design of static var compensator for power systems … · design of static var compensator for...

Design of Static Var Compensator for Power Systems Subject to Voltage Fluctuation Satisfying Bounding Conditions 297

Design of Static Var Compensator for PowerSystems Subject to Voltage Fluctuation

Satisfying Bounding Conditions

Suchin Arunsawatwong1 , Kittichai Tia2 , and Bundhit Eua-arporn3 , Non-members

ABSTRACTThis paper presents the design of a static var com-

pensator for power systems subject to voltage fluctu-ation by using a known framework comprising theprinciple of matching and the method of inequali-ties. In the design formulation, all possible voltagefluctuations are treated as persistent signals havinguniform bounds on magnitude and slope. The princi-pal design objective is to ensure that the rotor angle,the generator terminal voltage and the voltage of thenearby bus always deviate from their nominal valueswithin the allowable ranges for all time in the pres-ence of any possible voltage fluctuation. By virtueof the framework, it is clear that once a solution isfound, the system is guaranteed to operate satisfacto-rily in regard to the design objective. The numericalresults demonstrate that the framework adopted hereis suitable and effective, thereby giving a realistic for-mulation of the design problem.

Keywords: Power Systems Control, Static VarCompensator, Voltage Fluctuation, Design Formula-tion, Control Systems Design, Principle of Matching,Method of Inequalities

1. INTRODUCTION

In power system operation, there are occasionsin which large and rapid voltage fluctuations havea great influence on the system’s dynamic stabilityand performance. As a well-known example, the op-eration of an Electric Arc Furnace (EAF) can affectthe operation of other equipment or machines in thenearby factories and the nearby distribution systems.The instantaneous fluctuation with large amplitudesof EAF’s voltage is a source of power quality distur-bances in an electric power system [1–3]. Moreover,

Manuscript received on April 21, 2011 ; revised on June 14,2011.

1 The author to whom all correspondences should be ad-dressed. Tel: +66 2 2186503, Fax: +66 2 2518991.

1 The author is with Department of Electrical Engineering,Faculty of Engineering, Chulalongkorn University, Bangkok10330, Thailand. Email: [email protected].

2 The author is with Power Generation Systems ControlDepartment, Power System Company, Toshiba Corporation,Toshiba-Cho, Fuchu-Shi, Tokyo, 183-8511, Japan. Email: [email protected]

3 The author is with Department of Electrical Engineering,Faculty of Engineering, Chulalongkorn University, Bangkok10330, Thailand. Email: [email protected].

the stability and the performance of nearby genera-tors are deteriorated.

Compensators such as a power system stabilizer(PSS), a static var compensator (SVC) or a static syn-chronous (STATCOM) have been utilized to mitigatethe undesirable effects caused by EAF. In addition,they can be used for improving the power system’sstability (or damping) and dynamic performance. Inthis regard, the subject of tuning such compensatorshave been investigated by a number of authors, forexample, [1, 4–8] and the references cited therein. Inthese studies, the authors used the control theoriesthat are based on the system properties defined inconnection with deterministic test inputs (for exam-ple, step functions or sinusoids), which never happenin practice. Accordingly, their design problems werenot formulated in a realistic manner.

The objective of this paper is to present the designof an SVC for improving the dynamic performance ofa power system operating under load voltage fluctua-tion, in which all possible voltage fluctuations (definedas voltage fluctuations that can happen or are likelyto happen in practice) are modelled as persistent sig-nals having uniform bounds on magnitude and slope(see (6) and (29) for more details). Accordingly, thedesign problem is formulated realistically in the sensethat the uncertain characteristic of the possible volt-age fluctuations is explicitly taken into account.

For the system to have sufficient damping as wellas satisfactory performance, the SVC will be designedso as to ensure that the generator rotor angle, the gen-erator terminal voltage and the voltage of the nearbybus always deviate from their nominal values withincertain acceptable ranges for all the possible voltagefluctuations. The violation of the bound for the gen-erator rotor angle may give rise to the system’s in-stability and consequently may cause blackout in awidespread area.

The design methodology adopted in the paper isbased on Zakian’s framework [9, 10] (see also the ref-erences therein), which comprises the principle ofmatching and the method of inequalities. The frame-work has been successfully applied to various engi-neering problems (see, for example, [10–13]). See Sec-tion 2 for a recapitulation of the framework.

Recently, the framework has been used by [11] intuning a PSS to improve the stability and the perfor-mance of a power systems operating with load voltage

298 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.2 August 2011

fluctuation. The work reported in the present paper iscarried out in a more elaborate and refined way. TheSVC device is used, the configuration of the powersystem is more complex, and the representation ofthe voltage fluctuation is simpler.

In this work, attention is restricted only to the tun-ing of SVC devices. However, it should be noted thatthe method used here can readily be applied to thetuning of other types of power system compensatoras well.

The structure of the paper is as follows. Section 2provides necessary background of Zakian’s frameworkand also includes the fundamental theory used for for-mulating the design problem. Section 3 describes themodel of the power system used in the paper. InSection 4, the characterization of the set of the pos-sible inputs is explained. Section 5 explains how thedesign problem is formulated. In Section 6, the nu-merical results are presented. Finally, the conclusionsare given.

2. BACKGROUND

Section 2.1 briefly reviews the general concept ofZakian’s framework, and Section 2.2 provides themathematical results which are used specifically forsolving the design problem considered in the paper.

