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Available online at www.sciencedirect.com

Electric Power Systems Research 78 (2008) 484–493

Decentralized adaptive backstepping control of electric power systems

Ali Karimi, Ali Feliachi ∗Advanced Power & Electricity Research Center (APERC), West Virginia University, Morgantown, WV 26506-6109, USA

Received 24 January 2007; received in revised form 10 April 2007; accepted 11 April 2007Available online 30 May 2007

bstract

In this paper, a decentralized adaptive backstepping excitation controller, tuned using a Particle Swarm Optimization technique (PSO), is designedor stability enhancement of multi-machine power systems. To achieve decentralization, each machine is modeled as an independent uncertainynamic subsystem, where the uncertainty is a disturbance that represents the effects of the rest of the system on that particular machine. This

isturbance is expressed as a polynomial function of electric power deviation, and its parameters are adapted using PSO. The proposed technique isllustrated with a two-area benchmark power system. This system exhibits inter-area oscillations which are effectively damped with the proposedecentralized controllers under severe contingencies, for which traditional power system stabilizers fail.

2007 Elsevier B.V. All rights reserved.

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eywords: Power systems; Adaptive backstepping; Transient stability; Particle

. Introduction

Stability has been a major concern in power system for sev-ral years. As the complexity of system grows, the challengeo design controllers grows as well [1]. Linear controllers withead-lag structure are widely used either with remote or local

easurements inputs. These controllers are used to produceupplementary control signals for oscillation damping purposes2]. Since the design is based on linearized models of complexonlinear system, lead-lag controllers may not always guaran-ee stability especially when severe contingencies occur. Hence,onlinear control design techniques might be the only choice.n power systems, significant results were obtained by imple-enting different nonlinear controllers. Some, but not all, are

eviewed here starting with the tools implemented on simpleingle-machine-infinite-bus (SMIB) systems. Bazanella et al. [3]esigned state feedback control based on a Lyapounov approach,nown as LgV , to improve dynamic performance of the sys-

em. This controller requires the internal voltage of generator,hich is not measurable. More sophisticated and yet realistic

ontrol design technique, known as backstepping, have been

∗ Corresponding author. Tel.: +1 304 293 0405x2529; fax: +1 304 293 8602.E-mail addresses: akarimi@mix.wvu.edu, alfeliachi@mail.wvu.edu

A. Karimi).

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378-7796/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2007.04.003

m Optimization

eveloped based on Krstic’s approach [4–7]. First applicationn power systems is given in [8] for SMIB system with theoal of improving both transient stability and voltage regulation.lthough the control gains are obtained through trial and errorithout any optimization procedure, the results demonstrated

he effectiveness of this novel control technique.For multi-machine power systems, direct feedback lineariza-

ion (DFL) has been utilized. However, DFL has complex controlaws because their design is based on accurate system model toompensate for system nonlinearities [9,10]. Investigations werextended to adaptive nonlinear control for large disturbancesith structured uncertainties. Adaptive techniques were used to

ugment the DFL method [11,12]. Different adaptive versions ofeedback linearization control were presented by Jain et al. [13].n their analysis, equivalent reactances of the transmission linesere considered as unknown or varying parameters, then adap-

ation is used to estimate them and achieve an exact cancellationf terms by feedback linearization. The technique was imple-ented on a two-generator-infinite-bus system. Their approach

as been extended to a class of nonlinear systems with decentral-zed output feedback control, where the interconnection termsere expressed by polynomials [14].

Okou et al. [15,16] presented a hierarchical control structure

ased on wide area signals using input–output linearization andarameter adaptation. However, both local and remote signalsre assumed to be available. Local controllers dampen local rotor

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A. Karimi, A. Feliachi / Electric Pow

haft oscillations and a centralized controller decouples subsys-ems’ interactions. Final control is obtained with two objectivesne on voltage and the second on rotor speed regulator. Widerea control laws are derived from a reformulation of the multi-achine model, generator terminal voltages are used as state

ariables instead of internal field voltages, through complexransformations.

