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Coordinated Dynamic Voltage Stabilization basedon Model Predictive ControlLicheng Jin, Member, IEEE, and Ratnesh Kumar, Fellow, IEEE

Abstract—Keeping voltages of all buses within acceptablebounds is very important for power system operations. This paperpresents an approach for optimal coordination of static var com-pensators (SVCs), transformer under load tap changers (ULTCs)and load shedding to improve voltage performance followinglarge disturbances. The approach is based on model predictivecontrol (MPC) with a decreasing control horizon. The MPCformulation leads to a mixed integer quadratic programming(MIQP) problem, since both continuous and discrete controls areconsidered. Trajectory sensitivities are used to evaluate the effectof controllers on voltage performance. The iterative optimizationprocess of MPC helps ensure that errors introduced due to anymodel inaccuracies and approximations are minimized. The MPCbased coordination of these controllers is applied to a modifiedWECC system to enhance the voltage performance and to amodified 39-bus New England system to prevent voltage collapse.

Index Terms—Coordinated voltage control, model predictivecontrol, trajectory sensitivity, power system

I. INTRODUCTION

Voltage instability takes the form of a dramatic drop inbus voltages in a transmission system, which may result insystem collapse. Nowadays, voltage stability has become amajor concern in power system planning and operation [1], [2].The deregulation of power industry has created an economicalincentive to operate power systems closer to their limits.Voltage instability can occur under certain severe disturbances.In practice there exist various choices for exercising voltagecontrol, i.e., reactive power compensation devices, generatorreactive power control, transformer tap changer control andload shedding. As shown in Section V, in certain applications,a single type of control does not stabilize the system whereascoordination of multiple types of controls is able to stabilizethe system. Therefore, it is imperative that coordinated volt-age controls be in place to mitigate the catastrophic effectssuch as large scale shutdowns and collapses caused by suchdisturbances.

There exists prior work on coordinated voltage control.One approach is based on static analysis. A hierarchicalvoltage control system for the Italian transmission grid isintroduced in [3], [4]. The power plants adjust their reactive

The work was supported in part by the National Science Foundation underGrants NSF-ECCS-0424048, NSF-ECCS-0601570, NSF-ECCS-0801763, andNSF-CCF-0811541

L. Jin is with the Department of Electrical and Computer Engineering ofIowa State University, Ames, IA 50011, USA and also with California ISO,Folsom, CA, 95630 USA (email:ljin@caiso.com)

R. Kumar is with the the Department of Electrical and Com-puter Engineering of Iowa State University, Ames, IA 50011, USA(email:rkumar@iastate.edu)

power outputs based on voltages measured at pivot buses.Hierarchical control schemes are also studied in other coun-tries as shown in [5], [6], [7], [8] and [9]. [10] discussesthe coordination of distribution-level under load tap changers(ULTCs), mechanically-switched capacitors and static com-pensators (STATCOMs) to improve voltage profile and to re-duce the mechanical switching operations within a substation.[11] proposes a coordinated control method for ULTCs andcapacitors in distribution systems to reduce power loss andto improve voltage profiles during a day. [12] presents anartificial neural network based coordination control schemefor ULTCs and STATCOMs to minimize the amount of trans-former tap changes and STATCOM outputs while maintainingacceptable voltage magnitudes at substation buses.

Some work has also been done to design a coordinatedvoltage control strategy by considering dynamic response ofa power system. [13] presents a method of coordination ofload shedding, capacitor switching and tap changers usingmodel preventive control. The prediction of states is basedon the numerical simulation of nonlinear differential algebraicequations (DAEs) together with Euler state prediction. A treesearch method is adopted to solve the optimization. [14]proposes a coordination of generator voltage setting points,load shedding and ULTCs using a heuristic search and thepredictive control. The prediction of states is based on thelinearization of nonlinear DAEs. [15] presents an optimalcoordinated voltage control using model predictive control.The controls used include: shunt capacitors, load shedding, tapchangers and generator voltage setting points. The predictionof voltage trajectory is based on the Euler state prediction. Theoptimization problem is solved by a pseudo gradient evolution-ary programming (PGEP) technique. In [16] and [17], authorspresent a method to compute a voltage emergency controlstrategy based on model predictive control. The predictionof the output trajectories is based on trajectory sensitivity.However, in these two papers, the authors employ a simplifiedmodel predictive control, which computes the control actionsonly at the initial time and implements it over the entirecontrol horizon. A voltage stabilization control strategy is alsoproposed in [18] based on load shedding, where the objectivefunction is to minimize the amount of load shedding requiredto restore the voltages. It shows load shedding is an effectivevoltage control under emergency condition. [19] presents aMPC based voltage control design. The controls are referencevoltage of automatic voltage regulators and load shedding.