2.1 Zakian’s framework

Two essential and complementary constituents ofZakian’s framework [9, 10] are the method of inequal-ities (MoI) and the principle of matching (PoM). TheMoI [9, 14, 15] suggests that a multiobjective designproblem should be cast as a set of inequalities thatcan be solved in practice, whereas the PoM [9, 16]suggests what kind of design inequalities should beused so that the control objectives (2) are satisfied.A detailed account of the framework can be found in[9, 10] and also the references cited therein.

2.1.1 The method of inequalities

The MoI [9, 14, 15] is a general design principle re-quiring that a design problem should be formulatedas a set of inequalities

φi(p) ≤ Ci (i = 1, 2, ..., m) (1)

where p ∈ Rn is a vector of design parameters, φi :Rn → R ∪ {∞} represents a quality or a propertyor an aspect of the behaviour of the system, and thenumbers Ci is the largest value of φi(p) that can beaccepted. Any point p satisfying (1) characterizes anacceptable design solution.

The set of inequalities (1) includes two principalsubsets, one subset representing constraints and theother representing required performance. Whereasconstraints have traditionally been represented by in-equalities, the representation of desired performanceby a set of inequalities is a significant departure from

the tradition in which the performance is representedby a single objective function to be minimized. TheMoI recognizes that desired performance is appropri-ately stated by means of several distinct criteria, witheach criterion represented by one or more inequalities,thus allowing greater insight into the design process.

In many cases, some of the principal φi are non-convex functions of the design parameter p. Conse-quently, a numerical algorithm is usually employed todetermine a solution of the inequalities (1) by search-ing in the parameter space Rn. Considerable experi-ence over the last thirty years has shown that a widerange of practical problems can be formulated in theform (1) and can be solved by numerical methods.Because there are a large number of articles (morethan 40 articles) on the successful applications of theMoI, readers are referred to the book [10] for a com-prehensive list of references.

2.1.2 The principle of matching

The PoM [9, 16] is a general concept which requiresthat the system and the environment in which thesystem is to operate should be designed so that bothare matched in the sense defined below.

Assume that the environment subjects the systemto an input f which causes responses (or outputs) yi

(i = 1, 2, . . . , m) in the system. The input f gen-erated by the environment is assumed to be knownonly to the extent that it belongs to a set P, calledthe possible set. This set comprises all inputs thatcan happen or are likely to happen or are allowed tohappen in practice. Then the system and the envi-ronment are said to be matched [16] if the followingcriteria are satisfied.

|yi(t, f, p)| ≤ εi ∀t ∀f ∈ P (i = 1, 2, . . . , m) (2)

where yi(t, f, p) denotes the value of yi at time t inresponse to f with the design parameter p ∈ Rn, andεi is the largest value of yi that can be accepted. Thedesign problem is usually to determine any value ofp ∈ Rn that satisfies (2).

For economic reasons, it can be further requiredthat the set P should not contain too many fictitiousinputs, that is, inputs that never happen in practice.However, it is not within the scope of the paper todeal with this requirement; see [20] for further detailson this.

Indeed, the criteria (2) have been used in prac-tice by engineers to monitor the performance of thecontrol system. They are used to guarantee that theoutputs yi stay within the specified ranges ±εi for alltime and for all f ∈ P. The criteria (2), though con-ceptually useful, are not computationally tractable.

It is easy to show that the criteria (2) are equiva-lent to

yi(p) ≤ εi (i = 1, 2, . . . ,m) (3)

where the performance measures yi are called the peak


outputs for the set P and given by

yi(p) , supf∈P

supt≥0

|yi(t, f, p)| (i = 1, 2, . . . ,m). (4)

The criteria (3) become useful design inequalities,provided that yi are computable. In connection withthe PoM, a number of researches have been devotedto the development of readily computable inequalitiesthat can be used to ensure that (2) are satisfied forvarious models of the possible set. See, for example,[15, 17–20] and also the references therein.

When there are a large number of possible inputs,the usefulness of the criteria (2), as well as (3), be-comes very clear. Specifically, when the performancemeasures used in formulating the design problem arehardly related to (2), engineers will have to performsimulations with a number of input waveforms afterobtaining a design so as to check whether or not thesystem operates satisfactorily. This process usuallyinvolves repeated redesign and simulation of perfor-mance, a process that can be very time-consuming.It is clear that only the framework adopted in this pa-per can deal with the design problem in connectionwith the requirements (2) in a systematic manner.

2.2 Fundamental design theory

Consider a linear, time-invariant and nonantici-pative system whose input f and outputs yi (i =1, 2, . . . , m) are related by the convolution integral

yi(t, f, p) =∫ t

0

hi(t− τ, p)f(τ)dτ (5)

where hi denotes the impulse response of the outputyi and depends on the design parameter p. For therest of the paper, consider the possible set P charac-terized by

P , {f : ‖f‖∞ ≤M, ‖f‖∞ ≤ D} (6)

where the bounds M and D are positive numbersthat can be determined from the physical propertiesor the past records of the system.

In connection with (6), it should be noted that theset P contains an important class of bounded signalsthat persistently vary for all time (see [9, 10] for de-tails).

2.2.1 Computation of performance measures

For the possible set P in (6), there are availablemethods for computing the peak outputs yi. See, forexample, [19, 20] and the references therein. However,none of them are simple.

For clarity in explaining how to apply Zakian’sframework to the design problem, this paper consid-ers the use of the majorants yi instead of the peakoutputs yi. Note, in passing, that the majorants yi

have a property that yi(p) ≥ yi(p) for all p ∈ Rn.