In previous work, the authors have designed three other back-tepping controllers using different information [17,18], whichre centralized controllers, or [19] which handles the interfaceariables differently. The one proposed in this paper is com-letely decentralized and the interface variables are trackedetter. Specifically, in this paper interface variables are estimatedy a polynomial of electric power deviations which are availableocally at each generating unit. The adaptation laws are used tostimate the effects of the rest of the system on the particularachine, which allows for a decentralized control design. Per-

ormance of these controllers are compared with the proposedne in Section 4.2 of the paper.

The backstepping control design parameters and the adap-ation parameters are all obtained using a Particle Swarmptimization (PSO) search technique [20]. Alrashidi and El-awary have written a detailed survey on PSO applications

o large scale power systems [21]. PSO has been used byesearchers for tuning supplementary damping control withigenvalue-based objective functions to enhance system damp-ng of electromechanical modes [22,23]. But in this paper, the

odel is not linearized. The objective is to design nonlinearecentralized controllers that will guarantee asymptotically sta-le closed-loop system. This is demonstrated by obtaining ayapounov function for the closed-loop system.

The paper is organized as follows. In Section 2 a strict feed-ack generator model suitable for decentralized control design iseveloped. In Section 3 the adaptive backstepping control designlgorithm is presented. The control and estimation parametersre obtained using a Particle Swarm Optimization (PSO) searchechnique in Section 4. A case study using the inter-area oscil-ation benchmark two-area power system is given in Section 4.1o illustrate the effectiveness of the proposed controller and toompare it to power system stabilizers (PSS) that the authorsave previously published [24]. Comparison of proposed con-roller to all previous backstepping control designs are given inection 4.2. Sensitivity of proposed controller due to model-

ng errors are analyzed in Section 4.3. The simulation resultsre obtained using the Power Analysis Toolbox package (PAT)25]. Summary and conclusion are given in Section 5.

. Strict feedback generator model suitable forecentralized control design

To apply the design technique proposed in this paper, theenerator model is (1) cast in a strict feedback form [7], and (2)ach machine is modeled as an independent dynamic subsystem.

he starting point is the transient two-axis generator model given

n Appendix A. To obtain the strict feedback form model, electricower instead of direct and quadrature voltages is used as atate variable. The decoupling of the generator from the rest of

z

P

V

stems Research 78 (2008) 484–493 485

he system is obtained by considering the effect of the rest ofhe system on each generator as a disturbance. Therefore, eachenerator is modeled by the following state equations.

�δi = �ωi, �ωi = − Di

2Hi

�ωi − ωio

2Hi

�Pei,

Pei = −�Pei

1

T ′qoi

+ βi�Efldi+ di (1)

here the coupling term di is given by Eq. (51) in Appendix A.his term includes local and remote information. In this paper,

t is expressed as an uncertain polynomial function of electricower deviation, i.e. with parameters that will be estimated,hich are local informations:

i ≈ θ1i�Pei + θ2i�P2ei

(2)

here θ1i and θ2i are uncertain values which need to be estimatedhrough adaptation laws. In generic terms, the equation set (1)or ith generator is

˙1 = b1x2 (3)

˙2 = b2x3 + b3x2 (4)

˙3 = βu + b4x3 + d (5)

here disturbance d is

= θ1x3 + θ2x23 (6)

tate variables: x1 = �δ, x2 = �ω, x3 = �Pe. Parameters:1 = 1, b2 = −ω0/2H, b3 = −D/2H, b4 = −1/T ′

qoi. Controlnput: (u = �Efld) appears in the last equation. β, θ1, θ2 arencertainties, not known a priori. The quadratic polynomial esti-ate of the disturbance has given adequate results, and the global

ptimization that is performed later will counteract any errorsn this estimate.