In this paper, we design a coordinated control of SVCs,ULTCs and load shedding to improve voltage performancefollowing disturbances. Given the locations and capabilities

978-1-4244-3811-2/09/$25.00 ©2009 IEEE

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of SVCs, ULTCs and interruptible load, the control designproblem is to determine the control sequences and the controlamounts to satisfy voltage performance requirements. Thecomparison of our work and the prior works is summarizedas follows:

• Trajectory sensitivity is used to compute a 1st-order(linear) approximation of the effect of control withouthaving to linearize the system model. Further at eachcontrol step, the trajectory sensitivity is updated (basedon a prediction of system trajectory starting from anestimate of the current state under the control appliedin the past steps). This way of computing the effectof control provides a better approximation as comparedto [13], [14], [15], where either system linearization ornumerical simulation of DAEs was used.

• Optimization minimizes costs of control as well asvoltage-deviations. In contrast [18], only considers theamount of controls to restore the voltage.

• Optimization at each control step is a quadratic pro-gramming problem, and hence can be efficiently solved.In contrast [15] uses a pseudo gradient evolutionaryprogramming. [13], [14] use a tree search method.

• In contrast to [16] and [17], where the control action iscalculated only at the initial time, and remains the sameover the entire control horizon, a sequence of controlinputs is determined and only the first of them is appliedin our case.

• [13], [14], [15], [18], [19] are all based on a traditionalMPC with constant control horizon. However, our paperis based on a modified MPC with decreasing control hori-zon which facilitates the convergence of the optimizationand reduces the computation time.

II. METHODOLOGY

A. Model Predictive Control

Model predictive control is a class of algorithms that com-pute a sequence of control variable adjustments in order tooptimize the future behavior of a plant (system). MPC wasoriginally developed to meet the specialized control needsof petroleum refineries. Now it has been used in a widevariety of application areas including chemicals, food process-ing, automotive, aerospace, metallurgy, and power plants. Anintroduction to the basic concepts and formulations of MPCcan be found in [20]. The principle of MPC is graphicallydepicted in Fig. 1. Here x represents the state variable thatneeds to be controlled to a specific range. The available controlis represented by variable u.

At a current time tk, MPC solves an optimization problemover a finite prediction horizon [tk, tk + Tp] with respect to apredetermined objective function such that the predicted statevariable x(tk + Tp) can optimally stay close to a referencetrajectory. The control is computed over a control horizon[tk, tk + Tc], which is smaller than the prediction horizon(Tc ≤ Tp). If there were no disturbances, no model-plantmismatch and the prediction horizon is infinite, one couldapply the control strategy found at current time tk for all

Predicted state

Manipulated input

State trajectory

until current time, xk

Time

Magnitude

Input until

current time, uk

tk tk + Ts tk + Tctk + Tp

Control horizon Tc

Prediction horizon Tp

Fig. 1. Principle of MPC

times t ≥ tk. However, due to the disturbances, model-plant mismatch and finite prediction horizon, the true systembehavior is different from the predicted behavior. In order toincorporate the feedback information about the true systemstate, the computed optimal control is implemented only untilthe next measurement instant (tk + Ts), at which point theentire computation is repeated.

In a MPC, the optimization problem to be solved at timetk can be formulated as follows:

minu

∫ tk+Tp

tk

F (x(τ), u(τ))dτ (1)

subject to

˙x(τ) = f(x(τ), u(τ)), x(tk) = x(tk) (2)

umin ≤ u(τ) ≤ umax, ∀τ ∈ [tk, tk + Tc] (3)

u(τ) = u(tk + Tc), ∀τ ∈ [tk + Tc, tk + Tp] (4)

xmin(τ) ≤ x(τ) ≤ xmax(τ), ∀τ ∈ [tk, tk + Tp] (5)

Here, Tc and Tp are the control and prediction horizon withTc ≤ Tp. x denotes the estimated state and u represents“estimated” control (The true state may be different and thetrue control matches the estimated control only during the firstsampling period).

Equation (1) represents the cost function of the MPC opti-mization. Equation (2) represents the dynamic system modelwith initial state x(tk). Equations (3) and (4) represent theconstraints on the control input during the prediction horizon.Equation (5) indicates the state operation requirement duringthe prediction horizon.