It is known [10, 15] that the majorants yi are givenby

yi = |si,ss|M+ ‖si − si,ss‖1D (7)

where si are the responses yi when the input f is theunit step function, and si,ss denote the steady-statevalues of si. Expression (7) provides a useful andsimple formula for computing the majorants yi.

From the above, one can easily see that

yi(p) ≤ εi (i = 1, 2, . . . ,m) (8)

are readily computable inequalities that can be usedin practice to ensure that (3) are satisfied. Hence,in this paper, (8) are used as the design inequalitiesinstead of (3).

In using (7) to compute yi, once the step responsessi are obtained, the rest of the computation can becarried out very easily by standard numerical algo-rithms. Since the calculation of yi is very simple, onecan focus one’s attention mainly on understandingthe design methodology without having to worryingtoo much about the detail of how to compute yi.

2.2.2 Stabilization & finiteness of performancemeasures

Following previous works [14, 15, 21], it is readilyappreciated that in solving the inequalities

yi(p) ≤ εi (i = 1, 2, . . . ,m) (9)

by numerical methods, it is necessary to obtain a sta-bility point, that is, a point p ∈ Rn satisfying

yi(p) < ∞ ∀i. (10)

This is because numerical search algorithms, in gen-eral, are able to seek a solution of (10) only if theystart from such a point (see [14, 21] and also [10] fordetails).

It is important to note [14, 15, 21] that condition(10) is not soluble by numerical methods and there-fore needs to be replaced with conditions (usually inthe form of inequalities) that can be satisfied by nu-merical methods.

Assume that a state-space realization of the sys-tem is {A,B, C, D}. Then one can easily prove thatyi(p) < ∞ for all i if all the eigenvalues of A havenegative real parts; that is to say, if

α(p) < 0 (11)

where α(p) denotes the spectral abscissa of A, givenby

α(p) , maxi{Re λi(A)}

and λi(A) denotes an eigenvalue of A. Notice that(11) is in fact equivalent to many well-known stabil-ity conditions for finite dimensional systems, whichshows a close relationship between the finiteness ofyi(p) and the stability property of the system.


Fig.1: Power system configuration

In practice, especially when used in conjunctionwith the MoI, condition (11) is usually replaced with

α(p) ≤ −ε0 (12)

where a small positive bound ε0 is given (for example,ε0 = 10−4 is used here).

Accordingly, a stability point can readily be ob-tained by solving the inequality (12). Once the sta-bility point is obtained, the inequality (11) or (12) canalso be used to prevent the algorithm from steppingout of the stability region during the search process.See [14, 15, 21] for more details.

3. MODEL OF POWER SYSTEM

The power system considered in this study is mod-elled as a single generator connecting to an infinitebus with an SVC installed at bus 2. The single linediagram of the system is shown in Figure 1 where iTdenotes the generator current, eT the terminal volt-age of the generator, eB the voltage at the infinitebus, v2 the voltage at bus 2, vL the load voltage atbus 4, Ltr1 an inductance of the transformer connect-ing the generator to bus 2, Ltr2 an inductance of thetransformer connecting the bus 2 to bus 4 and Ll aninductance of the transmission line connecting bus 2to the infinite bus.

By assuming that the power system operates undera three-phase balanced condition, it follows that

eT (t) = ET (t) sin(ωt + ϕ1)v2(t) = V2(t) sin(ωt + ϕ2)vL(t) = VL(t) sin(ωt + ϕ3)iT (t) = IT (t) sin(ωt + ϕ4)

(13)

where ϕi (i = 1, . . . , 4) are the phase angles of eT , v2,vL and iT at the steady state.

Let (ET,d, V2,d, VL,d, IT,d) and (ET,q, V2,q, VL,q,IT,q) denote the d- and q-components of (ET , V2, VL,IT ), respectively. One can readily show that the vari-ables in the abc reference frame is related to the dqoreference frame by

ET (t) =√

E2T,d(t) + E2

T,q(t)

V2(t) =√

V 22,d(t) + V 2

2,q(t)

VL(t) =√

V 2L,d(t) + V 2

L,q(t)

IT (t) =√

I2T,d(t) + I2

T,q(t)

. (14)

The power system considered here consists of agenerator, an excitation system, a governor con-trol loop and an SVC. In the following subsections,the mathematical models of each component are de-scribed, grouped as an interconnected system, andlinearized about a nominal operating condition.

3.1 The synchronous generator model

The dynamics of the generator is described by thefollowing nonlinear differential equations [22].

dE′qdt = 1

T ′d0(−E′

q + (Xd −X ′d)IT,d + Efd)

dE′ddt = 1

T ′q0(−E′

d + (Xq −X ′q)IT,q)

dδdt = ωb∆ω

d∆ωdt = 1

2H (Tm − Te −KD∆ω)

Te = E′dIT,d + E′

qIT,q + (X ′q −X ′

d)IT,dIT,q

(15)where E′

d and E′q are the transient electromotive

forces, T ′d0 and T ′q0 are the open circuit field timeconstants, Xd and Xq are the reactances, X ′

d and X ′q

are the transient reactances, Efd is the field voltage,δ is the rotor angle of the generator, ∆ω is the devi-ation of the angular speed of the generator from thesynchronous speed, ωb is the base angular speed, His the inertia constant, Tm is the mechanical inputtorque, Te is the electrical input torque, and KD isthe damping coefficient of the generator.