. Adaptive backstepping control design

The objective is to stabilize the system (3)–(5) using back-tepping control which is to steer x1 to its desired value xd

1 =0 = constant, then find x2 to stabilize (3), and x3 to stabilize4) and finally u to stabilize (5) and hence the overall system.herefore, x2 and x3 will have virtual trajectories α1 and α2.efine the error variables:

i = xi − αi−1, i = 1, 2, 3 (7)

he problem then is to find α1, α2 and u to drive the errorariables zi to zero.

.1. Step 1: Find α1

Consider the dynamics of z1, using (7) and (3):

˙1 = x1 − α0 = x1 = b1x2 = b1(α1 + z2) (8)

ick the following Lyapounov function for this subsystem

1 = 12z2

1 (9)

4 er Sy

i

V

C

α

w

α

t

V

k

s

3

T

z

C

V

i

V

u

V

C

α

w

α

α

t

V

k

s

3

z

w

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z

C(

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wit

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w

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α

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86 A. Karimi, A. Feliachi / Electric Pow

ts derivative along the trajectory, using (7), is

˙1 = z1z1 = b1α1z1 + b1z2z1 (10)

hoose

1 = α11z1 = −k1z1

b1(11)

here

11 = −k1

b1(12)

hen

˙1 = −k1z21 + b1z2z1 (13)

1 and z2 will be chosen later to make V1 < 0 and hence sub-ystem (8) asymptotically stable.

.2. Step 2: Find α2

Consider the dynamics of z2, using (7), (4), (11), and (8).herefore

˙2 = x2 − α1, z2 = b2x3 + b3x2 + k1z1

b1

= b2x3 +(

−k21

b1− k1b3

b1

)z1 + (b3 + k1)z2 (14)

onsider the following Lyapounov function for (8) and (14):

2 = V1 + 12 (z2)2 (15)

ts derivative along the trajectory, using (10) and (14), is

˙2 = V1 + z2z2 (16)

sing (10) and (14) and let x3 = z3 + α2:

˙2 = −k1z21 + [b1z1 + b2α2

−(

k21

b1+ k1b3

b1

)z1 + (b3 + k1)z2

]z2 + b2z2z3 (17)

hoose α2

2 = α21z1 + α22z2 (18)

here

21 = −b1

b2+ k2

1

b2b1+ k1b3

b1b2,

22 = −k1 + k2 + b3

b2(19)

hen, V2 becomes

˙2 = −k1z21 − k2z

22 + b2z2z3 (20)

1, k2, and z3 will be chosen later to make V2 < 0 and henceubsystem (8), (14) asymptotically stable.

Fv

u

stems Research 78 (2008) 484–493

.3. Step 3: Find u

Consider the dynamics of z3, using (7), (5), (18), (8) and (14):

˙3 = x3 − α2 = βu + b4x3 + d − α2 (21)

here d is given in (6) and α2 is obtained using (8) and (14):

˙ 2 = α21z1 + α22z2 = (−k1α21 − b1α22)z1

+(b1α21 − k2α22)z2 + b2α22z3 (22)

q. (21) becomes

˙3 = βu + (b4α21 + k1α21 + b1α22)z1

+(b4α22 − b1α21 + k2α22)z2 + (b4 − b2α22)z3 + d

(23)

onsider the following Lyapounov function for (8), (14) and23)

= V2 + 12 (z3)2 + 1

2 (β − β)2γ−1

+1

2[(θ1 − θ1)(θ2 − θ2)]Γ −1[(θ1 − θ1)(θ2 − θ2)]

T(24)

here θ1, θ2, and β are estimate of θ1, θ2, β. Γ = diag[Γ1, Γ2]s an adaptation gain matrix, and γ is a scalar positive value.hen