B. Trajectory Sensitivity

Consider a differential algebraic equation (DAE) of a sys-tem,

x = f(x, y, u), x(0) = x0 (6)

0 = g(x, y, u) (7)

where x is a vector of state variables, y is a vector of algebraicvariables, and u is a vector of control variables. Trajectorysensitivity considers the influence of small variations in the

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control u (and any other variable of interest) on the solutionof the equations (6) and (7). Let u0 be a nominal value ofu, and assume that the nominal system in (8) and (9) has aunique solution x(t, x0, u0) over [t0, t1].

x = f(x, y, u0), x(0) = x0 (8)

0 = g(x, y, u0) (9)

Then the system in Equations (6) and (7) has a unique solutionx(t, x0, u) over [t0, t1] that is related to x(t, x0, u0) as:

x(t, x0, u) = x(t, x0, u0) + xu(t)(u − u0) + h.o.t.(10)

y(t, x0, u) = y(t, x0, u0) + yu(t)(u − u0) + h.o.t. (11)

Here xu(t) = ∂x(t,x0,u)∂u is called the trajectory sensitivities

of state variables with respect to variable u and yu(t) =∂y(t,x0,u)

∂u is the trajectory sensitivities of algebraic variableswith respect to variable u.

The evolution of trajectory sensitivities can be obtained bydifferentiating Equations (6) and (7) with respect to the controlvariables u and is expressed as:

xu(t) = fx(t)xu(t) + fy(t)yu(t) + fu(t) (12)

0 = gx(t)xu(t) + gy(t)yu(t) + gu(t) (13)

Detailed information about trajectory sensitivity theory canbe found in [21]. The trajectory sensitivity can be solvednumerically. [22] provides a methodology for the computationof trajectory sensitivity. When time domain simulation of apower system is based on trapezoidal numerical integration,the calculation of trajectory sensitivity requires solving a setof linear equations, thus costing a little time. In our work, weextended the Power System Analysis Tool [23] (a MATLABbased tool) to do trajectory sensitivity calculation and the MPCoptimization.

Fig. 2 illustrates the application of trajectory sensitivityin evaluating the effect of controls on system behavior. Thetrajectory xk of the nominal system represents the behaviorunder the control uk. When the control is increased by Δuk

1

at time tk, the change in predicted system behavior basedon sensitivity analysis at time tl, can be approximated asΔxkl

1 = xluk1Δuk

1 . Here xluk1

is the trajectory sensitivity ofthe state variable at time tl with respect to the control attime tk. Similarly if we increase the control by Δuk

n at timetk + (n − 1)Ts, the change in the state variable at time tl isrepresented by Δxkl

n = xluk

nΔuk

n. Here, xluk

nis the trajectory

sensitivity of the state variable at time tl with respect to thecontrol at time tk + (n − 1)Ts.

III. PROBLEM FORMULATION AND SOLUTION

For analyzing voltage performance following disturbances,we model generator and automatic voltage regulator (AVR)as well as aggregated exponential dynamic load models [24],[25]. The overall power system is represented by a set ofdifferential algebraic equations (DAE) as in Equations (6) and(7). Here x is a vector of states including state variables ingenerator dynamic models, AVR models and dynamic loadmodels such as, rotor angles and angular speeds of generators,outputs of AVRs, and active power recovery and reactive

Magnitude

Step 1

Time

Step n

ku1Δ

knuΔ

kl

u

kl uxx k 111Δ=Δ

kn

l

u

kln uxx k

nΔ=Δ

kx

ku

kt sk Tt + sk Tnt )1( −+ lt

Nominal system

trajectory

Fig. 2. Application of trajectory sensitivity in system behavior prediction

power recovery of dynamic load models. y is a vector ofalgebraic variables such as bus voltage magnitudes and phaseangles. The vector u includes all the control variables. In oursetting, it represents the position of under load tap changers,the susceptance output of SVCs and the amount of loadshedding. Whenever the occurrence of a certain pre-identifiedcontingency is detected and the system performance is notsatisfactory, for instance, voltages are out of the their limits,an optimal coordinated control strategy is identified based ona decreasing horizon MPC algorithm

Let Tp be the prediction horizon, Tc be the control horizon,Ts be the control sampling interval, and N = Tc

Tsbe the

total number of control steps. The procedure to determine thecontrol strategy at the kth sampling instant is as follows:(1) At time tk (i.e. the (k+1)th sampling instant), an estimate

of the current state x(tk) is obtained. The nominal powersystem evolves according to Equations (14) and (15).

x = f(x, y, uc, ud), x(0) = x0 (14)

0 = g(x, y, uc, ud) (15)

Here, uc = {C0m+

∑k−1i=0 ΔCi

m1}m=Mcm=1 is the continuous

control variable (e.g. amounts of SVC currently in use).C0

m is the amounts of continuous variables that existat time 0.