3.2 The excitation system

The excitation system comprises a voltage trans-ducer, an automatic voltage regulator (AVR) and apower system stabilizer (PSS) [22]. The block dia-gram of the excitation system is given in Figure 2and its state-space representation is

dm1dt = 1

TR(ET −m1)

dm2dt = Ks

d∆ωdt − 1

TWm2

dm3dt = 1

T2(T1

dm2dt + m2 −m3)

dm4dt = 1

T4(T3

dm3dt + m3 −m4)

Efd = KA(Vref −m1 − vs)

(16)

where mi and Ti (i = 1, . . . , 4) denote the state vari-ables and the lead-lag time constants of the excitationsystem, TR is the transducer time constant, TW is thewashout time constant, Ks is the PSS gain, and vs isthe output signal of the PSS described by

vs =

vmins , m4 < vmin

s

m4, vmins ≤ m4 ≤ vmax

s

vmaxs , m4 > vmax

s

(17)

where vmins and vmax

s are some constants.See Section 5 for further discussion on how to ob-

tain the PSS parameters Ks, TW and Ti (i = 1, . . . , 4)as shown in Table 1.


3.3 The governor control model

Figure 3 shows the block diagram of the governorcontrol loop. The differential equations describingthis subsystem are given by

dg1dt = 1

TG(∆ω

R − g1)dg2dt = 1

TP(g1 − g2)

Tm = Tref − g2

(18)

where gi (i = 1, 2) are the state variables, KI is theintegrator gain, R is the droop constant, TG is thegovernor time constant, and TP is the prime movertime constant.

3.4 The SVC model

The SVC model used in this study is taken from[23] and shown in Figure 4. The dynamic behaviourof the SVC is described by

ds1dt = 1

Tv(Kv(V20 − V2)− s1)

ds2dt = 1

Tv2(Tv1

ds1dt + s1 − s2)

dBdt = 1

Tb(Bref −B)

(19)

where V20 is the voltage of bus 2 at the steady state,Kv is the SVC gain, Tv is the SVC time constant,s1 and s2 are state variables, Tv1 and Tv2 are thelead-lag time constants, Tb is the thyristor firing timeconstant, B is the susceptance of the SVC and Bref isthe output susceptance of the voltage regulator givenby

Bref =

Bmin, s2 < Bmin

s2, Bmin ≤ s2 ≤ Bmax

Bmax, s2 > Bmax

. (20)

It is worth noting that, in order to ensure that thePSS and SVC models always operate in the linearranges, the control signals m4 and s2 need to satisfy

vmins ≤ m4(t) ≤ vmax

s

Bmin ≤ s2(t) ≤ Bmax

}for all t (21)

where vmins , vmax

s , Bmin and Bmax are specified con-stants.

3.5 The overall system equations

Owing to the very fast transient responses of thetransmission network [22], it suffices to represent the

KssTW

1+sTW

1+sT11+sT2

1+sT31+sT4

KA1

1+sTR

g

- - - -6- --- ?

££

PSS

∆ω m2 m3 vsm4

vmins

vmaxs

Vref

m1ETEfd+

−+

Fig.2: Block diagram of excitation system

1R

11+sTG

11+sTP

h- - - - ? -∆ωg1 g2

Tref

Tm

+−

Fig.3: Block diagram of governor loop

transmission network with the algebraic equation

I(x, V ) = YNV (22)

where x is the state vector of the system, V is the busvoltage vector and I is the current injection vector.

Accordingly, the overall system equations, includ-ing the differential equations for all the devices andthe algebraic equations for the transmission network(22), are expressed as

dx

dt= h(x, V ) (23)

where h is a vector of corresponding nonlinear func-tion.

Define the output vector of interest y, the inputvector u and the state vector x as follows:

y ,[δ ET V2 m4 s2

]T

u , VL∠θ

x ,[δ ∆ω E′

q E′d m1 m2

m3 m4 g1 g2 s1 s2 s3

]T

(24)

where VL∠θ is the phasor of the voltage of bus 2.The generator parameters (in per unit with respect

to 2220 MVA base) are shown in Table 1, and thepower system is in the steady state with the followingconditions:

P0 = 0.9 pu, Q0 = 0.436 pu,

ET0 = 1.0 pu, EB0 = 0.9 pu.

By applying the steady-state analysis given in [22],one obtains a nominal operating point

x0 =[δ0 ∆ω0 E′

q0 E′d0 m10 m20

m30 m40 g10 g20 s10 s20 s30

]T (25)

where δ0 = 46.26◦, E′q0 = 0.3170 pu, E′

d0 =1.0875 pu, all others are zero and the nominal voltageof the load bus VL0 = 0.732 pu.

Equation (23) is linearized about the nominal op-erating point given in (25) and the incremental linearmodel is given by

d∆xdt = A∆x + B∆u

∆y = C∆x + D∆u(26)

h Kv

1+sTv

1+sTv11+sTv2

11+sTb

-6

- - - -¢¢V20

V2

+−s1 s2 Bref

Bmin

Bmax

B

Fig.4: Block diagram of SVC


Table 1: Generator parameters in per unit baseKD = 10 H = 3.5 sec

KA = 100 Ks = 19.03

Ltr = 0.15 pu Ll1 = 0.5 pu

Ll2 = 0.93 pu T ′d0 = 8.0 sec

T ′q0 = 1.0 sec TG = 0.2 sec

TP = 0.3 sec TR = 0.015 sec

TW = 0.844 sec T1 = 0.137 sec

T2 = 0.058 sec T3 = 0.225 sec

T4 = 0.158 sec R = 0.05

Xd = 1.81 pu Xq = 1.76 pu

Rs = 0.003 pu X ′d = 0.30 pu

X ′q = 0.65 pu ωb = 120π rad/sec

where A, B, C and D are constant matrices, the in-cremental output vector ∆y is defined by

∆y = [∆δ ∆ET ∆V2 ∆m4 ∆s2] (27)

and ∆u, ∆x are the incremental input and state vec-tors respectively.

