the already conducting thyristor and the triggered thyristor, leadingto the case in figure 2.4. This is called the overlap angle µ.

The rest of the period is called the extinction angle or commutationmargin, γ. This leads to the following well known HVDC expression:

α+ µ+ γ = 180 (2.2)

2.1.2 Control system

The basic HVDC system consists of two connected converter stations, calledthe rectifier and the inverter.

Figure 2.5: Basic HVDC system

Figure 2.5 shows a basic HVDC system consisting of a rectifier and aninverter connecting two AC networks. The DC power at the rectifier PdcR

is given byPdcR = UdcRIdc (2.3)

The DC current Idc is given by the voltage drop across the DC line dividedby the resistance R which inserted into equation 2.3 gives

PdcR = UdcRUdcR − UdcI

R(2.4)

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where UdcR and UdcI are the DC-voltages at the rectifier and the inverterrespectively. The power at the inverter is calculated in the same manner. Itis clear that the power transmitted depends on the voltage in the rectifierand the inverter, which is controlled by varying α.

The requested DC power is compared to the actual DC voltage Udc andthe required current order Iorder is calculated. This current order is sent tothe current control amplifier (CCA) which can be seen in figure 2.6.

Figure 2.6: Basic HVDC control system

The CCA is a PI controller designed for a stable and responsive currentcontrol. The CCA calculates the necessary firing angle αord in order to keepthe DC current Idc at the requested level. The firing angle αord is sent tothe firing control (FC) and control pulse generator (CPG) which translateαord into firing pulses that are sent to the thyristors.

2.1.3 Current Control Amplifier

The Current Control Amplifier (CCA) is a slightly modified PI controller.Its main objective is to keep the current at a desired level. It is tuned tomatch the dynamics of the system to give a fast yet stable response. Thetransfer function can be written as [2]

GCCA(s) = G1 +KpTis

Tis. (2.5)

G is called the gain, Kp is called the proportional factor and Ti is the timeconstant. These are design variables and are tuned to give the system theproper characteristics.

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2.1.4 Voltage Dependent Current Order Limiter

The Voltage Dependent Current Order Limiter (VDCOL) is a protectivefunction located before the CCA. Its objective is to limit the current orderwhen the DC voltage decreases. This is to avoid instability during ACdisturbances in the inverter network. It also provides safe restarts afterfault clearances. The characteristics for the VDCOL can be seen in figure2.7.

Figure 2.7: VDCOL function

IO ABS MIN and IO ABS MAX set the global minimum and maximumfor the current order respectively. UD LOW , UD HIGH and IO LIM de-cides the location and steepness of the slope.

2.1.5 Rectifier Alpha Minimum Limiter

If the rectifier AC voltage decreases, the firing angle will decrease in orderto make up for this loss. If the voltage suddenly goes back to normal, itcan cause spikes in the DC current. The Rectifier Alpha Minimum Limiter(RAML) function is used to prevent these spikes by detecting disturbancesand increasing the minimum allowed firing angle αmin.

The RAML has two different functions for handling three phase and sin-gle phase faults. Three phase faults are detected via the RAML REF param-eter and single phase faults are detected via the CRAML REFparameter. Ifa fault is detected, αmin is increased to an angle specified via the DL LEVELand CDL LEVEL parameters. When the fault is cleared, αmin will slowlydecrease to its original value. The rate at which it decreases is controlledby the RAML DECR parameter.

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2.2 Dynamic Performance Studies

The Dynamic Performance Studies (DPS) are carried out to ensure that theHVDC system meets the system specifications with regard to performanceand stability during transient conditions. The parameters of the differentfunctions of the control system are tuned until these specifications are met.This section explains how the DPS are carried out for the CCA, VDCOLand RAML.

2.2.1 CCA

The CCA is tuned by step responses. The parameters tuned are G and Kp.The time constant Ti is normally not changed [3].

A value ∆I is added or subtracted to the current order of the rectifier.The value of ∆I is usually in the magnitude of 0.1 p.u. This change inthe current order causes a step in the direct current. When the currenthas settled, the step ∆I is removed and the current returns to its normaloperating point. Figure 2.8 shows a typical current step used when tuningthe CCA.

Figure 2.8: Current step for tuning of the CCA

The performance of the CCA is determined by the rise-time and over-shoot. Rise-time is the time it takes for the current to reach 90 % of thereference step. The rise-time of both the positive and negative steps need tobe taken into account. To differentiate between the cases, the rise-time andovershoot of the negative steps will be denoted as fall-time and undershoot,respectively. The desire is to minimize rise-time, fall-time, overshoot andundershoot.

Stability of the CCA is ensured by inspection of the Nyquist curve. TheNyquist curve is drawn by performing a frequency sweep of the system. Themain factor taken into account is the phase margin.

The frequency sweep needed to draw the Nyquist curve usually takes alot of time. To save time, experts use a rule of thumb to make sure thephase margin is acceptable. This rule sets maximum limits on G · Kp andKp and these limits are then used throughout the tuning of the CCA.

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2.2.2 VDCOL

The VDCOL is tuned by performing voltage drops in the inverter AC net-work. The drops can vary in magnitude and time. Both single phase andthree phase faults are tested. Figure 2.9 shows a typical case for tuning ofthe VDCOL. Shown is the recovery of the DC power after a three phasefault in the inverter AC network. The fault occurs at 0.1 s and is cleared at0.2 s.

Figure 2.9: DC power recovery after a three phase fault in the inverter ACnetwork

To measure the performance of the VDCOL, the recovery time of the DCpower is monitored. The recovery time is calculated from when the voltagedrop is cleared, until the DC power has reached 90 % of its pre-fault value.The maximum recovery time is stated in the system specification.

Stability is of outmost importance when tuning the VDCOL. To ensurestability, several variables are monitored. The recovery of the DC powershould be controlled and not have severe overshoot or dip after recovery, seefigure 2.9.

Figure 2.10: DC voltage recovery after a single phase fault in the inverterAC network

Furthermore, the DC voltage should not spike during recovery, althoughsome overshoot is generally acceptable (usually about 10 % above pre-faultvalue [4]). Figure 2.10 shows a DC voltage recovery after a single phase faultwhich would be regarded as acceptable.

The extinction angle γ of the inverter is monitored closely when tuningthe VDCOL. It should not drop too low during recovery. Usually a fewdegrees below its stationary value is acceptable [5]

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If γ drops too low, commutation failures could occur. This is because ofthe physical properties of the thyristors. When commutation failures occur,the ability to control the firing of the thyristors is lost for a short period oftime.

Figure 2.11: Extinction angle during recovery after single phase fault in theinverter AC network

Figure 2.11 shows the extinction angle after a single phase AC fault.After the fault is cleared at 0.2 s, γ decreases in a slow and stable mannerwhich is what is aimed for.

For tuning of the VDCOL, the parameters UD HIGH, UD LOW,TC UP REC and TC UP INV are used. UD HIGH and UD LOW are ex-plained in section 2.1.4. The TC UP REC and TC UP INV parameters arepart of a low pass filter acting on Ud prior to the VDCOL function for therectifier and inverter, respectively.

2.2.3 RAML

The RAML function is tuned by applying voltage drops in the rectifier ACnetwork. Faults of different magnitudes and durations are tested. Bothsingle phase and three phase faults are tested.