∑k−1i=0 ΔCi

m1 is the amounts of the contin-uous variable that were added over time [0, tk − Ts].ud = {D0

m+∑k−1

i=0 Sim1ΔDi

m1}m=Mc+Md

m=Mc+1 is the discretecontrol amount. D0

m is the amounts of discrete variablesthat exist at time 0.

∑k−1i=0 Si

m1ΔDim1 is the amount of

the discrete control that were added over time [0, tk−Ts].Here Si

m1 is the step size of the discrete actuator m atsampling point ti, and ΔDi

m1 is the number of steps ofthe discrete actuator at time ti.Time domain simulation is used to obtain the trajectory ofthe nominal system (14) and (15), starting from the statex(tk) at time tk to the end of prediction horizon tk +Tp.At the same time, the trajectory sensitivities of bus volt-ages with respect to the continuous and discrete controlsto be added at instants tk +(n−1)Ts, n = 1 . . .N−k areobtained and denoted as V kj

Cmn(t), V kj

Dmn(t) (see below for

the explanation of notation).(2) At time tk, solve the quadratic integer programming opti-

mization problem over the prediction horizon [tk, tk+Tp]and a control horizon [tk, tk + (N − k)Ts] as stated in(16)-(23).

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Minimize (with respect to ΔCkmn and ΔDk

mn)∫ tk+Tp

tk

(V k(t) − Vref )′R(V k(t) − Vref )dt

+m=Mc∑m=1

n=N−k∑n=1

WmnΔCkmn

+m=Mc+Md∑m=Mc+1

n=N−k∑n=1

WmnSkmnΔDk

mn (16)

Subject to

ΔCminm ≤ ΔCk

mn ≤ ΔCmaxm , (17)

ΔDminm ≤ ΔDk

mn ≤ ΔDmaxm , (18)

Cminm ≤ C0

m +k−1∑i=0

ΔCim1 +

N−k∑n=1

ΔCkmn ≤ Cmax

m (19)

Dminm ≤ D0

m+k−1∑i=0

Sim1ΔDi

m1+N−k∑n=1

SkmnΔDk

mn ≤ Dmaxm

(20)

V kjmin(t) ≤ V kj(t) +

Mc∑m=1

N−k∑n=1

V kjCmn

(t)ΔCkmn

+Mc+Md∑

m=Mc+1

N−k∑n=1

V kjDmn

(t)SkmnΔDk

mn ≤ V kjmax(t) (21)

ΔCkmn ≥ 0, m = 1, · · · , Mc (22)

ΔDkmn is an integer, m = Mc + 1, · · · , Mc + Md (23)

Here,– R is the weighting matrix.– V k(t) is the voltage vector at time t ∈ [tk, tk + TP ]

as predicted at the sampling instant tk.– Wmn is the weight for the cost of control m to be

added at time tk + (n − 1)Ts.– Mc is the total number of continuous control vari-

ables, i.e. the number of available SVCs.– Md is the total number of discrete control variables,

i.e. the number of available under load tap changerplus the number of load shedding candidate loca-tions.

– N is the total number of control steps.– ΔCk

mn is the amount of continuous actuator m to beadded at time tk + (n − 1)Ts in iteration k.

– ΔDkmn is the number of steps of discrete actuator m

to be added at time tk + (n− 1)Ts in iteration k. Itis an integer.

– Skmn is the step size of discrete actuator m at time

tk + (n − 1)Ts in iteration k.– ΔCmin

m ∈ � is the minimum amount of continuouscontrol m to be added at control sampling points,typically 0.

– ΔCmaxm ∈ � is the maximum amount of continuous

control m to be added at control sampling points.

– ΔDminm ∈ � is the minimum number of steps of

discrete control m to be added at control samplingpoints, typically 0.

– ΔDmaxm ∈ � is the maximum number of steps of

discrete control m to be added at control samplingpoints.