4. CHARACTERIZATION OF POSSIBLESET

This section describes how to characterize the pos-sible set P, which contains all the possible load volt-age fluctuations in the power system. From the pre-vious section, one can see that the input in the lin-earized model is the incremental load voltage ∆VL.Assume that the power system is in the steady statefor t ≤ 0 and the load voltage changes for t > 0. Thatis,

vL(t) ={

VL0 sin(ωt + θ), t ≤ 0(VL0 + ∆VL(t)) sin(ωt + θ), t > 0 (28)

where VL0 is constant and depends upon the nominaloperating condition of the system. Accordingly, theset P is defined as

P ,{

∆VL : ‖∆VL‖∞ ≤M, ‖∆VL‖∞ ≤ D}

(29)

Governor Loop

Excitation System

(AVR & PSS)

Generator

Network

SVC

-

-

--

-

-

-

-

-

-

-

¾

-

¤¡

¤¡

•

•

Efd

Tm

VL

ET

∆ω

δ

IT

V2BSVC

Fig.5: Configuration of the power system

0 100 200 300 400 500−0.2

−0.1

0

0.1

0.2

time (sec)

∆V

L(p

u)

M = 0.2 pu,D = 0.2 pu/s

Fig.6: Test input waveform

where M and D are positive numbers specified bydesigners.

At this point, the possible set P is to be defined.Accordingly, the bound M is chosen with respect tothe change of real power of the load bus that usedto happen or is likely to happen in practice. Thebound D is related to the rate of change of the powerflow from the transmission network to the fluctuatingload. In this study, the bounds used are chosen, forexample, by

M = 0.2 pu and D = 0.2 pu/s. (30)

To shed some light on the physical meaning ofM = 0.2 pu, suppose that the system was in sinu-soidal steady-state. By performing load-flow calcu-lation, it is found that if the load voltage decreased(or increased) 0.2 pu from its nominal value, then thereal power would be drawn from (or injected into) theload bus by 275.66 MW (or 210.90 MW).

5. DESIGN FORMULATION

This section explains how the design problem isformulated according to the PoM and the MoI.

Assume that the power system is subjected to theincremental load voltage ∆VL(t) for t > 0 where ∆VL

is characterized by (28)–(30). The SVC will be usedfor improving the performance of the power systemwith the signal V2 being fed back to the SVC. Theconfiguration of the feedback control system is shownin Figure 5.

To ensure good performance for the power system,the design problem considered here is to determinethe design parameters so that the following require-ments are fulfilled for any disturbance ∆VL ∈ P andfor all time t ≥ 0.

R1. The incremental rotor angle (∆y1) should notbe too large so as to ensure that the system has asufficient amount of damping and that the generatoris always in stable operation (that is, the rotor an-gle δ does not reach the stability limit). From thelinearization in Section 3, δ0 = 46.26◦ and thus weset

ε1 = 20◦. (31)

R2. The incremental terminal voltage of the genera-tor (∆y2) and the incremental voltage of the nearby


bus (∆y3) should remain strictly within ±3% and±5%, respectively, so that both the generator andthe nearby bus have good voltage regulation. Hence,

ε2 = 0.03 pu and ε3 = 0.05 pu. (32)

R3. The signals ∆m4 and ∆s2 (defined as ∆y4 and∆y5, respectively) should remain within their linearranges of operation so as to ensure that the linearmodel used is valid throughout the design process.Accordingly, the bounds ε4 and ε5 are chosen as

ε4 = 0.2 pu and ε5 = 1 pu. (33)

From the above discussion, the principal designspecifications to be considered in Section 6 are

∆y1 ≤ ε1

∆y2 ≤ ε2

∆y3 ≤ ε3

∆y4 ≤ ε4

∆y5 ≤ ε5

(34)

where ∆yi denotes the majorant of ∆yi and is givenby the formula (7), ∆yi is the peak value of the output∆yi for the possible set P in (29)–(30), the bound εi

is given in (31)–(33). Notice that (34) clearly showsthat the design problem is multiobjective.

6. NUMERICAL RESULTS

This section presents the numerical results of thedesign problem formulated in Section 5.

To verify the design results, a test input ∆V ∗L is

generated randomly as a piecewise linear functionsuch that ∆VL(0) = 0 pu, ||∆VL||∞ = 0.2 pu and||∆VL||∞ = 0.2 pu/s. Its waveform is displayed inFigure 6. All the simulations in this section are per-formed by using the actual nonlinear systems.

In this work, the inequalities (34) are solved by anumerical search algorithm called the moving bound-aries process (MBP). See [10, 14] for details of thealgorithm. Note, in passing, that other algorithmsfor solving inequalities can be found, for example, in[24, 25].

6.1 Tuning the SVC by the PoM and the MoI

Before employing the SVC, we attempt to tune thePSS in the excitation system so that the followingdesign criteria are satisfied.

∆y1(p) ≤ ε1

∆y2(p) ≤ ε2

∆y3(p) ≤ ε3

∆y4(p) ≤ ε4

(35)

where p = [Ks, TW , T1, T2, T3, T4] denotes the vec-tor of the PSS parameters.