˙ = −k1z21 − k2z

22 + [βu + α31z1 + α32z2 + α33z3 + d]z3

−(β − β)γ−1 ˙β − (θ1 − θ1)Γ −1

1˙θ1 − (θ2 − θ2)Γ −1

2˙θ2

(25)

here

31 = b4α21 + k1α21 + b1α22,

32 = b2 − b1α21 + b4α22 + k2α22,

33 = b4 − b2α22 (26)

hoose adaptation laws ˙β,

˙θ1,

˙θ2 in (25) as

˙1 = Γ1z3x3,

˙θ2 = Γ2z3x

23,

˙β = γz3u (27)

q. (25) becomes

˙ = −k1z21 − k2z

22 + [βu + α31z1 + α32z2 + α33z3 + d]z3(28)

here d = θ1x3 + θ2x23. The controller u is then designed to

ake V < 0. This is achieved with the following controller andositive values k1, k2, k3:

= β−1[−α31z1 − α32z2 − α33z3 − k3z3 − d] (29)

ˆ �= 0 for normal operating conditions, and make the controllereasible. In fact, with this controller, V is expressed by

˙ = −k1z21 − k2z

22 − k3z

23 (30)

inally, this control law is written in terms of the original stateariables as

= β−1[F1x1 + F2x2 + F3x3 − d] (31)

er Sy

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F33 +

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A. Karimi, A. Feliachi / Electric Pow

here

1(k1, k2, k3) =[−α31 + α21α33 + k3α21 + −k1α32 + k1α22α

b1

2(k1, k2, k3) = [−α32 + α33α22 + k3α22], F3(k1, k2, k3)

his controller u(x1, x2, x3, θ1, θ2, β) (31) is a nonlinear func-ion that is affected by the choice of the control parameters:

= [ k1 k2 k3 k4 k5 k6 ] (33)

here k4 = Γ1, k5 = Γ2, k6 = γ

The setting of these control parameters or gains is the topicf the next section.

. Optimal settings of controllers gains

The controller designed in the previous section is done at eachachine or subsystem level called station i, and the control gains

33) are denoted Ki = [ ki1 ki2 ki3 ki4 ki5 ki6 ]. If theystem comprises n substations, then there are N = 6n controlarameters that need to be selected simultaneously so that eachubstation is asymptotically stable. A vector Lyapounov Func-ion for the entire system is taken as V = [ V1 V2 · · · Vn ]hich is positive definite and its derivative along the trajectory

s [ V1 V2 V3 · · · Vn ]. Vi is the derivative along the tra-ectory of the Lyapounov function of subsystem (i) given by30). Therefore, the goal is to minimize the objective function= [ J1 J2 · · · Jn ], where Ji = Vi for i = 1, 2, . . . , n, i.e.

o make each entry of J as negative as possible by tuning theontrol gains. The problem is formulated as follows:

mini such that Vi<0,i=1:n

‖J‖∞ (34)

iven the size of the problem, and the system complexities,his problem is solved here using the Particle Swarm Optimiza-ion (PSO) search, which is a heuristic search technique that isffective for large scale nonlinear systems [20]. The solutions assumed to lie in a range of an N-dimensional space, whereach potential solution is called a particle. It has a position and aelocity and moves in the search space toward an optimal solu-ion. Here the particles represent the control gains that are sought22,23,17,18]. Some definitions are given below:

Particle Ki(t): A candidate solution for ith controller at iter-ation t.Population: A set of n particles {K1(t), K2(t), . . . , Kn(t)},where n is total number of controllers.Individual bestK∗

i (t): This is the best value of the performanceindex J that this particle has ever achieved up to t th iteration.

K∗i (t) = {Ki(t) : Ji(K

∗i (t)) ≤ Ji(Ki(τ)), τ ≤ t},

J∗i (t) = Ji(K

∗i (t)) (35)

Global best K∗∗(t): Among all individual best positions

achieved so far, the best position for all particles is calledglobal best.

