Coordinated Dynamic Voltage Stabilization based on ... - Ece.iastate.edu
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Coordinated Dynamic Voltage Stabilization based on Model Predictive Control Licheng Jin, Member, IEEE, and Ratnesh Kumar, Fellow, IEEE
Abstract—Keeping voltages of all buses within acceptable bounds is very important for power system operations. This paper presents an approach for optimal coordination of static var compensators (SVCs), transformer under load tap changers (ULTCs) and load shedding to improve voltage performance following large disturbances. The approach is based on model predictive control (MPC) with a decreasing control horizon. The MPC formulation leads to a mixed integer quadratic programming (MIQP) problem, since both continuous and discrete controls are considered. Trajectory sensitivities are used to evaluate the effect of controllers on voltage performance. The iterative optimization process of MPC helps ensure that errors introduced due to any model inaccuracies and approximations are minimized. The MPC based coordination of these controllers is applied to a modified WECC system to enhance the voltage performance and to a modified 39-bus New England system to prevent voltage collapse. Index Terms—Coordinated voltage control, model predictive control, trajectory sensitivity, power system
I. I NTRODUCTION Voltage instability takes the form of a dramatic drop in bus voltages in a transmission system, which may result in system collapse. Nowadays, voltage stability has become a major concern in power system planning and operation [1], [2]. The deregulation of power industry has created an economical incentive to operate power systems closer to their limits. Voltage instability can occur under certain severe disturbances. In practice there exist various choices for exercising voltage control, i.e., reactive power compensation devices, generator reactive power control, transformer tap changer control and load shedding. As shown in Section V, in certain applications, a single type of control does not stabilize the system whereas coordination of multiple types of controls is able to stabilize the system. Therefore, it is imperative that coordinated voltage controls be in place to mitigate the catastrophic effects such as large scale shutdowns and collapses caused by such disturbances. There exists prior work on coordinated voltage control. One approach is based on static analysis. A hierarchical voltage control system for the Italian transmission grid is introduced in [3], [4]. The power plants adjust their reactive The work was supported in part by the National Science Foundation under Grants NSF-ECCS-0424048, NSF-ECCS-0601570, NSF-ECCS-0801763, and NSF-CCF-0811541 L. Jin is with the Department of Electrical and Computer Engineering of Iowa State University, Ames, IA 50011, USA and also with California ISO, Folsom, CA, 95630 USA (email:[email protected]) R. Kumar is with the the Department of Electrical and Computer Engineering of Iowa State University, Ames, IA 50011, USA (email:[email protected])
power outputs based on voltages measured at pivot buses. Hierarchical control schemes are also studied in other countries as shown in [5], [6], [7], [8] and [9]. [10] discusses the coordination of distribution-level under load tap changers (ULTCs), mechanically-switched capacitors and static compensators (STATCOMs) to improve voltage profile and to reduce the mechanical switching operations within a substation. [11] proposes a coordinated control method for ULTCs and capacitors in distribution systems to reduce power loss and to improve voltage profiles during a day. [12] presents an artificial neural network based coordination control scheme for ULTCs and STATCOMs to minimize the amount of transformer tap changes and STATCOM outputs while maintaining acceptable voltage magnitudes at substation buses. Some work has also been done to design a coordinated voltage control strategy by considering dynamic response of a power system. [13] presents a method of coordination of load shedding, capacitor switching and tap changers using model preventive control. The prediction of states is based on the numerical simulation of nonlinear differential algebraic equations (DAEs) together with Euler state prediction. A tree search 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 the predictive control. The prediction of states is based on the linearization of nonlinear