After a large number of trials, it appears that theMBP algorithm cannot a vector p ∈ R6 satisfying

(35). This may indicate that the inequalities (35) areunlikely to have a solution. By performing a numberof iterations, the best possible p that can be found is

p = [19.03, 0.844, 0.137, 0.058, 0.225, 0.158] (36)

and the corresponding performance measures of thesystem are

∆y1(p) = 14.62◦ (≤ 20◦)∆y2(p) = 0.023 pu (≤ 0.03 pu)∆y3(p) = 0.092 pu (> 0.05 pu)∆y4(p) = 0.015 pu (≤ 0.2 pu)

. (37)

The PSS parameters in (36) are listed in Table 1 andused throughout the paper.

The time responses ∆yi due to the test input ∆V ∗L

for the system without the SVC are plotted in Figure7. It can be seen from (37) and Figure 7 that only therequirement on the voltage regulation of the nearbybus cannot be achieved. Hence, the use of SVC isinevitable.

Now, the SVC (19) is installed at the nearby bus.Let p ∈ R4 denote the design parameter vector of theSVC and be given by

p1 ,[Kv Tv Tv1 Tv2

]T. (38)

The SVC parameters are determined by solving theinequalities (34). By starting from p1 = [0, 0, 0, 0]T

and after a number of iterations, the MBP algorithmlocates a design solution p1 where

p1 =[28.0 0.150 0.169 0.033

]T (39)

and the corresponding performance measures are

∆y1(p1) = 5.16◦ (≤ 20◦)∆y2(p1) = 0.008 pu (≤ 0.03 pu)∆y3(p1) = 0.034 pu (≤ 0.05 pu)∆y4(p1) = 0.004 pu (≤ 0.2 pu)∆y5(p1) = 0.94 pu (≤ 1 pu)

. (40)

Figure 8 shows the time responses ∆yi due to thetest input ∆V ∗

L for the system with the SVC (39). Itis clear from (40) and Figure 8 that the obtained SVCnot only significantly reduces the fluctuation in thevoltage of the near by bus and the terminal voltage,but also provides a better damping to the system.

6.2 Comparison with the design obtained bythe MoI

To illustrate what can go wrong when the PoMis not used, the design p1 in (39) will be comparedwith another design obtained by using the MoI inconjunction with the step-response criteria.

In the following, assume that the incremental loadvoltage ∆VL is a step disturbance given by

∆VL(t) = 0.2 pu for t ≥ 0. (41)


0 100 200 300 400 500−20

0

20

time (sec)

0 100 200 300 400 500

−0.030

0.03

time (sec)

0 100 200 300 400 500

−0.050

0.05

time (sec)

0 100 200 300 400 500−0.2

0

0.2

time (sec)

Fig.7: Responses of the system without SVC due tothe test input ∆V ∗

L

Note that ∆VL does not belong to the possible setP. The step responses of the power system with theSVC (39) are shown in Figure 9.

For the purpose of comparison, define φi (i =1, 2, . . . , 5) as the two norm of the transient devia-tion of the step response of ∆yi. That is to say,

φi ,√∫ ∞

0

{Yi(t)− Yi,ss}2 dt (42)

where Yi denotes the step response of ∆yi due to the∆VL given in (41) and Yi,ss denotes the steady-statevalue of Yi.

Then the MoI is used to determine another set ofthe SVC parameters, denoted by p2, so that φi(p2) isnot worse than φi(p1) for every i where p1 is givenin (39). That is to say, p2 is obtained by solving thefollowing criteria:

φ1(p2) ≤ φ1(p1) , 1.6535φ2(p2) ≤ φ2(p1) , 0.0029φ3(p2) ≤ φ3(p1) , 8.8515× 10−2

φ4(p2) ≤ φ4(p1) , 7.4501× 10−3

φ5(p2) ≤ φ5(p1) , 0.0123

. (43)

By using the MBP algorithm, the following solu-tion p2 is obtained where

p2 =[

31.7 0.21 0.142 0.016]T (44)

and the corresponding φi(p2) are given by

φ1(p2) = 1.6314φ2(p2) = 0.0028φ3(p2) = 8.1724× 10−2

φ4(p2) = 7.1697× 10−3

φ5(p2) = 0.0117

. (45)

0 100 200 300 400 500−20

0

20

time (sec)

0 100 200 300 400 500

−0.030

0.03

time (sec)

0 100 200 300 400 500

−0.050

0.05

time (sec)

0 100 200 300 400 500−0.2

0

0.2

time (sec)

0 100 200 300 400 500

−1

0

1

time (sec)

Fig.8: Responses of the system with SVC (39) dueto the test input ∆V ∗

L

The step response of the system with the SVC (44)are displayed in Figure 10.

From (45), it is evident that the step-response ofthe system with p2 is slightly better than the responseof the system with p1 in the sense of the performancemeasures φi. By comparing Figures 9 and 10, the stepresponses of both systems are not much different.

In connection with the actual specifications (34),we compare the performance of the SVC (39) withthat of the SVC (44). To this end, the time responses∆yi of the system with the SVC (44) due to the testinput ∆V ∗

L are plotted in Figure 11. From this, it fol-lows that when the SVC (44) is used, the incrementalvoltage of the nearby bus violates the requirementR2, and thereby the specifications (34) are not sat-isfied. From Figures 8 and 11, one can see that byusing only the MoI with the step-response criteria,the specifications (34) are not explicitly taken intoaccount in formulating the design problem. In thiscase, it is very difficult to determine the design pa-rameters so that (34) are satisfied. By contrast, theproblem can be solved effectively by using the PoMin conjunction with the MoI. Accordingly, the valueof the design method used here is evident.