The performance and stability is measured in the same way as for theVDCOL, with the exception of the extinction angle, which is not monitoredfor the RAML.

The parameters tuned in the RAML function are RAML DECR,CRAML REF, RAML REF, CDL LEVEL and DL LEVEL.

2.3 Nelder-Mead Simplex Algorithm

The Nelder-Mead Simplex Algorithm is an algorithm first published in 1965by J. A. Nelder and R. Mead [6]. The goal of the algorithm is to mini-mize a function of n variables, usually called the objective function (OF).It accomplishes this by forming a simplex which iteratively changes shapeand location in order to locate the minimum of the OF. It should not beconfused with the Simplex Algorithm of Dantzig, an algorithm for linearprogramming.

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An n-simplex is defined as an n-dimensional polytope, which is the con-vex hull of n + 1 vertices. For example, a simplex in 1 dimension is a linesegment, a simplex in 2 dimensions is a triangle and so on. For each iterationthe algorithm replaces the vertex with the highest OF value with a vertexof lower OF value. This is performed until a minimum of the OF is found.

The algorithm starts with a user defined simplex of any size. Eachiteration of the algorithm include the following steps:

• Ordering of vertices

• Calculation of centroid

• Simplex transformation

• Termination test

The algorithm will continue this loop until the termination criteria havebeen fulfilled

2.3.1 Ordering of vertices

In this step, the algorithm orders the vertices according to the objectivefunction value at these points so that OF (x1) ≥ OF (x2) ≥ · · · ≥ OF (xn+1).

2.3.2 Calculation of centroid

The centroid on the opposite side of the vertex with the worst OF value iscalculated. The centroid is calculated as

c =1

n

n+1∑i=2

xi

Figure 2.12 shows a simplex in two dimensions with centroid c. In thefigure, OF (x1) ≥ OF (x2) ≥ OF (x3)

Figure 2.12: Simplex of two dimensions, centroid c

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2.3.3 Simplex transformation

This step contain different operations, depending on the OF value at thespecific points. It starts with the reflection operation.

Reflection The reflection point xr and corresponding OF value is calcu-lated. The reflection point can be expressed as

xr = c+ α(c− x1)

where α is a design constant. In most implementations of the algorithm,α = 1.

Upon reflection, there exist three outcomes that lead to different actions:

• OF (xr) > OF (x2): Here, the reflection point is worse than the secondworst vertex. If this is the case, contraction is performed.

• OF (x2) ≥ OF (xr) ≥ OF (xn+1): Here, the reflection point is betterthan, or equal to, the second worst vertex, but not better than the bestvertex. If this happens, x1 is replaced by xr and the transformationis complete.

• OF (xn+1) > OF (xr): Here, the reflection point is better than thebest vertex, i e. a new objective function minimum is found. If thishappens, expansion is performed.

Figure 2.13: Simplex using reflection point xr, dashed line showing the orig-inal simplex

Expansion The expansion point is expressed as

xe = c+ γ(xr − c)

where γ is a design constant defined by the user. In most implementations,γ = 2.

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Figure 2.14: Simplex showing expansion point xe, dashed line showing theoriginal and reflected simplex

Figure 2.14 shows the simplex after expansion. If OF (xn+1) > OF (xe),i.e. the expansion point is better than the current best point, x1 is replacedwith xe and the transformation is complete. Otherwise, x1 is replaced byxr and the transformation is complete.

Contraction Contraction is performed using the better of the two pointsx1 and xr. The contraction point is defined as (assuming xr is the betterpoint)

xc = c+ β(xr − c)

where β is a constant defined by the user. In most implementations, β = 12

(a) (b)

Figure 2.15: Simplex, performing the contraction operation using (a) xr and(b) x1 with the dashed line showing the original simplex

If OF (xc) is better than the current worst vertex, it replaces it andthe transformation is complete. Otherwise, the reduction operation is per-formed.

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Reduction During the reduction operation, n new vertices are calculatedas

xi = xn+1 + δ(xi − xn+1)

for i = 1 . . . n. This simplex is then accepted and transformation is com-pleted. Figure 2.16 shows the reduction operation.

Figure 2.16: Simplex showing reduction operation around best point, x3,dashed line shows original simplex

2.3.4 Termination test

When a new simplex has been formed, some termination criterion is testedfor stopping the algorithm. Without it, the algorithm would continue untilit is stopped manually. Several different termination criteria exist [7]. Forexample, the algorithm could terminate when the simplex has shrunk to acertain size, or the objective function values of the vertices are close enough,or the number of iteration has reached a certain limit, or a combination ofcriteria.

The PSCAD implementation uses the objective function termination cri-teria [8]. It terminates when the difference in objective function values be-tween iterations becomes less than a value specified by the user.

3 Method

This section describes the methodology used for this thesis. The methodol-ogy can be divided into two steps:

• Finding an objective function

• Testing the objective function

3.1 Finding an objective function

To make an algorithm optimize the performance of the system, the impor-tant factors that define the performance need to be represented mathemat-ically. These factors then form the objective function and minimization is

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performed on this function. The minimum of the objective function corre-sponds to what the user has defined as the optimal system.

The most important feature of the objective function is that its minimumcorresponds to what normally is considered optimal for an HVDC system.To verify that this is the case, the solution found by the algorithm is com-pared to what an expert HVDC designer would consider optimal. Thesecriteria can be found in section 2.2.

It is also important that the objective function is smooth and containfew local minima, as otherwise the algorithm could converge to non-optimalsolutions. To evaluate the smoothness, several different start values aretested. If the objective function converges to the same minimum, usingseveral different start values, the likelihood increases that it is the globalminimum.

The process of finding a good objective function takes an evolutionarypath. A similar approach is taken in [9]. The idea is to start with anobjective function and evaluate how it performs. The objective function isthen modified if needed.

For evaluating the different objective functions, the Nelder-Mead Sim-plex algorithm is used. It has been used previously for similar problems withsuccess [9] [10]. It also seems to be the best alternative for multi-variableoptimization in PSCAD. The alternative in PSCAD is the Hooke-Jeeves al-gorithm, proposed by R. Hooke and T. A. Jeeves in 1961 [11]. However,the Hooke-Jeeves algorithm tend to converge more slowly than the Nelder-Mead Simplex algorithm, due to its need to evaluate more objective functionvalues per iteration.

An alternative approach is to use some external application, such asMATLAB for example, for the actual optimization. The application wouldthen receive the objective function value from PSCAD, evaluate the newparameter values, and send these values back to PSCAD each run.

Another approach is to write a new optimization module in PSCAD.This makes it possible for the user to choose algorithm freely.

Given the time span and scope of this thesis, it was decided to go withthe PSCAD built-in optimization module.

3.1.1 CCA

The CCA is one of the most important functions to design to get the properdynamics for the system. It is also the function that is most often describedin papers on HVDC optimization.

Objective functions widely used for similar problems is the IntegralSquare Error (ISE)

OF (Id) =

∫(Iord − Id)

2dt

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and the Integral Absolute Error (IAE)

OF (Id) =

∫|Iord − Id|dt

Both these functions integrate the error between the reference and actualvalue of the direct current. It is easy to understand why such functions canbe used to tune a step response. Slow step responses would render a bigerror in the early part of the step, too quick step responses will have a bigovershoot. Both these scenarios will render a big integral value. Hence, theminima of the functions will be at some trade-off between slow solutions andquick solutions with much overshoot.