DAEs. [15] presents an optimal coordinated voltage control using model predictive control. The controls used include: shunt capacitors, load shedding, tap changers and generator voltage setting points. The prediction of voltage trajectory is based on the Euler state prediction. The optimization problem is solved by a pseudo gradient evolutionary programming (PGEP) technique. In [16] and [17], authors present a method to compute a voltage emergency control strategy based on model predictive control. The prediction of the output trajectories is based on trajectory sensitivity. However, in these two papers, the authors employ a simplified model predictive control, which computes the control actions only at the initial time and implements it over the entire control horizon. A voltage stabilization control strategy is also proposed in [18] based on load shedding, where the objective function is to minimize the amount of load shedding required to restore the voltages. It shows load shedding is an effective voltage control under emergency condition. [19] presents a MPC based voltage control design. The controls are reference voltage of automatic voltage regulators and load shedding. In this paper, we design a coordinated control of SVCs, ULTCs and load shedding to improve voltage performance following disturbances. Given the locations and capabilities
2
Magnitude
of SVCs, ULTCs and interruptible load, the control design problem is to determine the control sequences and the control amounts to satisfy voltage performance requirements. The comparison of our work and the prior works is summarized as follows: •
•
•
•
•
Trajectory sensitivity is used to compute a 1st-order (linear) approximation of the effect of control without having to linearize the system model. Further at each control step, the trajectory sensitivity is updated (based on a prediction of system trajectory starting from an estimate of the current state under the control applied in the past steps). This way of computing the effect of control provides a better approximation as compared to [13], [14], [15], where either system linearization or numerical simulation of DAEs was used. Optimization minimizes costs of control as well as voltage-deviations. In contrast [18], only considers the amount of controls to restore the voltage. Optimization at each control step is a quadratic programming problem, and hence can be efficiently solved. In contrast [15] uses a pseudo gradient evolutionary programming. [13], [14] use a tree search method. In contrast to [16] and [17], where the control action is calculated only at the initial time, and remains the same over the entire control horizon, a sequence of control inputs is determined and only the first of them is applied in our case. [13], [14], [15], [18], [19] are all based on a traditional MPC with constant control horizon. However, our paper is based on a modified MPC with decreasing control horizon which facilitates the convergence of the optimization and reduces the computation time. II. M ETHODOLOGY
A. Model Predictive Control Model predictive control is a class of algorithms that compute a sequence of control variable adjustments in order to optimize the future behavior of a plant (system). MPC was originally developed to meet the specialized control needs of petroleum refineries. Now it has been used in a wide variety of application areas including chemicals, food processing, automotive, aerospace, metallurgy, and power plants. An introduction to the basic concepts and formulations of MPC can be found in [20]. The principle of MPC is graphically depicted in Fig. 1. Here x represents the state variable that needs to be controlled to a specific range. The available control is represented by variable u. At a current time tk , MPC solves an optimization problem over a finite prediction horizon [t k , tk + Tp ] with respect to a predetermined objective function such that the predicted state variable xˆ(tk + Tp ) can optimally stay close to a reference trajectory. 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-plant mismatch and the prediction horizon is infinite, one could apply the control strategy found at current time t k for all
Predicted state
State trajectory until current time, xk
Manipulated input
Input until current time, uk
tk
t k + Ts
tk + Tp
tk + Tc
Time
Control horizon Tc Prediction horizon Tp
Fig. 1.
Principle of MPC