– ΔCim1 is the amount of control m implemented at

the control sampling point ti, i = 0, ..., k − 1.– Cmin

m ∈ � is the minimum amount of continuouscontrol m that must be used, typically 0.

– Cmaxm ∈ � is the maximum available amount of

continuous control m.– Dmin

m is the minimum amount of discrete control m.– Dmax

m is the maximum available amount of discretecontrolm.

– V kj(t) ∈ � is the voltage of bus j at time t(tk ≤t ≤ tk + Tp) of the nominal system at time tk.

– V kjmin(t) is the minimum voltage at bus j desired at

time tk ≤ t ≤ tk + Tp.– V kj

max(t) is the maximum voltage at bus j desired attime tk ≤ t ≤ tk + Tp.

– V kjCmn

(t) is the trajectory sensitivity of the voltage atbus j at time tk ≤ t ≤ tk + Tp with respect to thecontinuous control m added at time tk + (n− 1)Ts.

– V kjDmn

(t) is the trajectory sensitivity of the voltage atbus j at time tk ≤ t ≤ tk + Tp with respect to thediscrete control m added at time tk + (n − 1)Ts.

The objective of the optimization is to minimize thevoltage deviation and cumulative cost of continuous anddiscrete controls as shown in Equation (16). Equation (17)constraints the amount of the continuous control m tobe added at time tk + (n − 1)Ts. Equation (18) is thecontrol step constraints on discrete actuators. Equation(19) constraints the total amount of continuous controlm to be added over [tk, tk + (N − k)Ts]. Equation(20) constraints the total amount of discrete control mto be added over [tk, tk + (N − k)Ts]. Equation (21)constraints the voltage fluctuation at time t ∈ [tk, tk+Tp].The number of candidate control locations and theirupper limits are determined through a prior planningstep (see for example [26]). The total number of controlvariables in the optimization is the number of candidatecontrol locations times the number of control steps. Theoptimization problem is solved in Matlab, and it doesconverge to a global minimum.

(3) At time tk, the solution of the optimization problem (16)-(23) computes a sequence of controls ΔCk

mn, ΔDkmn.

Add only the first control ΔCkm1, S

km1ΔDk

m1 at timetk and obtain the system state x(tk+1) at time tk+1 =tk + Ts.

(4) Increase k to k + 1 and repeat steps (1)-(3) until k =N − 1.

IV. IMPLEMENTATION

The functional structure of implementing the MPC basedcoordinated voltage control is shown in Figure 3. Line flow,bus voltage information, switch status measured by phase

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measurement units (PMUs) and collected by Phasor DataConcentrators (PDCs) are sent to a control center throughcommunication channels. These measurements plus a networkmodel are used by the state estimator (SE) for filtering out thenoise and making best use of the measured data. The resultsfrom the state estimator are used for power flow analysis.A power flow solution is then used by an on-line dynamicsecurity assessment program to initialize the state variablesof the dynamic models. Further, it uses system models anddisturbance information to perform the contingency analysisto evaluate the security margin of the power system. Ifa contingency is identified where the system will becomeunstable, MPC based computation will get triggered at thetime an identified critical contingency occurs. A final step isto implement the real time control computed to improve thesecurity of the power system.

The steps of the MPC computation in the kth iterationinclude:

• Estimate static variables such as voltage magnitudes andangles at time tk as well as the dynamic variablesx(tk) such as generator angles, velocities and real andreactive load recovery. The values of the static variablesis provided by the state-estimator. As far as the dynamicvariables are concerned, they can be classified into short-term dynamic variables (such as generator angles andvelocities) and long-term dynamic variables (such as realand reactive load recovery). The values of the long-termdynamic variables can be directly measured and hence areknown, whereas the short-term dynamic variables are inquasi steady-state (QSS) with respect to the long-termvoltage/frequency stability phenomenon investigated inthis proposal. Thus the values of the short-term dynamicvariables can be obtained by solving an equilibriumequation of the form:

0 (= xs) = fs(xs, xl, y, u), (24)

where xs is the short-term dynamic variable vector (tobe computed by solving (24)), xl is long-term dynamicvariable vector (which is measured and hence known), yis the static variable vector (which is provided by thestate-estimator and hence known), and u is the inputvariable vector (which is of course known). Then inequation (24), the number of unknowns (dimension ofxs) is the same as the number of equations (dimensionof fs), and so the short-term dynamic variables can becomputed by solving (24).

• Run time-domain simulation to compute the system tra-jectory given the current state.