Another objective function that has been tested is a very intuitive func-tion consisting of only the recovery time and overshoot. They are simplyadded together to form the objective function. The idea is that minimiz-ing this will also optimize the system. This will also lead to some form oftrade-off between the two parameters.

Along with performance, stability needs to be taken into account. Asdescribed in section 2.2.1, when tuning the CCA, the phase margin has tobe taken into account. The way this is handled in this thesis is by limitingG ·Kp and Kp as per the rule of thumb.

The need of limiting G ·Kp and Kp brings a problem, The Nelder-MeadSimplex algorithm has no formal way of handling constraints. A way tosolve this problem is to punish solutions where the limits are violated. Thiscan be done by adding a piecewise function to the objective function. Thisfunction adds a weight whenever the limits are violated. In the case ofthe CCA, it adds a weight whenever G · Kp or Kp are greater than theirmaximum values as specified by the user. An equation for such a functionis

OFconstraints(G,Kp) =

C if G ·Kp > (G ·Kp)max

or Kp > Kpmax

0 otherwise

where C is a constant. For this function to have the needed effect, C has tobe significantly larger than the objective function. This creates a big stepin the objective function when the parameters are outside of their limits,making sure the minimum of the objective function lies inside of the limits.

Adding the constraints for stability, the complete expressions for theinitial objective functions for tuning of the CCA are the following:

Integral square error

OF (Id, G,Kp) =

∫(Iord − Id)

2dt +OFconstraints(G,Kp) (3.1)

15

where



or Kp > Kpmax

0 otherwise

Integral absolute error

OF (Id, G,Kp) =

∫|Iord − Id|dt +OFconstraints(G,Kp) (3.2)

where



or Kp > Kpmax

0 otherwise

Recovery time and overshoot

OF (Id, G,Kp) = Wrecovery(Tf (Id) + Tr(Id)

+Wovershoot(Yu(Id) + Yo(Id))

+OFconstraints(G,Kp)

(3.3)

where



or Kp > Kpmax

0 otherwise

The functions Tf (Id) and Tr(Id) represent the time it takes for the currentto reach 90 % of the negative and positive step respectively. The functionsYu(Id) and Yo(Id) represent the undershoot and overshoot of the negativeand positive step, respectively. Wrecovery andWovershoot are weights. The ra-tio of the weights decide how much the two terms contribute to the objectivefunction.

3.1.2 VDCOL

The focus when tuning the VDCOL is a bit different compared to that ofthe CCA. The point of the VDCOL is to provide stability. An objectivefunction focusing mainly on performance would not be appropriate.

A way of optimizing the VDCOL is described in paper [9]. The way it isdone in this paper is by using ISE to optimize the DC current recovery. Thereference is a user-defined ramp function. This solution has problems withinstability. The way the instabilities are handled in this paper is by addinga piecewise function to the objective function that adds a weight when theseinstabilities occur.

16

The approach taken in this thesis is to put stability first, to make sureinstabilities do not happen in the first place. Stability is handled by mini-mizing the overshoot in Ud and undershoot in γ as described in section 2.2.2.To perform this, modified ISE functions are used. For the Ud function

OFUd(Ud) =

∫Yo(Ud)dt (3.4)

where

Yo(Ud) =

{(Udlimit

− Ud)2 if Ud ≥ Udlimit

0 if Ud < Udlimit

and for the γ function

OFγ(γ) =

∫Yu(γ)dt

where

Yu(γ) =

{0 if γ ≥ γlimit

(γlimit − γ)2 if γ < γlimit

γlimit and Udlimitare user defined constants.

In this thesis the performance of the VDCOL is measured by the DCpower recovery, see section 2.2.2. To optimize the power recovery, integralerrors are used. By using integral errors, a quick recovery is ensured whilepunishing overshoot. Both ISE and IAE are tested. For power reference,a normal reference step is used. By minimizing the overshoot in Ud andundershoot in γ, stability is upheld during the step.

Adding both performance and stability to the objective function, theexpressions for the initial objective functions used for tuning of the VDCOLare the following:


OF (Pd, Ud, γ) = WP

∫(Pref − Pd)

2dt

+WU

∫Yo(Ud)dt

+Wγ

∫Yu(γ)dt

(3.5)

where

Yo(Ud) =

{(Udlimit


0 if Ud < Udlimit

and

Yu(γ) =



17

WP , WU and Wγ are weights that decide how much the corresponding termscontribute to the objective function.


OF (Pd, Ud, γ) = WP

∫|Pref − Pd|dt

+WU

∫Yo(Ud)dt

+Wγ

∫Yu(γ)dt

(3.6)

where

Yo(Ud) =

{(Udlimit


0 if Ud < Udlimit

and

Yu(γ) =



WP , WU and Wγ are weights that decide how much the corresponding termscontribute to the objective function.

3.1.3 RAML

Due to the similarities between the tuning of the VDCOL and the RAML,see section 2.2.2 and 2.2.3, the same discussion for optimizing it can bemade.

For performance, ISE and IAE are used on the DC power, with a stepfunction as reference. For keeping stability during the recovery, the over-shoot in UDC will be monitored the same way as for the VDCOL.

For tuning of the RAML, the following objective functions are tested:


OF (Pd, Ud) = WP

∫(Pref − Pd)

2dt

+WU

∫Yo(Ud)dt

(3.7)

where

Yo(Ud) =

{(Udlimit


0 if Ud < Udlimit

WP and WU are weights that decide how much the corresponding termscontribute to the objective function.

18


OF (Pd, Ud) = WP

∫|Pref − Pd|dt

+WU

∫Yo(Ud)dt

(3.8)

where

Yo(Ud) =

{(Udlimit


0 if Ud < Udlimit

WP and WU are weights that decide how much the corresponding termscontribute to the objective function.

3.2 Testing the objective function

The previous steps were about finding a good objective function to optimizethe system. In this step, the objective function is tested and the results arecompared to the reference solution, tuned by experts.

For the CCA, the standard current step, used throughout the objectivefunction development, is used to compare the two solutions

For the VDCOL and RAML, six critical cases are used to compare theoptimized parameters with the reference parameters. These cases are:

• Single phase fault, 10% remaining voltage, 100ms




• Three phase fault, 10% remaining voltage, 100ms

• Three phase fault, 70% remaining voltage, 100ms

For the VDCOL, these faults are applied to the inverter AC network andfor the RAML, they are applied to the rectifier AC network.

To compare the solutions, both performance and stability will be takeninto account, using the criteria discussed in section 2.2

4 Simulation setup

4.1 PSCAD’s Optimum Run

Optimum Run is a module available in PSCAD which gives the user thepossibility to use optimization algorithms to optimize a set of parameters.

19

PSCAD 4.2.1 Professional, which was used for this paper, includes two algo-rithms for multi-variable optimization: the Nelder-Mead Simplex algorithmand Hooke-Jeeve’s algorithm.

4.1.1 Initial step size

The initial step size of the algorithms is decided by the user via the InitialStep Size variable. A bigger initial step size means that the algorithm willsearch a wider area, which could lead to a higher probability that the globalminimum is found, but it will also lead to slower convergence and increasesthe possibility of running into unstable solutions.

It was found experimentally that an initial step size of 10-25 % of theinitial parameter values seemed to give a good trade-off among these at-tributes.