K∗∗(t) = {K∗i (t) : J(K∗∗(t))≤Ji(K

∗i (t)), i=1, . . . , n} (36)

stems Research 78 (2008) 484–493 487

k1k3α22]

,

α33 − k3] (32)

The steps of the PSO algorithm are

Step 1: Initialization(a) Given a number of controllers n, number of gains for each

controller m, maximum number of iterations tmax, max-imum number of performance evaluation counter cmax,admissible range for gains kmin

i,j , kmaxi,j

kmini,j ≤ ki,j ≤ kmax

i,j , i=1, 2, . . . , n, j = 1, 2, . . . , m

(37)

and number of intervals in jth dimension (μ). Set speedrange

−νmaxi,j ≤ νi,j ≤ νmax

i,j (38)

where

νmaxi,j = kmax

i,j − kmini,j

μ(39)

(b) Initialize position and speed using uniformly distributedrandom numbers, and evaluate the objective function Ji(t)of each particle.

(c) Let J∗i (t) = Ji(t), K∗

i (t) = Ki(t) and J∗∗(t, c) =[J∗

1 (t)· · ·J∗n (t)], K∗∗(t) = [K∗

1(t)· · ·K∗n(t)].

(d) Set t = t + 1, c = c + 1Step 2: Velocity Update

Velocity is updated by the following Eq. [26]:

νi,j(t) = Ψ [νi,j(t − 1) + ε1r1(k∗i,j(t − 1)

− ki,j(t − 1)) + ε2(k∗∗i,j(t − 1) − ki,j(t − 1))] (40)

Ψ = 2

|2 − ϕ −√

ϕ2 − 4ϕ|, where ϕ = ε1 + ε2, ϕ > 4

(41)

where positive constants ε1, ε2 are weighting factors, andr1, r2 are randomly generated numbers between 0 and 1. Incase the velocity violates its range, it will be set to its limit(38).Step 3: Position update

For each particle, update each gain using the velocity Eq.(40):

ki,j(t) = νi,j(t) + ki,j(t − 1) (42)

Update the position Ki(t) = [ki1, . . . , kim] for i = 1, . . . , n

Step 4: Performance evaluation

Using the updated position, evaluate the objective function:

[J1(t)· · ·Jn(t)] = [J1(K1(t))· · ·Jn(Kn(t))]

488 A. Karimi, A. Feliachi / Electric Power Systems Research 78 (2008) 484–493

nchm

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•

•

4

sdT

sTptcsaua

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•

Fig. 1. Two area be

Step 5: Individual best updateFind individual best using (35), i.e., find J∗

i (t) and theassociated K∗

i (t) for each i.Step 6: Global best update

Find global best using (36), i.e. find J∗∗(t, c) = J(K∗∗(t))and K∗∗(t). If:

J∗∗(t, c) ≤ J∗∗(t − 1, c − 1) (43)

then the objective function has not improved, the gains donot need to be updated, set c = c + 1 and go to next step.Otherwise, update J∗∗(t, c), K∗∗(t) and set c = 0. Then go tonext step.Step 7: Stopping criteria(a) If the best global solution J∗∗(t, c) can no longer be

improved and the counter has reached its maximum num-ber, c = cmax, the optimal solution is then the currentK∗∗(t). Exit.

(b) If t reaches (tmax), the maximum number of allowed itera-tions has been reached and no feasible solution was found,stop. Otherwise: t = t + 1 and go to Step 2.

.1. Two-area system

The system chosen is two-area power system [27] whosechematic is shown in Fig. 1. Synchronous generator and exciterata are given in Appendix A, Tables A.2 and A.3, respectively.he actuator consists of an exciter gain KA, an exciter time con-

Fig. 2. Supplementary controller (adaptive ba

ark power system.

tant TA, a filter time constant TR and a saturation hard limit.he proposed adaptive backstepping controller acting as sup-lementary signal to exciter (Fig. 2). Scenarios are presentedo illustrate the effectiveness of the proposed controller and toompare it to existing PSS. PSSs at generators 2 and 3 have beenuccessfully designed by the authors previously to damp localnd inter-area oscillations [24]. Hence, in this paper the samenits are being controlled. The following control strategies arenalyzed:

Two PSSs, whose transfer functions are given below, areimplemented on generators 2 and 3.