• Obtain trajectory sensitivities of voltage with respect tothe control variables as a by-product of the time-domainsimulation performed in the previous step.

• Solve the quadratic programming optimization problemand implement the first step of the control.

V. APPLICATION

The proposed method is illustrated using the modifiedWECC 9-bus system and New England 39-bus system. The

PMU

measurements

State estimator

Power flow

analysisSystem model

Disturbance

Recorder

On-line dynamic

security program

System

secure?

MPC based coordinated

voltage controller

Disturbance

happens?

End

Yes

No

Yes

No

Control signal to

power system

Fig. 3. Structure of implementing a MPC based Voltage stabilization

exponential recovery load model is used in both cases. Theparameters of the load model are as following:

TP = TQ = 30, αs = 0, αt = 1, βs = 0, βt = 4.5.

The parameters in MPC optimization are determined basedon the following considerations. Any voltage instability fol-lowing a contingency must be stabilized in a certain timeduration (typically the time in which voltage will decrease by15%). This is the prediction horizon Tp. The control shouldbe exercised on a time horizon Tc, which is shorter than theprediction horizon, typically the time in which voltage willdecrease by 10% (if no control is applied). A discrete-timecontrol must be applied within this duration Tc at a sample-rate high enough to adequately react to the changing voltagetrajectory, as well as to allow accurate enough predictionsof the voltage trajectory based on the linearization of thetrajectory-sensitivity. This dictates the sampling duration Ts.The number of sampling point N is then determined as theratio of Tc and the sampling duration Ts.

The voltage control means in the test cases include SVCs,ULTCs, and load shedding. To avoid over-voltage problems,the maximum amount of the controls is limited at eachsampling point. For SVCs, the maximum control amount is0.1 p.u.. The maximum number of under load tap changersteps is 3. And the maximum load shedding at one samplingpoint is 10%. The step size of ULTCs is 0.006 p.u.. The stepsize of load shedding is 5%.

A. Modified WECC 3-Generator Test System

1) System description: Fig. 4 is a representation of themodified WECC 3-generator 9-bus system. Transformer bankswith under load tap changers are connected to bus 6 and bus8 to regulate the voltages of load buses 10 and 11. A fourth-order generator model is used. The state variables include rotorangle δ, rotor speed ω, q-axis transient voltage e′q , and d-axis transient voltage e′d. Generator 1 is equipped with anautomatic Voltage Regulator (AVR). The continuously actingregulator and exciter model [27] are employed in the study.It is represented by a four-dimensional state equation. Theloads at buses 5, 10 and 11 are represented by the exponentialrecovery dynamic model. Thus, each load is described by atwo-dimensional state equation. Therefore, the total dimensionof the state space is 22. The voltage control mechanismsinclude the followings:

6

2

7 8 9

5 6

3

4

1

Gen 2 Gen 3

Gen 1

10

11

Fig. 4. Modified WECC 3-generator 9-bus test system

0 50 100 150 200 250 300 350 400

0.75

0.8

0.85

0.9

0.95

1

Time (second)

Voltage magenitude (p.u.)

Voltage without any control

Bus 5

Bus 7

Bus 9

Fig. 5. Voltage behavior of the modified WECC case without MPC control

• The SVCs at bus 5, bus 7, and bus 8;• The under load tap changer of transformer banks con-

necting bus 8 and bus 11, bus 6 and bus 10;• The load shedding at bus 5, bus 10, and bus 11.

2) Fault scenario: We consider a three-phase-to-groundfault at bus 5 at t = 1.0 second, which is cleared at t = 1.2seconds by tripping of the line between bus 4 and bus 5.Based on the time domain simulation, the voltage performanceis not satisfactory as shown in Fig. 5. At t = 1.0 second,the voltages begin to drop dramatically due to the fault. Att = 1.2 seconds, the voltages start to recover since the faultis cleared. However, the voltages begin to oscillate. Fifteenseconds later, voltages begin to decline gradually. The dynamicload models result in slightly recovery of load consumption,which deteriorates the voltage condition. Assume that thepost-transient load bus voltages must be above 0.95 p.u.Therefore, some control actions are required to satisfy thevoltage performance requirement.