times t ≥ tk . However, due to the disturbances, modelplant mismatch and finite prediction horizon, the true system behavior is different from the predicted behavior. In order to incorporate the feedback information about the true system state, the computed optimal control is implemented only until the next measurement instant (t k + Ts ), at which point the entire computation is repeated. In a MPC, the optimization problem to be solved at time tk can be formulated as follows: tk +Tp minuˆ F (ˆ x(τ ), uˆ(τ ))dτ (1) tk
subject to x ˆ˙ (τ ) = f (ˆ x(τ ), uˆ(τ )), x ˆ(tk ) = x(tk )
(2)
ˆ(τ ) ≤ umax , ∀τ ∈ [tk , tk + Tc ] umin ≤ u
(3)
u ˆ(τ ) = u ˆ(tk + Tc ), ∀τ ∈ [tk + Tc , tk + Tp ]
(4)
ˆ(τ ) ≤ xmax (τ ), ∀τ ∈ [tk , tk + Tp ] xmin (τ ) ≤ x
(5)
Here, Tc and Tp are the control and prediction horizon with Tc ≤ Tp . x ˆ denotes the estimated state and uˆ represents “estimated” control (The true state may be different and the true control matches the estimated control only during the first sampling period). Equation (1) represents the cost function of the MPC optimization. Equation (2) represents the dynamic system model with initial state x(tk ). Equations (3) and (4) represent the constraints on the control input during the prediction horizon. Equation (5) indicates the state operation requirement during the prediction horizon. B. Trajectory Sensitivity Consider a differential algebraic equation (DAE) of a system, x˙
= f (x, y, u),
0
= g(x, y, u)
x(0) = x0
(6) (7)
where x is a vector of state variables, y is a vector of algebraic variables, and u is a vector of control variables. Trajectory sensitivity considers the influence of small variations in the
3 Magnitude
control u (and any other variable of interest) on the solution of the equations (6) and (7). Let u 0 be a nominal value of u, and assume that the nominal system in (8) and (9) has a unique solution x(t, x 0 , u0 ) over [t0 , t1 ]. x˙ = 0 =
f (x, y, u0 ), g(x, y, u0 )
x(0) = x0
Step n
Δx = x Δu nk kl n
Step 1
(8) (9)
x(t, x0 , u0 ) + xu (t)(u − u0 ) + h.o.t. (10) y(t, x0 , u0 ) + yu (t)(u − u0 ) + h.o.t. (11)
0 ,u) is called the trajectory sensitivities Here xu (t) = ∂x(t,x ∂u of state variables with respect to variable u and y u (t) = ∂y(t,x0 ,u) is the trajectory sensitivities of algebraic variables ∂u with respect to variable u. The evolution of trajectory sensitivities can be obtained by differentiating Equations (6) and (7) with respect to the control variables u and is expressed as:
x˙ u (t) = fx (t)xu (t) + fy (t)yu (t) + fu (t) 0 = gx (t)xu (t) + gy (t)yu (t) + gu (t)
(12) (13)
Detailed information about trajectory sensitivity theory can be found in [21]. The trajectory sensitivity can be solved numerically. [22] provides a methodology for the computation of trajectory sensitivity. When time domain simulation of a power system is based on trapezoidal numerical integration, the calculation of trajectory sensitivity requires solving a set of linear equations, thus costing a little time. In our work, we extended the Power System Analysis Tool [23] (a MATLAB based tool) to do trajectory sensitivity calculation and the MPC optimization. Fig. 2 illustrates the application of trajectory sensitivity in evaluating the effect of controls on system behavior. The trajectory xk of the nominal system represents the behavior under the control u k . When the control is increased by Δu k1 at time tk , the change in predicted system behavior based on sensitivity analysis at time t l , can be approximated as l k l Δxkl 1 = xuk Δu1 . Here xuk is the trajectory sensitivity of 1 1 the state variable at time tl with respect to the control at time tk . Similarly if we increase the control by Δu kn at time tk + (n − 1)Ts , the change in the state variable at time t l is l represented by Δx kl Δukn . Here, xluk is the trajectory n = xuk n n sensitivity of the state variable at time t l with respect to the control at time tk + (n − 1)Ts . III. P ROBLEM 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 of differential algebraic equations (DAE) as in Equations (6) and (7). Here x is a vector of states including state variables in generator dynamic models, AVR models and dynamic load models such as, rotor angles and angular speeds of generators, outputs of AVRs, and active power recovery and reactive
Nominal system trajectory
Δx1kl = xul k Δu1k
xk
1
Δunk Δu1k
Then the system in Equations (6) and (7) has a unique solution x(t, x0 , u) over [t0 , t1 ] that is related to x(t, x0 , u0 ) as: x(t, x0 , u) = y(t, x0 , u) =
l u nk
uk tk
Fig. 2.