KwsTw

1 + sTw︸ ︷︷ ︸wash−out

(1 + sT1)2

(1 + sT2)2︸ ︷︷ ︸lead−lag

(44)

where Tw wash-out time constant, Kw wash-out gain, andT1, T2 are lead-lag time constants. PSS parameters are givenin Table 1.Two adaptive backstepping controllers, designed using theproposed approach, are implemented on generators 2 and 3.Gains, tuned by PSO, are given in Table 1. The scenarios thathave been analyzed are tabulated in Table 2. A three phase

fault is applied at bus 3 for each scenario. In scenario #1,simulation results show that relative rotor angle oscillations�δ31, �δ41 are damped fast enough, in less than 8 s (Fig. 3).In scenario #2, when one transmission line between buses

ckstepping or PSS) with static exciter.

A. Karimi, A. Feliachi / Electric Power Systems Research 78 (2008) 484–493 489

Table 1Adaptive backstepping and PSS control gains

Generator #2 Generator #3

Adaptive backsteppingk1 2.40 10.73k2 1.27 16.76k3 19.34 28.20k4 13.50 1.15k5 14.41 3.81k6 10.72 11.34

PSSTw 20 20K 2.36 15

F(

Table 2Analyzed scenarios for two-area system

Scenarios #

1 2 3

Transmission line (13–101) Double Single SingleFault occurrence time (s) 1.00 1.00 1.00FL

steady state and post-fault steady state are equal (Fig. 5).

w

T1 0.7109 0.15T2 0.155 0.0843

13–101 is removed, the system is under stress. Adaptive back-stepping controller dampen rotor oscillation in approximately8 s, while it takes almost twice that time for PSS to suppressthe oscillations (Fig. 4). In scenario #3, unlike aforementioned

ig. 3. Relative rotor angles δ31, δ41. Adaptive backstepping (solid line), PSSdash-dotted line) and uncontrolled system (dotted line) for Scenario 1.

F(

ault clearance time (s) 1.026 1.046 1.06ine re-closing time (s) 1.03 1.05 10.00

scenarios, the fault duration is increased, near end of the lineis opened at 1.05 s and completely removed at 1.06 s. Thetransmission line is reconnected at 10 s. Proposed controllersstabilize the system which returns to its pre-fault equilibriumpoint. Relative rotor angle δ31 depicts the fact that pre-fault

Instability, with PSS controller, is the result of low frequencyinter-area oscillations that arise due to weak interconnectiontie line and long fault duration. Finally Fig. 6 compares the

ig. 4. Relative rotor angles δ31, δ41. Adaptive backstepping (solid line), PSSdash-dotted line) and uncontrolled system (dotted line) for Scenario 2.

490 A. Karimi, A. Feliachi / Electric Power Systems Research 78 (2008) 484–493

Fig. 5. Relative rotor angle δ31, speed deviations �ω3, terminal voltage at buses11 and excitation field voltage Efld3 for generator 3. Adaptive backstepping(solid line), and PSS (dash-dotted line) in Scenario 3.

Fig. 5 (Continued).

Fig. 6. Coupling terms d3, d4 (solid line) and estimated d3, d4 (dash-dotted line)for generators 2 and 3.

A. Karimi, A. Feliachi / Electric Power Systems Research 78 (2008) 484–493 491

F[l

4b

tadim(diTi[tsatb

4

taviFop

Fl

ii

5

iedraastb

ig. 7. Relative rotor angles δ41. Proposed control (solid line), control design19](dash line), control design [18](dotted line), control design [17](dash-dottedine).

estimated and actual (d and d) coupling terms. The adapta-tion laws adjust the fraction of incoming power to decoupleeach generator from the effects of the rest of the system, andproduce decentralized control signals.