4.1.2 Normalization of parameters

The Optimum Run module applies the same initial step size to all parame-ters. This will lead to imbalance among the different parameters due to theirdifferent values. Some parameters will have a larger relative step size com-pared to that of other parameters. With the big difference in magnitudesbetween parameters, this can be quite significant.

To balance this out, it is necessary to normalize the parameters. Thiscan be done by initiating normalized parameters in the module, so that theoptimization algorithm sees the parameter as having the same magnitude.Before the HVDC model receives the parameters, they are multiplied withappropriate factors to give them their true values.

4.1.3 Tolerance

The termination criterion for the optimization algorithm is set via the tol-erance variable. The objective function value is compared to the objectivefunction value of the previous iteration. If the difference between thesevalues becomes less than the tolerance, the algorithm terminates [8].

Using a large tolerance can lead to the optimization terminating pre-maturely, even when it hasn’t found a minimum. Using a small Tolerancecan lead to unnecessary fine tuning of the parameters, which increases thenumber of iterations. It was found experimentally that a tolerance of about10−4 times the expected value of the objective function provided a goodtrade off.

4.2 HVDC test system

The HVDC model used in this thesis is a back-to-back system with a shortcircuit ratio of about 3 in both the rectifier and inverter.

20

5 Results and discussion

5.1 CCA

5.1.1 Initial objective functions

Integral Square Error Figure 5.1 shows the step response of the op-timized CCA using the Integral Square Error objective function, equation3.1. Here, G = 128.5 and Kp = 0.7. The optimized system is very quickwith recovery times for the negative and positive step being 8ms and 10msrespectively. The optimal solution possesses a bit of overshoot. The max-imum overshoot was measured to be 0.0216 p.u. or 27 % which is abovewhat normally is acceptable (15-20 %)

Figure 5.1: Step response with CCA tuned using the ISE objective function

Table 5.1: Test runs using ISE objective function

Ginitial Kpinitial Gfinal Kpfinal Runs Obj. function

15 1 127.5 0.71 177 0.6704 · 10−4

30 2 115.9 0.78 219 0.6740 · 10−4

50 1.5 55.0 1.64 62 0.7120 · 10−4

80 1 128.5 0.70 151 0.6703 · 10−4

120 0.5 125.3 0.72 88 0.6704 · 10−4

170 0.4 147.2 0.61 132 0.6710 · 10−4

200 0.3 204.9 0.44 73 0.6940 · 10−4

Table 5.1 shows the different test runs used when evaluating the perfor-mance of the ISE objective function. Given that 3 runs with starting valuesfar apart converged to practically the same minimum, it can be concludedthat it likely is the global minimum. The number of runs until convergencevaried from 62 to 219. Some runs converged to a local minimum very closeto the initial guess while some runs converged very far from the initial guess.

Integral Absolute Error Figure 5.2 shows the step response of the opti-mized CCA using the Integral Absolute Error objective function (equation3.1). The values of G and Kp were found to be 300.1 and 0.3 respectively.The solution found is very quick where the recovery times are about 6 and

21

9 ms. However, the solution has even worse overshoot than the ISE solu-tion. It also has some oscillations after the negative step. The maximumovershoot was measured to 0.046 p.u or 57.5 % of the total step.

Figure 5.2: Step response with CCA tuned using the IAE objective function

Table 5.2: Test runs using IAE objective function


15 1 194.0 0.44 98 0.2267 · 10−2

30 2 187.7 0.47 180 0.2258 · 10−2

80 1 190.7 0.47 78 0.2261 · 10−2

150 0.6 164.6 0.54 140 0.2269 · 10−2

200 0.3 204.8 0.43 48 0.2259 · 10−2

250 0.2 253.5 0.33 53 0.2199 · 10−2

300 0.2 300.1 0.30 78 0.2188 · 10−2

Table 5.2 shows the test runs used to evaluate the IAE objective function.It had problems with local minima, even more so than the ISE case, with onlyone start guess converging to the minimum objective function value. Theruns often converged to solutions very near the initial guess. The numberof runs until convergence ranged from 48 to 180.

Recovery time and overshoot Figure 5.3 shows the step response ofthe optimized CCA using the objective function consisting of the sum ofthe recovery time and overshoot, equation 3.3. The algorithm found theobjective functions minimum to be at G = 144.2 and Kp = 0.62. Theresulting step response has recovery times of 7 and 9 ms and a maximumovershoot of 0.022 p.u or 27.5 % of the total step which is comparable tothe ISE case but a lot lower than the IAE case.

Table 5.3 shows the test runs used to evaluate the recovery time andovershoot objective function. All runs converged to different minima. Thenumber of runs until convergence ranged from 35 to 119.

22

Figure 5.3: Step response with CCA tuned using the recovery time and over-shoot objective function

Table 5.3: Test runs using recovery time and overshoot objective function


15 1 33.1 2.19 76 0.5976 · 10−1

25 2 33.3 2.20 84 0.5980 · 10−1

50 1.5 42.6 1.96 35 0.6027 · 10−1

80 1 82.6 1.07 81 0.5510 · 10−1

120 0.5 136.8 0.66 81 0.5500 · 10−1

170 0.4 144.2 0.62 60 0.5487 · 10−1

200 0.4 135.2 0.63 119 0.5507 · 10−1

5.1.2 Objective function modification

It is clear from looking at these step responses that these objective func-tions do not produce good results. The most important feature of a goodobjective function, that it finds the best solution for the problem, is missing.The resulting step responses have too much overshoot. They are also verysensitive to the initial start guesses and have a tendency to converge to localminima.

The overshoot needs to be reduced. The only objective function of thethree that can perform this task explicitly is the recovery and overshootobjective function via the weight constants. Increasing the weight of theovershoot should reduce the overshoot. This was experimented with, withpoor results. Changing the weight did indeed reduce the overshoot, but itstill had the problems of convergence to local minima. Using weights also hasanother disadvantage. The weights will be very system dependent. Differentsystems have different characteristics and would need different weights tofind the optimal solution. Because of this, the user would have to find newweights for every new system tested. This would lead to an optimizationprocess in itself and would take extra time.

An alternative approach that was tested was to remove the use of over-shoot in the objective function and use it as a constraint instead, the wayG · Kp and Kp is constrained. Using the objective function consisting ofthe recovery time, the user can find the quickest solution with a specified

23

amount of overshoot. To test this, the overshoot was set to be under anarbitrary value, in this case 0.016 p.u or 20 % of the total step, and therecovery time was optimized. Figure 5.4 and Table 5.4 show the results ofthis test.

Figure 5.4: Step response with CCA tuned using recovery time, overshootconstrained to 20% of step

Table 5.4: Test runs using recovery time objective function, overshoot con-strained to 20% of step


10 2 50.0 1.52 34 31.65 · 10−3

15 1.75 49.9 1.50 48 31.65 · 10−3

20 2 50.0 1.60 34 31.70 · 10−3

25 1.75 50.1 1.54 42 31.65 · 10−3

30 1.5 49.8 1.72 48 31.70 · 10−3

35 1.75 49.8 1.53 35 31.70 · 10−3

40 1.5 50.2 1.55 40 31.65 · 10−3

Using this setup, all the initial guesses found the same minimum whichmakes it very likely it is the global minimum. This minimum correspondsto a recovery time of 0.03165 s for the positive and negative step combined.Some runs converged to 0.0317 s, which is a difference of 50 microsecondswhich also is the time step of the simulated system. This difference is sosmall it is regarded as negligible. The value of G ranges from 49.8 to 50.2and Kp ranges from 1.5 to 1.72 which has to be regarded as fairly narrow.The amount of runs until convergence showed consistency and averagedaround 40 runs. The initial guess does not seem to have much influence onthe convergence speed, with the starting guess furthest from the final valuebeing the quickest with 34 runs.