7. CONCLUSIONS

This paper describes a general procedure for de-signing a device such as an SVC for power systemssubject to load voltage fluctuation by Zakian’s frame-work, comprising the PoM and the MoI. The PoMprovides a set of design inequalities that can be used


0 0.5 1 1.5 2 2.5−5

0

5

time (sec)

∆δ

(deg)

0 0.5 1 1.5 2 2.5−0.1

0

0.1

time (sec)

∆E

T(p

u)

0 0.5 1 1.5 2 2.5−0.1

0

0.1

time (sec)

∆V

2(p

u)

0 0.5 1 1.5 2 2.5−0.02

0

0.02

time (sec)

∆m

4(p

u)

0 0.5 1 1.5 2 2.5−2

−1

0

time (sec)

∆s2

(pu)

Fig.9: Step responses of the system with SVC (39)

in practice to ensure the satisfaction of the criteria(2). The resultant inequalities then give rise to themulti-objective design problem that are solved effec-tively by the MoI. The framework facilitates design-ers to arrive at a realistic formulation of the designproblem.

The study in the paper shows that the frameworkcan effectively handle the design problem in whichthe system possesses conditionally linear elements.Specifically, in spite of the saturations in the SVC andthe excitation system, the power system with the ob-tained design solution is guaranteed to always operatewithin the linear ranges of the devices. Consequently,ubiquitous theories for linear systems can be utilizedin the analysis and the design.

The numerical results clearly show that all the out-puts ∆yi of interest (namely, the rotor angle, the ter-minal voltage and the voltage of the nearby bus) canbe ensured to remain strictly within the prescribedbounds as long as the incremental load voltage ∆VL

belongs to the possible set P. Evidently, the criteria(29) or (2) are realistic in the senses that they are ac-tually used by the plant engineers to monitor the per-formance of the system in practice, and that all theinputs that happen in reality are explicitly taken intoaccount in the formulation. It is interesting to note,however, that in power systems operation, the rotorangle, the generator terminal voltage and the nearbybus voltage can be allowed to exceed the bounds fora short period of time. In this case, the use of thecriteria (2) may cause some conservatism, which canusefully provide a safety margin for the design pro-

0 0.5 1 1.5 2 2.5−5

0

5

time (sec)

∆δ

(deg)

0 0.5 1 1.5 2 2.5−0.1

0

0.1

time (sec)

∆E

T(p

u)

0 0.5 1 1.5 2 2.5−0.1

0

0.1

time (sec)

∆V

2(p

u)

0 0.5 1 1.5 2 2.5−0.02

0

0.02

time (sec)

∆m

4(p

u)

0 0.5 1 1.5 2 2.5−2

−1

0

time (sec)

∆s2

(pu)

Fig.10: Step responses of the system with SVC (44)

vided that it is not too excessive.The comparison in Section 6.2 shows what can go

wrong if the design problem is formulated by usingthe step-response criteria, which do not represent theactual requirements (34).

In this work, the possible inputs are considered inthe form of the deviation of the voltage at the load busfrom its nominal value. However, it is worth pointingout that in power systems operation, the informationof the real power and the reactive power of the load iseasily obtainable (for example, in the case of EAF’s).Because of the potential of the framework, it may befruitful to develop in the future a practical methodfor determining the bounds M and D of ∆VL due tothe load fluctuation using such information so thatthe possible inputs are characterized more accuratelyand the conservatism can therefore be reduced.

ACKNOWLEDGMENT

Kittichai Tia would like to acknowledge withthanks Chulalongkorn University scholarship of theHonor Graduate Program for Electrical EngineeringStudents. The authors would like to thank the anony-mous referee(s) whose valuable suggestions helped toimprove the presentation of the paper.

References

[1] C.-J. Wu, Y.-S. Chuang, C.-P. Huang and Y.-P.Chang (2007). Tuning of static excitation sys-tems to improve dynamic performance for gen-erators serving electric arc furnaces loads, IEEE


0 50 100 150 200 250 300 350 400 450 500−20

0

20

time (sec)

∆δ

(deg)

0 50 100 150 200 250 300 350 400 450 500

−0.03

0

0.03

time (sec)

∆E

T(pu)

0 50 100 150 200 250 300 350 400 450 500

−0.050

0.05

time (sec)

∆V

2(pu)

0 50 100 150 200 250 300 350 400 450 500−0.2

0

0.2

time (sec)

∆m

4(pu)

0 50 100 150 200 250 300 350 400 450 500

−1

0

1

time (sec)

∆s2

(pu)

Fig.11: Responses of the system with SVC (44) dueto the test input ∆V ∗

L

Transactions on Energy Conversion, vol. 22, pp.350–357.

[2] C. S. Chen, H. J. Chuang, C. T. Hsu, S. M.Tseng (2001). Mitigation of voltage fluctuationfor an industrial customer with arc furnace, inproceedings of IEEE Power Engineering SocietySummer Meeting, vol. 3, pp. 1610–1615.

[3] L. Zhang, Y. Liu, C. Han, Z. Du, A. Q. Huangand M. R. Ingram (2006). Mitigation of EAFinduced problems, in proceedings of IEEE PESPower Systems Conference and Exposition, pp.1448–1453.

[4] A. R. Mahran, B. W. Hogg and M. L. El-Sayed(1992). Co-ordinated control of synchronous gen-erator excitation and static var compensator,IEEE Transactions on Energy Conversion, vol.7, pp. 615–622.