3) Simulation result: In this example, we have chosenprediction horizon Tp to be 60 seconds (the time in whichvoltage drops by nearly 15% at bus 5). Tc has been chosento be 50 seconds. We found that a sample duration of Ts =10 seconds works well for this example, and so we have thenumber of control steps: N = Tc

Ts= 50

10 = 5. The modelpredictive control approach determines a coordinated controlstrategy to recover the bus voltages. During the optimization,we set the lower bound of all bus voltages to be 0.95 p.u.and the upper bound of load bus voltages to be 1.05 p.u. Forgenerator buses, we set the maximum voltage magnitude tobe 1.08 p.u., which is slightly higher than load buses. These

770 50 100 150 200 250 300 350 400

0.8

0.85

0.9

0.95

1

1.05

1.1

Time (second)

Vol

tag

e m

agni

tude

(p.u

.)

Voltage with coordinated control

Bus 5

Bus 7

Bus 9

Fig. 6. Voltage behavior of the modified WECC with MPC control

TABLE ITHE RESULTING CONTROL STRATEGY FOR THE WECC SYSTEM

Time(second) 20 30 40 50 60ULTC between buses 6 and 10 (step) 3 3 3 3 0ULTC between buses 8 and 11 (step) 3 3 0 0 0Load shedding at bus 5(%) 0 0 0 0 0Load shedding at bus 10 (%) 0 0 0 0 0Load shedding at bus 11(%) 0 0 0 0 0SVC capacitor 0.1 0.1 0.1 0 0change at bus 5 (p.u.)SVC capacitor change 0.1 0.1 0.022 0.025 0.001at bus 7 (p.u.)SVC capacitor change 0.1 0.059 0 0 0at bus 8 (p.u.)

settings are practical. Fig. 6 shows the bus voltages after MPCbased control was implemented. From the figure, we can seethat all the bus voltages were restored to be above 0.95 p.u.

The control strategy is shown in Table I. The first row hasthe time information of the 5 control sampling points, i.e. 20seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds.Each column corresponding to the control sampling point hasthe information of the control actions. For example, at time 20second, both under load tap changers increase their tap ratiosby 3 steps, which is 0.018 p.u.. No load shedding has beentaken. All the existing three SVCs increase their susceptanceoutput by 0.1 p.u..

B. Modified New England 9-Generator 39-Bus Test System

1) System description: Fig. 7 shows the modified NewEngland 9-generator 39-bus system. There are totally 41 busesand 9 generators. Two transformer banks with under load tapchangers are added between bus 8 and bus 40, bus 4 and bus41. A fourth-order generator model is used. The exceptionis that a third-order model is used for the generator at bus39. In addition, all generators excluding those at bus 34 andbus 37 have automatic voltage regulators (AVRs), which arerepresented by fourth-order models. The loads are representedby the exponential recovery dynamic models. The controlvariables are as follows:

• The SVCs at buses 1, 6, 14, and 28;• The under load tap changers at the transformer banks

between bus 8 and bus 40, bus 4 and bus 41;• The load shedding at bus 15 and bus 16.

7

G G

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

1011

13

14

15

8 31

126

32

7

G

G

40

41

Fig. 7. Modified New England 10-generator 39-bus test system

0 20 40 60 80 100 1200.8

0.85

0.9

0.95

1

1.05

Time (second)

Vol

tag

e m

agni

tude

(p.u

.)

Voltage behavior without control

Bus 34

Bus 21Bus 20

Bus 16

Fig. 8. Voltage behavior of the modified New England system without MPCcontrol

2) Fault scenario: The contingency considered here is athree-phase-to-ground fault at bus 21 at t = 1.0 second,which is cleared at t = 1.2 seconds by the tripping of thetransmission line between bus 21 and bus 22. Bus voltagesdrop dramatically when the fault occurs as shown in Fig. 8.After the fault is cleared at 1.2 seconds, the voltages recovergreatly whereas some oscillations follow. About 20 secondslater, the oscillations are damped out, but the voltages startto decline slowly because of the exponential recovery of theloads. Around 2 minutes later, the voltages collapse.

3) Simulation result with only SVC control: In this testcase, there are three types of voltage control options. Theyare ULTCs, SVCs and load shedding. This subsection studiesthe effect of SVCs on the restoration of the voltage behavior.There are four SVCs, which locate at bus 1, bus 6, bus 14 andbus 28. The upper limit of these SVCs is 0.3 p.u.. The controlstrategy is to switch all the available capacity of SVCs at 20seconds. The voltage behavior is presented in Fig. 9. Fromthis figure, we find that even if all the SVCs are put into use,the voltage can not be stabilized following the contingency.

4) Simulation result: In this example, we have chosenprediction horizon Tp to be 90 seconds (the time in which

0 20 40 60 80 100 1200.8

0.85

0.9

0.95

1

1.05

Time (second)

Vol

tag

e m

agni

tude

(p.u

.)