The obvious downside to this approach with limiting the overshoot isthat the user must know what value to set it to. Because of the differentcharacteristics of systems, this could be difficult. Usually however, the de-signer performing the DPS has a good idea what this value should be. If thesystem is totally unknown, it would be possible to try different values of theovershoot, and see how the recovery time changes. At some point, a small

24

decrease in overshoot will lead to a big increase in recovery time and viceversa. Between these points there is a range where the trade-off betweenrecovery time and overshoot is good, and the designer could use this rangefor the CCA.

5.1.3 Testing the objective function

To test this objective function and proposed method, the recovery time wasoptimized while the overshoot was constrained. This was done for severalvalues of the overshoot, ranging from 10 % to 25 %. The results can be seenin figure 5.5 and table 5.5. Recovery time here is the combined recovery timeof the positive and the negative step. Maximum overshoot is the maximumovershoot of either the positive or negative step.

Figure 5.5: Recovery time as a function of overshoot

It can be seen that the recovery time decreases fairly linearly in theinterval of about 10 % to 17 % overshoot. From 17 % and up it flattens outsome and then declines quickly again. There is no obvious range where thetrade-off between recovery and overshoot stands out as being particularlygood. Because of the linear characteristics, it is difficult from this test, tomake any intelligent choice of G and Kp. The reference solution (tunedmanually by experts) has G = 28 and Kp = 2, which according to the table,has an overshoot of around 16 %.

25

Table 5.5: Test runs using recovery time objective function, overshoot con-strained

Max Overshoot (%) Recovery time (ms) G Kp

10 76.25 7 2.111.25 72.65 8 2.212.5 66.25 10 2.213.75 56.45 14 2.215 49.95 21 2.2

16.25 40.50 27 2.217.5 35.05 35 2.118.75 32.85 43 2.020 31.65 50 1.6

21.25 29.45 57 1.522.5 23.75 73 1.223.75 19.70 81 1.125 19.00 94 1.0

5.2 VDCOL

It was found experimentally that weights of WP = 1, WU = 10−4 andWγ = 1 provided a good trade-off between performance and stability andwas used along with Udlimit

= 1.1 and γlimit = 16 throughout this section.

5.2.1 Initial objective functions

Integral square error Figure 5.6 shows the recovery after a voltage dropin the inverter AC network using the VDCOL parameters obtained by usingthe Integral Square Error objective function (equation 3.5). The DC powerrecovers in about 90 ms which is well under the required recovery time of120 ms. The DC power has a bit of a dip after recovery which is probablydue to its quick recovery time. The DC voltage is below its limit of 1.1 p.uat all times. The extinction angle is kept above 16 degrees.

Table 5.6 shows the different test runs used to evaluate the ISE objec-tive function for tuning of the VDCOL parameters. The parameters arepresented in order TC UP RE, TC UP INV, UD HIGH, UD LOW. As canbe seen, all runs converge to different minima. The number of runs untilconvergence ranged from 100 to 123.

26

Figure 5.6: AC fault recovery with VDCOL tuned using ISE objective func-tion


Initial values Final values Runs Obj. function

0.05, 0.065, 0.8, 0.2 0.036, 0.079, 0.75, 0.15 119 0.4300 · 10−1

0.04, 0.05, 0.9, 0.3 0.036, 0.061, 0.73, 0.32 115 0.4748 · 10−1

0.02, 0.03, 0.7, 0.15 0.024, 0.045, 0.78, 0.16 123 0.4320 · 10−1

0.06, 0.08, 0.6, 0.25 0.048, 0.078, 0.66, 0.34 100 0.4760 · 10−1

Integral absolue error Figure 5.7 shows the recovery after an AC voltagedrop in the inverter network using the VDCOL parameters obtained usingthe Integral Absolute Error objective function (equation 3.6). The DC powerrecovery time is about 80 ms and holds steady above 90 % with only a minordip. Ud and γ stay within their limits during the recovery, which is essential.

Table 5.7 shows the different runs used to evaluate performance of theIAE objective function for tuning of the VDCOL. The number of runs rangedfrom 74 to 216 which is rather wide compared to the ISE range.

27

Figure 5.7: AC fault recovery with VDCOL tuned using IAE objective func-tion



0.05, 0.065, 0.8, 0.2 0.034, 0.079, 0.83, 0.14 74 0.6925 · 10−1

0.04, 0.05, 0.9, 0.3 0.029, 0.060, 0.89, 0.32 125 0.6200 · 10−1

0.02, 0.03, 0.7, 0.15 0.034, 0.080, 0.84, 0.14 88 0.6923 · 10−1

0.06, 0.08, 0.6, 0.25 0.051, 0.092, 0.70, 0.34 216 0.6205 · 10−1

5.2.2 Modification of the objective function

Comparing the two objective functions it can be seen that the DC powerrecovery is very similar in the two cases. The IAE recovery rises a bit slowercompared to the one of ISE. Ironically, this makes the recovery time faster(the time it takes for the power to reach 90 % of pre-fault value). The IAErecovery is smoother, it does not dip as much as the ISE one. However, thedifferences are very small.

The fact that the objective functions are calculated from the instant offault clearing, means that these objective functions will tend to favor solu-tions that are quick to get rid of this error. This has one big disadvantage;it goes against the purpose of the VDCOL. The point of the VDCOL is todelay and control the recovery for a smoother and safer restart.

A way to solve this problem would be to not calculate the error in theDC power where its not necessary, in this case during the ramp up. This

28

is controlled via the lower bound of the integral. Changing this value wastested and it appeared to delay the restart.

Figure 5.8: AC fault recovery with VDCOL tuned using ISE objective func-tion, lower integral bound 0.3 s

Figure 5.8 shows the same fault case as earlier, but with the integralcalculated from time 0.3 s and onwards. The ISE objective function wasused. Compared to the other case (figure 5.6) where the lower bound was at0.2 s, just at the fault clearing, it can be seen that the recovery is delayed,and the dip is essentially gone.

Because the integral is calculated from 0.3 s and onwards, the algorithmwill search for solutions that have a small error in this range. It is obviousthat solutions with a slow recovery will have a large error in this rangebecause they have not recovered at 0.3 s. According to the test earlier,solutions with a quick recovery will have a dip (or possible overshoot) inthis range, and will be prevented as well.

Figure 5.9 shows the same fault case using IAE, here the lower bound isalso 0.3 s. The results are almost identical to the ISE case which suggeststhat there is not much difference between them when using the lower integralbound modification. Because of this, only the ISE solution is investigatedfrom here on.

From these tests, it appears that the user can essentially choose the re-covery speed of the system by changing the lower integral bound. Obviously,setting it too low would lead to the unwanted case of too quick solutions,as described earlier. Setting it too high could lead to several different sce-

29

Figure 5.9: AC fault recovery with VDCOL tuned using IAE objective func-tion, lower integral bound 0.3 s

narios because too many solutions would be considered optimal, i.e. have azero error within the integral range. The limits of Ud and γ also affect therecovery time. Tightening these limits would reduce the possible recoverytime, but increasing stability.