.2. Comparison of proposed controller to previous designsy authors

In this subsection the performance of the proposed con-roller is compared with previous control designs by theuthors [17–19] using scenario #2. The proposed controller is aecentralized control which uses only local information. Moremportantly the interface is modeled easily by simple polyno-

ial function of electric power. In Fig. 7, the proposed controllersolid line) dampen the oscillations more effectively than theesign presented in [19](dash line). In control design [18], thenterface is modeled as an external disturbance (dotted line).he disadvantage of this control is the high rate of control effort

n order to counteract the disturbance effect. In control design17], it is assumed that interface variables are available and con-roller is centralized (dash-dotted line – .). Local and remoteignals are used to produce the control signal. In summary,ll designs work well, but the proposed one has the advan-age of being decentralized and tracks the interface variablesetter.

.3. Modeling errors

Performance of proposed controller is evaluated with respecto generator modeling error. For scenario 2, given in Table 2,

random change of generator parameters from their nominalalues (within the range of ±50%) are considered for the follow-

ng parameters of each generator (�x′

q, �T ′do, �x′

d, �T ′do, �H).

ig. 8 shows speed deviation for generator 3 for several numberf simulations. Proposed controller numerically shows robusterformance due to modeling errors. The mean (solid line), max-

uAtc

ig. 8. Speed deviations �ω3 for several simulations, minimum (dash-dottedine), mean (solid line), and maximum (dotted line) values for �ω3.

mum (dash-dot line) and minimum (dotted line) values are givenn Fig. 8.

. Conclusion

Decentralized adaptive backstepping control is presented andmplemented to stabilize multi-machine power systems throughxcitation control. Each generator is modeled as an uncertainynamic subsystem. The uncertainty represents the effects of theest of the system on the particular generator, and it is expresseds a polynomial of electric power deviation. Controller gainsre tuned simultaneously through a PSO technique. A two-areaystem is used to illustrate the performance of proposed con-rol strategy. Test results show the effectiveness of the adaptiveackstepping control in improving dynamic stability of system

nder large disturbances in comparison with conventional PSS.counter example demonstrates that conventional PSSs, unlike

he proposed one, are only valid in the vicinity of the operatingonditions and fails during large disturbances.

4 er Systems Research 78 (2008) 484–493

A

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Table A.1Generators and transmission lines variables

δi Rotor angle in radiansωi Speed in radians per secondω0 Rated speed in radians per secondEfld Excitation field voltagePei

Active power in per unitQei

Reactive power in per unitHi Inertia constant in secondsDi Damping constant in per unitP0

mi Mechanical powerIdi Direct axis currentIqi Quadrature axis currentδij Relative rotor angle in radianE′

qi Transient EMF in quadrature axisE′

diTransient EMF in direct axis

Eqi EMF in quadrature axisEdi EMF in direct axisT ′

do Direct open circuit time constantT ′

do Quadrature open circuit time constantXq Quadrature axis reactanceXd Direct axis reactanceX′

d Direct axis transient reactanceX′

q Quadrature axis transient reactanceBij Elements of susceptance matrixG′

ij Elements of conductance matrix

Table A.2Parameters for synchronous machine

Generator # 1, 2, 3, 4S base MVA 900xl Leakage reactance (p.u.) 0.2ra resistance (p.u.) 0.0025xd d-axis synchronous reactance (p.u.) 1.8x′

d d-axis synchronous transient reactance (p.u.) 0.3T ′

d d-axis open circuit time constant (s) 8xq q-axis synchronous reactance (p.u.) 1.7x′

q q-axis synchronous transient reactance (p.u.) 0.55T ′

q q-axis open circuit time constant (s) 0.4H Inertia constant (s) 6.5D Damping coefficients 0