Voltage behavior with SVC control

Bus 34

Bus 21

Bus 20

Bus 16

Fig. 9. Voltage behavior of the modified New England system with SVCcontrol

TABLE IITHE CONTROL STRATEGY FOR THE MODIFIED NEW ENGLAND SYSTEM

Time(second) 20 35 50 65 75SVC at bus 1 (p.u.) 0 0 0 0 0SVC at bus 6 (p.u.) 0 0 0.1 0 0.0544SVC at bus 14 (p.u.) 0 0.0806 0.1 0.0274 0.092SVC at bus 28 (p.u.) 0 0 0.1 0 0ULTC between buses 3 3 3 3 38 and 40 (steps)ULTC between buses 3 3 3 3 34 and bus 41 (steps)Load shedding 5 10 10 10 0at bus 15 (%)Load shedding 0 0 0 0 0at bus 16 (%)

voltage drops by nearly 12% at bus 20). Tc has been chosento be 75 seconds. We found that a sample duration of Ts =15 seconds works well for this example, and so we have thenumber of control steps: N = Tc

Ts= 75

15 = 5. The controlaction determined by the MPC based algorithm starts around20 seconds to recover voltage. The system response with MPCin place is shown in Fig. 10. With the MPC implemented,the voltages are stabilized at a value between [0.95, 1.05]p.u.. The corresponding control strategy is shown in Table II.From that table, we find that the under load tap changers areat the maximum change steps at each sampling point. Loadshedding is also used to stabilize the system. The table showsa coordinated control strategy between under load tap changer,static var compensators as well as load shedding.

VI. CONCLUSION AND DISCUSSION

This paper presents a coordinated voltage control strategy.The design is based on a modified MPC method with adecreasing control horizon. Trajectory sensitivity is used toevaluate the effect of controls on the voltage improvement. Theiterative optimization process of MPC helps ensure that errorsintroduced due to trajectory sensitivity linearizations and anymodel inaccuracies are minimized. The coordination of staticvar compensators, under load tap changers and load sheddingis achieved by solving a quadratic mixed integer optimizationformulation. The test cases indicated that the proposed MPC-based coordinated control strategy can effectively improve thesystem performance. The proposed methodology also works

8

0 20 40 60 80 100 1200.8

0.85

0.9

0.95

1

1.05

Time (Second)

Vol

tag

e m

agni

tude

(p.u

.)

Voltage behavior with coordinated controls

Bus 34

Bus 21Bus 20

Bus 16

Fig. 10. Voltage behavior of the modified New England system with MPC-based coordinated voltage control

for controls such as generator voltage setting points, shunt ca-pacitors. The proposed approach is applicable to industrial-sizesystems. The control computation at each control step requires(i) estimation of static and dynamic variables, (ii) time-domainsimulation to predict system trajectory starting from newlyestimated state under the controls applied in the past steps, (iii)trajectory-sensitivity computation, (iv) quadratic mixed-integerprogramming solution. The most time-consuming component,dominating the other components, is time-domain simulation.Currently there already exist on-line dynamic security assess-ment (DSA) programs, e.g., the on-line version of DSA ofPower Tech. It runs stability study for 3000 contingenciesfor a 12000 bus system based on a time domain simulationin a 10 minute cycle using around 10 servers. We believetherefore that it should be possible to design controls basedon the proposed method for on-line real-time system protectionagainst a single contingency.

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Licheng Jin received the bachelor’s degree in 2000 and master’s degree ofscience in 2003 from Zhejiang University, Hangzhou, China. She is currentlyworking towards the Ph.D. from Department of Electrical and ComputerEngineering, Iowa State University, Ames. She joined California ISO, Folsom,CA, as a Network Application Engineer in 2007.

Ratnesh Kumar received the B.Tech. degree in Electrical Engineering fromthe Indian Institute of Technology at Kanpur, India, in 1987, and the M.S. andthe Ph.D. degree in Electrical and Computer Engineering from the Universityof Texas at Austin, in 1989 and 1991, respectively. From 1991-2002 he wason the faculty of University of Kentucky, and since 2002 he has been onthe faculty of the Iowa State University. He has held visiting positions atUniv. of Maryland at College Park, NASA Ames, Applied Research Lab. atPenn. State, Argonne National Lab.—West, and United Technology ResearchCenter. He is a Fellow of the IEEE.