When tuning the VDCOL, several different test cases have to be tested.Optimizing only one fault case and using these parameters on a differentcase, it is very unlikely that it would provide the same optimal result. Opti-mizing with regard to all necessary cases is therefore necessary to properlyoptimize the system as a whole.

A way to run several faults is to simply place them one after another.Figure 5.10 shows such a case consisting of two faults run in succession.

Figure 5.10: Two faults run in succession, single phase and three phase

The first fault is a single phase fault, the same that has been used so

30

far. The second is a three phase fault in the inverter AC network with 10 %remaining voltage and duration 0.1 s. The two faults are similar in the sensethat the tuning is performed the same way. This makes it probable that theobjective function obtained thus far translates well to a three phase faultcase. To optimize both fault cases at the same time, the objective function iscalculated for both cases individually. After each run, the objective functionsof the two cases are added together and this combined objective function issubject to minimization.

Figure 5.11: Single phase fault recovery with VDCOL tuned using two faults,objective function ISE with lower integral bound 0.3 s

Figure 5.11 shows the single phase fault recovery using the parametersobtained after performing optimization on two faults. It can be seen thatthe recovery time is slightly reduced compared to tuning only the single faultcase, which is expected. However, the reduction is not that big. There isno dip in the DC power but there is a slight halt in the recovery just afterit reaches 90 %. Since it does not dip below the 90 % mark, the recoverytime is still kept low at just over 100 ms. Ud and γ are a bit more dampedthan the single fault case which is expected with the recovery time being abit slower.

Figure 5.12 shows the three phase fault using the same set of parameters.The recovery of the DC power is very smooth and does not dip below 90 %after it reaches it. The recovery time is about 130 ms. Ud and γ both reachtheir limits during recovery.

Table 5.8 shows the simulations used to evaluate the two-fault test case.

31

Figure 5.12: Three phase fault recovery with VDCOL tuned using two faults,objective function ISE with lower integral bound 2.3 s

Table 5.8: Test runs using two faults, ISE objective function with increasedlower integral bound


0.05, 0.065, 0.8, 0.2 0.042, 0.074, 0.84, 0.16 78 0.3396 · 10−2

0.04, 0.05, 0.9, 0.3 0.032, 0.059, 0.82, 0.36 67 0.3975 · 10−2

0.02, 0.03, 0.7, 0.15 0.041, 0.075, 0.85, 0.14 85 0.3274 · 10−2

0.06, 0.08, 0.6, 0.25 0.051, 0.088, 0.72, 0.34 88 0.3766 · 10−2

Two starting guesses converged to the same set of parameters. The numberof runs ranged from 67 to 85 which shows good consistency.


To test the modified objective function, the lower integral bound was set tothe desired recovery time, in this case 120 ms after fault clearing. All sixcritical cases were used, which include:

• Single phase fault inverter AC network, 10% remaining voltage, 100ms




32

• Three phase fault inverter AC network, 10% remaining voltage, 100ms

• Three phase fault inverter AC network, 70% remaining voltage, 100ms

The faults were placed in succession, 5 seconds apart. The objectivefunction for all the faults were added together and optimization was carriedout on this combined objective function. Default start values were used. Thefirst run, it was found that fault with 10 % remaining voltage and duration300 ms had severe overshoot in Ud. A few different start values were testedbut the overshoot persisted. To be able to perform the optimization, it wasdecided to remove the limit of Ud for this particular fault.

Table 5.9: VDCOL parameters before and after optimization, compared toreference values

Parameters

Start 0.05 0.06 0.8 0.25Final 0.0372 0.0648 0.862 0.260Reference 0.034 0.049 0.91 0.35

Table 5.9 shows the VDCOL parameters before and after the optimiza-tion, it also includes the reference parameters which were obtained by ex-perts.

The optimization completed in 139 runs. The objective function valueusing the initial parameters was 0.884 · 10−2 compared to 0.324 · 10−2 forthe optimized parameters.

Single phase fault to ground, inverter side, 10 % remaining voltage,duration 100 ms The complete characteristics of this fault, using thethree sets of parameters can be seen in appendix A.1. It can be seen that theoptimization has greatly improved the recovery speed of this fault, comparedto the initial values. It has reduced the recovery time from 150ms to about100. The recovery time of the reference solution is around 120 ms whichis in line with the desired speed. None of the solutions have any problemswith overshoot in Ud or undershoot in γ.

Single phase fault to ground, inverter side, 70 % remaining volt-age, duration 100 ms The characteristics of this fault can be seen inappendix A.2. The recovery speed of this fault is reduced by about 10 msafter the optimization, making it about as quick as the reference solution.All solutions recover within 120 ms. Ud and γ recover without issues for allcases.

Single phase fault to ground, inverter side, 10 % remaining voltage,duration 300 ms Appendix A.3 shows the characteristics of this fault.

33

It can be seen that all solutions have problems with this fault. None ofthe solutions recover within 120 ms, although the optimized solution is thequickest at about 150 with the other two being slightly slower. All solutionshave overshoot in Ud. The reference solution has the least initial spike in Ud

but possesses some oscillations. The extinction angle dips dangerously low,in both the optimized and the reference case.

Single phase fault to ground, inverter side, 70 % remaining voltage,duration 300 ms The characteristics of this fault can be seen in appendixA.4. Both the initial solution and the optimized solution recover within 120ms. The reference solution experiences some oscillations which makes itunable to recover within the desired 120 ms. None of the solutions have anyproblems with Ud or γ.

Three phase fault to ground, inverter side, 10 % remaining voltage,duration 100 ms Appendix A.5 shows the characteristics of this fault.The optimized parameters greatly improve the recovery speed of the system,reducing the recovery time from about 200ms to about 130ms. By increasingthis recovery speed it also increases the overshoot in Ud. The referencesolution is quicker than the initial solution but not as quick as the optimized.

Three phase fault to ground, inverter side, 70 % remaining voltage,duration 100 ms Appendix A.6 shows the characteristics of this fault forthe three cases. The initial solution has a recovery time of about 100ms.Both the reference and the optimized solution has a recovery time of about60 ms which is a big reduction. None of the solutions had problems withovershoot in Ud or undershoot in γ.

5.3 RAML

It was found experimentally that weights of WP = 1 and WU = 10−4 pro-vided a good trade-off between performance and stability and was used alongwith Udlimit

= 1.1 throughout this section.

5.3.1 Initial objective function

Integral Square Error Figure 5.13 shows a single phase fault in therectifier AC network, duration 0.1 seconds and remaining voltage 10 %.The RAML parameters used were obtained with the integral square er-ror objective function (equation 3.7). The parameters optimized here wereCRAML REF and CDL LEVEL. The optimal values obtained were 0.94and 34. It can be seen that the DC power recovery starts with a fast riseimmediately after the fault is released, which also can be seen in Ud, then

34

flattens out. The recovery time is about 130 ms which is slower than the120 ms required.