Table A.3Parameters for static exciter

Exciter 1, 2, 3, 4KA, regulator gain (p.u.) 200TA, regulator time constant (p.u.) 0.05Tr, filter time constant (p.u.) 0.01V

V

Sgft

92 A. Karimi, A. Feliachi / Electric Pow

cknowledgments

This research is sponsored in part by a US DoE EPSCoR WVtate Implementation Award, and in part by grant from the USEPSCoR and ONR (DOD/ONR N000 14-031-0660).

ppendix A

The generators are modeled by [28]:

δi = �ωi, �ωi = − Di

2Hi

�ωi − ωio

2Hi

�Pei, �E′di

= 1

T ′qoi

[−Iqi(Xqi − X′qi) − �E′

di − E′di0], �E′

qi

= 1

T ′doi

[−�E′qi + Idi(Xdi − X′

di) − E′q0 + �Efldi

] (45)

here

ei = E′diIdi + E′

qiIqi (46)

di = E′di + IqiX

′di, Eqi = E′

qi − IdiX′di (47)

qi(t) =n∑

j=1

E′qj(Bij sin δij(t) + Gij cos δij(t)), Idi(t)

=n∑

j=1

E′qj(Gij sin δij(t) − Bij cos δij(t)) (48)

he subsystem dynamics for each generator are nonlinear andoupled through nonlinear coupling equations Idi(t) and Iqi(t).ij and Gij are elements of susceptance and conductanceatrix, respectively (48). The equilibrium point of the system

s shifted to the origin by assuming that the initial conditionsi0, E

′di0, E

′qi0 are available by solving a load flow problem:

�ωi = ωi(t) − ωi0, �E′di = E′

di(t) − E′di0,

E′qi = E′

qi(t) − E′qi0 (49)

ntegration of �ωi is used to obtain �δi with zero initial value.o apply backstepping method, one needs to model the system

n a specific form called strict feedback model [7]. Dynamicquation for electric power deviation, �Pei (t) = Pei (t) − P0

m,s derived by differentiating (46) and substituting the dynamicquations for direct and quadrature voltage. The equation willecome

Pei = −�Pei

1

T ′qoi

+ βi�Efldi+ di (50)

here di and βi are defined by βi = (Iqi)/(T ′doi):

i = −IqiIdi(Xqi − X′qi)

′ + E′qiIqi

(1′ − 1

′

)− P0

mi′

Tqoi Tqoi Tdoi Tqoi

+ E′diIdi + E′

qiIqi + IqiIdi(Xdi − X′di)

T ′doi

+ E◦fldi

Iqi

T ′doi

(51)

R

Rmax (p.u.) 10

Rmin (p.u.) −10

ynchronous generators and transmission lines variables areiven in Table A.1. Values for generators and excitation systemor two-area system are given in Tables A.2 and A.3, respec-ively.

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li Karimi received his B.S. (EE) in 1999, Tehran, Iran and M.S. degree fromest Virginia (WVU) University in 2003. He is currently a Ph.D. candidate andGraduate Research Assistant in the Lane Department of Computer Science andlectrical Engineering, WVU. His main interests are application of nonlinearontrol theories to electric power systems, stability enhancement of large scaleystems, and approximation theories.

li Feliachi received the Diplome d’Ingenieur en Electrotechnique from Ecoleationale Polytechnique of Algiers, Algeria, in 1976, and the M.S. and Ph.D.egrees in Electrical Engineering from the Georgia Institute of Technology,tlanta, in 1979 and 1983, respectively. Currently, he is a Full Professor and

he holder of the endowed Electric Power Systems Chair position in the Laneepartment of Computer Science and Electrical Engineering at West Virginianiversity, where he has been since 1984. He is also the Director of the Advancedower and Electricity Research Center at WVU. He has been working in theeld of large-scale systems and power systems for about 30 years.