[5] P. Rao, M. L. Crow and Z. Yang (2000). STAT-COM control for power system voltage controlapplications, IEEE Transactions on Power De-livery, vol. 15 , pp. 1311–1317.

[6] P. S. Sensarma, K. R. Padiyar and V. Ra-manarayanan (2001). Analysis and performanceevaluation of a distribution STATCOM for com-pensating voltage fluctuations, IEEE Transac-tions on Power Delivery, vol. 16, pp. 259–264.

[7] Q. Zhao and J. Jiang (1995). Robust SVC con-troller design for improving power system damp-ing, IEEE Transactions on Power Systems, vol.10, pp. 1927–1932.

[8] R. You and M. H. Nehrir (2004). A systematic

approach to controller design for SVC to enhancedamping of power system oscillations, in proceed-ings of IEEE PES Power System Conference andExposition, vol. 2, pp. 1134–1139.

[9] V. Zakian (1996). Perspectives of the principleof matching and the method of inequalities, In-ternational Journal of Control, vol. 65, pp. 147–175.

[10] V. Zakian, editor (2005). Control Systems De-sign: A New Framework, Springer-Verlag, Lon-don.

[11] K. Tia, S. Arunsawatwong and B. Eua-arporn(2009). Design of compensators for power sys-tems operating under load voltage fluctuationsatisfying bounding conditions, in proceedings ofECTICON 2009, pp. 252–255.

[12] J. F. Whidborne (1993). EMS control system de-sign for a maglev vehicle – A critical system, Au-tomatica, vol. 29, pp. 1345–1349.

[13] S. Arunsawatwong (2005). Critical control ofbuilding under seismic disturbance, Chapter 13in V. Zakian (editor), Control Systems Design:A New Framework, Springer-Verlag, London.

[14] V. Zakian and U. Al-Naib (1973). Design of dy-namical and control systems by the method ofinequalities, Proceedings of the IEE, vol. 120, pp.1421–1427.

[15] V. Zakian (1979). New formulation for themethod of inequalities, Proceedings of the IEE,vol. 126, pp. 579–584.

[16] V. Zakian (1991). Well matched systems, IMAJournal of Mathematical Control and Informa-tion, vol. 8, pp. 29–38 (see also Corrigendum,1992, vol. 9, p. 101).

[17] B. J. Birch and R. Jackson (1959). The be-haviour of linear systems with inputs satisfy-ing certain bounding conditions, Journal of Elec-tronics and Control, vol. 6, pp. 366–375.

[18] I. Horowitz (1962). Analysis and synthesis oflinear systems with inputs satisfying certainbounding conditions, Journal of Electronics andControl, vol. 12, pp. 195–208.

[19] P. G. Lane (1992). Design of Control Sys-tems with Inputs and Outputs Satisfying Cer-tain Bounding Conditions, PhD thesis, UMIST,Manchester, UK.

[20] W. Silpsrikul and S. Arunsawatwong (2010).Computation of peak output for inputs satisfy-ing many bounding conditions on magnitude andslope, International Journal of Control, vol. 83,pp. 49–65.

[21] V. Zakian (1987). Design formulations, Interna-tional Journal of Control, vol. 46, pp. 403–408.

[22] P. Kunder (1994). Power System Stability andControl, McGraw-Hill, New York.

[23] IEEE Special Stability Controls Working Groups(1994). Static var compensator models for powerflow and dynamic performance simulation, IEEE


Transactions on Power Systems, vol. 9, pp. 229–240.

[24] J. F. Whidborne (2005). A simulated annealinginequalities solver, Chapter 7 in V. Zakian (ed-itor), Control Systems Design: A New Frame-work, Springer-Verlag, London.

[25] T.-K. Liu and T. Ishihara (2005). Multi-objective genetic algorithms for the method ofinequalities, Chapter 8 in V. Zakian (editor),Control Systems Design: A New Framework,Springer-Verlag, London.

Suchin Arunsawatwong received hisB.Eng. and M.Eng. degrees in electri-cal engineering from Chulalongkorn Uni-versity, Thailand, in 1985 and 1988, re-spectively, and Ph.D. degree in controlengineering from the Control SystemsCentre, University of Manchester Insti-tute of Science and Technology, U.K., in1995. He is currently an assistant pro-fessor at the Department of ElectricalEngineering, Chulalongkorn University.

His main research interests include delay differential systems,numerical solution of differential equations, and control sys-tems design by the method of inequalities and the principle ofmatching.

Kittichai Tia received the B. Eng. andM. Eng. degrees in electrical engineeringfrom Chulalongkorn University, Thai-land in 2007 and 2009, respectively. Heis currently an engineer at Power Gen-eration Systems Control Department,Power System Company, Toshiba Cor-poration, Japan. His work responsibilityis the design of Automatic Voltage Reg-ulator (AVR) for power plant. His cur-rently undertaking projects include nu-

clear power plants in Japan, thermal and nuclear power plantsin various countries outside Japan.

Bundhit Eua-arporn received theB.Eng. and M.Eng. degrees in electricalengineering from Chulalongkorn Univer-sity, Thailand. In 1992, he received thePh.D. degree from Imperial College ofScience Technology and Medicine, Lon-don, U.K. He is currently a professorat the Electrical Engineering Depart-ment, and the director of the Energy Re-search Institute, Chulalongkorn Univer-sity, Thailand.

design of static var compensator for power systems … · design of static var compensator for...

Documents