Figure 5.13: AC fault recovery with RAML tuned using ISE objective func-tion


Initial values Final values Runs Objective function

0.9, 60 0.94, 34 45 0.1888 · 10−1

0.9, 35 0.91, 34 25 0.1889 · 10−1

0.6, 60 0.58, 35 45 0.1923 · 10−1

0.6, 35 0.62, 35 37 0.1923 · 10−1

Table 5.10 shows the simulations used for evaluating the integral squareerror objective function). It can be seen that the CDL LEVEL parameterconverges to nearly the same value in every run, but CRAML REF barelychanges from its starting value, which likely means that CDL LEVEL ismore crucial for the end result than CRAML REF. The rate of convergenceis good with the number of runs ranging from 25-45.

Integral Absolute Error Figure 5.14 shows the fault recovery using theparameters obtained by using the integral absolute error objective function(equation 3.8). It is nearly identical to the ISE case. The recovery time isthe same, around 150 ms, which does not meet the desired recovery time.

Table 5.11 shows the simulations used to evaluate the IAE objective func-tion. As evident by the little difference in the objective function, all simula-tions give roughly the same performance. Like the ISE case, CDL LEVELconverges to a tight range of values, while CRAML REF does not, this im-plies that the result does not depend heavily on the value of CRAML REF.

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Figure 5.14: AC fault recovery with RAML tuned using IAE objective func-tion



0.9, 60 0.93, 36 51 0.5712 · 10−1

0.9, 35 0.95, 36 43 0.5713 · 10−1

0.6, 60 0.49, 34 60 0.5755 · 10−1

0.6, 35 0.62, 35 40 0.5781 · 10−1

5.3.2 Objective function modification

It is obvious that the two objective functions tested do not meet the systemrequirements. They are too slow which, like the VDCOL case, could dependon the integral favoring solutions that remove the big error at the instantafter fault clearing.

The same argument can be made for the RAML as for the VDCOL.Both the VDCOL and RAML are included for a slow safe restart of thesystem. To delay the start, the same method that worked for the VDCOLwas tested. The lower bound of the DC power error integral was increasedby 100 ms.

Figure 5.15 shows the fault recovery using the parameters obtained usingthe ISE objective function with the lower bound of the integral set to 0.3s, which corresponds to 100 ms after fault clearing. It can be seen that thespike after fault clearing is greatly reduced. The overshoot in Ud is reducedby over 50 %. The DC power recovers to 90 % in around 120 ms with onlya minor dip after.

Table 5.12 shows the simulations used to evaluate the modified objectivefunction. Again CRAML REF does not change much from its initial value,and CDL LEVEL converges to values fairly close. The differences in the

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Figure 5.15: AC fault recovery with RAML tuned using ISE, lower integralbound 0.3 s

Table 5.12: Test runs using ISE objective function, lower integral bound 0.3s


0.9, 60 0.96, 46 27 0.1242 · 10−2

0.9, 35 0.92, 42 23 0.1133 · 10−2

0.6, 60 0.70, 46 24 0.1297 · 10−2

0.6, 35 0.64, 46 25 0.1300 · 10−2

final values most likely depend on the oscillation of the DC power duringrecovery which creates local minima in the objective function. What standsout the most is the quick and consistent convergence. The number of runsvaries from 23 to 27 runs.

This test shows the efficiency of increasing the lower bound of the inte-gral calculating the error in the DC power. It is even more evident in theRAML than the VDCOL. The test also shows that an optimization usingthis objective function gives a result very close to what is desired, for thistype of fault.

When tuning the RAML, both single phase and three phase faults aretested. The objective function developed so far was tested using singlephase faults. Three phase faults are tuned similarly, so the same objectivefunction that worked for the single phase case was tested to start with.The parameters used to tune the three phase faults were RAML DECR andDL LEVEL.

Figure 5.16 shows the recovery of a three phase fault in the rectifierAC network. The parameters used were obtained using the ISE objectivefunction with lower integration bound 0.3 s. The DC power recovers in areliable manner with a recovery time of 140 ms, which is 20 ms above the

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desired time.

Figure 5.16: AC fault recovery with RAML tuned using ISE objective func-tion, lower integral bound 0.3 s

Table 5.13: Test runs using ISE objective function, lower integral bound 0.3s


0.85, 35 0.87, 44 33 0.2217 · 10−2

0.60, 60 0.66, 66 95 0.6411 · 10−2

0.60, 35 0.65, 45 33 0.2194 · 10−2

Table 5.13 shows the simulation used to evaluate the ISE objective func-tion with lower integral bound 0.3 s using a three phase fault. One of thesimulations found a solution with very poor performance with an objec-tive function three times as big as the other two which ended up roughlythe same. For these two solutions the DL LEVEL values are close but notRAML REF which is similar to the single phase case.

This test shows that the objective function that gives good results in thesingle phase case also gives a good result in the three phase case.

Just like for the VDCOL, it is essential that the RAML is tuned so thatit works for all the necessary fault cases. Optimization with regard to allthese cases is therefore necessary. The same approach as in the VDCOLcase was tested. A three phase and a single phase fault were placed insuccession. Close enough for a quick run time, but far enough to reachsteady state before the other fault goes active, see figure 5.17. The objectivefunction of the two faults were calculated individually and added togetherfor optimization. For this test, RAML DECR, RAML REF, CRAML REF,DL LEVEL and CDL LEVEL were tuned simultaneously.

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Figure 5.17: Two faults in succession, three phase and single phase

Table 5.14: Test runs using two faults, ISE objective function, increasedlower integral bound


0.85, 0.6, 0.85, 35, 60 0.90, 0.61, 0.9, 45, 60 54 0.7527 · 10−2

0.85, 0.6, 0.6, 45, 45 0.87, 0.66, 0.64, 51, 52 56 0.4395 · 10−2

0.6, 0.85, 0.85, 60, 60 0.66, 0.94, 0.90, 51, 46 66 0.4180 · 10−2

0.6, 0.6, 0.6, 60, 60 0.67, 0.67, 0.64, 44, 40 85 0.2599 · 10−2

Table 5.14 shows the simulation used to evaluate the case using twofaults using the ISE objective function with lower integral bound at 0.1seconds after fault clearing. Parameters presented in order RAML DECR,RAML REF, CRAML REF, DL LEVEL, CDL LEVEL. It can be seen thatjust like the earlier cases, RAML REF and CRAML REF does not changemuch from the initial values. RAML DECR follows the same behavior. Themore important parameters, DL LEVEL and CDL LEVEL vary a lot eachrun, with all runs converging to different solutions. The performance ofthese solutions vary a lot as well, as can be seen by the big difference in theobjective function.

Figure 5.18 shows the three phase fault with parameters obtained usingthe two fault case. It can be seen that the DC power recovers quickly andin a stable manner in about 130 ms.

Figure 5.19 shows the single phase fault with parameters obtained usingthe two fault case. The DC power recovers quickly in about 110 ms. Thesetests suggest that using the ISE objective function, with lower bound setto a value near the desired recovery time, it is possible to tune the RAMLparameters so that it gives desirable results for different fault cases.

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Figure 5.18: Three phase fault recovery with RAML tuned using two faults

Figure 5.19: Single phase fault recovery with RAML tuned using two faults


For this test, the ISE objective function was used with the lower integralbound set to the time of desired recovery time (120 ms after fault clearing).All six critical cases for tuning of the RAML were used:

• Single phase fault to ground, rectifier side, 10 % remaining voltage,duration 100 ms




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• Three phase fault to ground, rectifier side, 10 % remaining voltage,duration 100 ms

• Three phase fault to ground, rectifier side, 70 % remaining voltage,duration 100 ms

Table 5.15: RAML parameters before and after optimization, compared toreference values

Parameters

Start 0.7 0.9 0.8 55 60Final 0.742 0.923 0.847 40.3 59.0Reference 0.75 0.9 0.6 50 55

Table 5.15 shows the initial and the optimized RAML parametersRAML DECR, RAML REF, CRAML REF, DL LEVEL and CDL LEVEL.Also shown are the reference parameters obtained manually by experts usingthe trial and error technique.

The optimization completed in 79 runs. The objective function valueusing the initial parameters was 0.611 · 10−2 compared to 0.372 · 10−2 forthe optimized parameters.

Single phase fault to ground, rectifier side, 10 % remaining volt-age, duration 100ms Appendix B.1 shows this fault for all three setsof parameters. Using the starting values the system has a recovery time ofabout 140 ms which does not meet the requirements. The optimized sys-tem has a recovery time of about 120 ms. The reference system also has arecovery time of around 120 ms.

Single phase fault to ground, rectifier side, 70 % remaining voltage,duration 100ms The characteristics for this fault can be seen in appendixB.2. The initial solution has a recovery time of about 90 ms which is wellbelow the desired time. The optimized solution is slower with a recoverytime of 110 ms but is still within the limit of 120 ms. The reference recoversin about 90 ms.

Single phase fault to ground, rectifier side, 10% remaining voltage,duration 300ms Appendix B.3 shows the results of this fault case. Usingthe initial values, the system has a recovery time of about 140 ms whereasthe optimized system has a recovery time of about 120 ms but the recoveryaround 90%. The reference solution clears 90% easily in about 120 ms.

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Single phase fault to ground, rectifier side, 70% remaining voltage,duration 300ms The results from this fault case can be seen in appendixB.4. It can be seen that the optimization has reduced the recovery timefrom about 130 ms to about 110 ms. The reference has a recovery time ofabout 120 ms.

Three phase fault to ground, rectifier side, 10% remaining voltage,duration 100ms Appendix B.5 shows the result of this fault case. Theinitial solution has a recovery time of about 150 ms while the optimizedsolution around 160, a decrease in recovery speed. The reference recovers inaround 140 ms.

Three phase fault to ground, rectifier side, 70% remaining voltage,duration 100ms The results can be found in appendix B.6. The initialvalues give a recovery time of around 150 ms and the reference around140 ms. The optimized parameters give a recovery time around 110 ms.Worth noting here is that during the fault, the optimized solution has AL-PHA ORD at 42 degrees, which is equal to the CDL LEVEL. This meansthat the fault is seen as a single phase fault and not a three phase fault.

6 Conclusions

The goal of this thesis was to find a way to tune the HVDC control systemparameters by using optimization algorithms instead of manual trial anderror. The Nelder-Mead Simplex algorithm and PSCAD software was used.The focus was on three essential parts of the control system, the CCA,VDCOL and RAML.

For the CCA, all three initial objective functions tested had problemswith having too much overshoot. Modifying one of the objective functions,by optimizing recovery time and limiting the overshoot produced desirablestep responses for the current. It also had excellent convergence properties.However, this approach assumes the user knows what overshoot the systemshould have. The method proposed to use different values of the overshootto find a good trade-off between overshoot and recovery time failed to finda good trade-off for the test system used. It is concluded that due to theexcellent convergence properties of the chosen objective function, using thisobjective function could help the designer tune the CCA. However, it re-quires that the user knows what overshoot to aim for.

For the VDCOL, it is clear by looking at the results that using themethod proposed, the VDCOL can be tuned with good results. The recoveryspeeds were near equal or better compared to the reference solution for allfault cases, while limiting Ud and γ makes sure that stability is upheld. Thebiggest issue that was seen throughout the section was that the objective

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functions had problems with convergence. The start guesses converged tovery different solutions for almost all cases. This suggests that the user willneed to try several initial values for the parameters to increase the chanceof finding the global minimum.

Optimizing the RAML produced similar results to the VDCOL, with theoptimized solution being comparable to the reference solution for most cases.Besides the convergence issue, another critical issue arose. For one threephase fault, the RAML detected it as a single phase fault. This has to dowith the DL LEVEL and CDL LEVEL parameters. It is likely better to tunethem manually, and only optimize the remaining three RAML-parameters.Another approach worth investigating is whether tuning three phase andsingle phase faults separately, yields better results.

A subject only briefly touched on in this thesis is that of the initial stepsize and termination criteria for the algorithm. An idea for future work is tomore thoroughly investigate the effect that changing these parameters have.

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References

[1] Ake Ekstrom, High Power Electronics HVDC and SVC. The RoyalInstitute of Technology, 1990.

[2] B. Nordstrom, Functional description: Converter Firing Control. ABB,June 2003. 1JNL100090-917 Rev. 02.

[3] ABB, Guidelines: Dynamic Performance Study (DPS) for HVDC Clas-sic in PSCAD, February 2009. 1JNL000424.

[4] ABB, Memorandum: Automating Result Analysis for DPS, November2011. 11TST0387.

[5] ABB, Minutes of Meeting: Study Outline for evaluation of optimizationtools for HVDC control tuning, September 2011. 11TST0113.

[6] J. Nelder and R. Mead, “A simplex method for function minimization,”The Computer Journal, vol. 7, no. 4, pp. 308–313, 1965.

[7] J. Nelder and S. Singer, “Nelder-mead algorithm,” Scholarpedia, vol. 4,no. 2, p. 2928, 2009.

[8] PSCAD User Guide.

[9] S. Filizadeh, A. M. Gole, D. A. Woodford, and G. D. Irwin,“An optimization-enabled electromagnetic transient simulation-basedmethodology for hvdc controller design,” IEEE Transactions on PowerDelivery, vol. 22, no. 4, pp. 2559–2566, 2007.

[10] A. M. Gole, S. Filizadeh, and P. L. Wilson, “Inclusion of robustness intodesign using optimization-enabled transient simulation,” IEEE Trans-actions on Power Delivery, vol. 20, no. 3, pp. 1991–1997, 2005.

[11] R. Hooke and T. A. Jeeves, “Direct search solution of numerical andstatistical problems,” Journal of the ACM, vol. 8, no. 2, pp. 212–229,1961.

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A VDCOL Comparisons 1

A.1 Single phase fault to ground, inverter side, 10% remain-ing voltage, duration 100ms

A.1.1 Initial parameters

1Discussion regarding these results can be found in section 5.2.3

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A.1.2 Optimized parameters

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A.1.3 Reference parameters

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A.5 Three phase fault to ground, inverter side, 10% remain-ing voltage, duration 100ms


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A.6 Three phase fault to ground, inverter side, 70% remain-ing voltage, duration 100ms


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B RAML Comparisons 2

B.1 Single phase fault to ground, rectifier side, 10% remain-ing voltage, duration 100ms

B.1.1 Initial parameters

2Discussion regarding these results can be found in section 5.3.3

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B.1.2 Optimized parameters

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B.5 Three phase fault to ground, rectifier side, 10% remain-ing voltage, duration 100ms


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B.6 Three phase fault to ground, rectifier side, 70% remain-ing voltage, duration 100ms


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optimized tuning of parameters for hvdc dynamic...

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