tk + Ts
t k + (n − 1)Ts
tl
Time
Application of trajectory sensitivity in system behavior prediction
power recovery of dynamic load models. y is a vector of algebraic variables such as bus voltage magnitudes and phase angles. The vector u includes all the control variables. In our setting, it represents the position of under load tap changers, the susceptance output of SVCs and the amount of load shedding. Whenever the occurrence of a certain pre-identified contingency is detected and the system performance is not satisfactory, for instance, voltages are out of the their limits, an optimal coordinated control strategy is identified based on a decreasing horizon MPC algorithm Let Tp be the prediction horizon, T c be the control horizon, Ts be the control sampling interval, and N = TTcs be the total number of control steps. The procedure to determine the control strategy at the k th sampling instant is as follows: (1) At time tk (i.e. the (k+1)th sampling instant), an estimate of the current state x(t k ) is obtained. The nominal power system evolves according to Equations (14) and (15). x˙ = f (x, y, uc , ud ),
x(0) = x0
(14)
0 = g(x, y, uc , ud ) (15) m=Mc 0 i + k−1 is the continuous Here, uc = {Cm i=0 ΔCm1 }m=1 control variable (e.g. amounts of SVC currently in use). 0 Cm is the amounts of continuous variables that exist k−1 i at time 0. i=0 ΔCm1 is the amounts of the continuous variablethat were added over time [0, t k − Ts ]. k−1 i 0 i c +Md ud = {Dm + i=0 Sm1 ΔDm1 }m=M is the discrete m=Mc +1 0 control amount. D m is the amounts of discrete variables k−1 i i that exist at time 0. i=0 Sm1 ΔDm1 is the amount of the discrete control that were added over time [0, t k −Ts ]. i Here Sm1 is the step size of the discrete actuator m at i sampling point t i , and ΔDm1 is the number of steps of the discrete actuator at time t i . Time domain simulation is used to obtain the trajectory of the nominal system (14) and (15), starting from the state x(tk ) at time tk to the end of prediction horizon t k + Tp . At the same time, the trajectory sensitivities of bus voltages with respect to the continuous and discrete controls to be added at instants t k +(n−1)Ts , n = 1 . . . N −k are obtained and denoted as V Ckjmn (t), VDkjmn (t) (see below for the explanation of notation). (2) At time tk , solve the quadratic integer programming optimization problem over the prediction horizon [t k , tk +Tp ] and a control horizon [t k , tk + (N − k)Ts ] as stated in (16)-(23).
4
k k Minimize (with respect to ΔC mn and ΔDmn ) tk +Tp (V k (t) − Vref ) R(V k (t) − Vref )dt tk
+ +
m=M c n=N −k
k Wmn ΔCmn
m=1 n=1 m=M c +Md n=N −k m=Mc +1
k k Wmn Smn ΔDmn
(16)
n=1
Subject to min k max ΔCm ≤ ΔCmn ≤ ΔCm ,
(17)
min k max ΔDm ≤ ΔDmn ≤ ΔDm ,
(18)
min 0 Cm ≤ Cm +
k−1
i ΔCm1 +
i=0 min 0 Dm ≤ Dm +
k−1
N −k n=1
i i Sm1 ΔDm1 +
N −k
≤V
kj
k k max Smn ΔDmn ≤ Dm
n=1
i=0
kj (t) Vmin
k max ΔCmn ≤ Cm (19)
(t) +
Mc N −k
(20)
k VCkjmn (t)ΔCmn
m=1 n=1
+
M −k c +Md N m=Mc +1 n=1
k k kj VDkjmn (t)Smn ΔDmn ≤ Vmax (t)
k ΔCmn ≥ 0, m = 1, · · · , Mc
(21) (22)
k ΔDmn 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 ∈ [t k , tk + TP ] as predicted at the sampling instant t k . – 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 variables, 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 changer plus the number of load shedding candidate locations. – N is the total number of control steps. k is the amount of continuous actuator m to be – ΔCmn added at time tk + (n − 1)Ts in iteration k. k – ΔDmn is the number of steps of discrete actuator m to be added at time t k + (n − 1)Ts in iteration k. It is an integer. k is the step size of discrete actuator m at time – Smn tk + (n − 1)Ts in iteration k. min – ΔCm ∈ is the minimum amount of continuous control m to be added at control sampling points, typically 0. max ∈ is the maximum amount of continuous – ΔCm control m to be added at control sampling points.
min ∈ is the minimum number of steps of – ΔDm discrete control m to be added at control sampling points, typically 0. max ∈ is the maximum number of steps of – ΔDm discrete control m to be added at control sampling points. i is the amount of control m implemented at – ΔCm1 the control sampling point t i , i = 0, ..., k − 1. min ∈ is the minimum amount of continuous – Cm control m that must be used, typically 0. max – Cm ∈ is the maximum available amount of continuous control m. min is the minimum amount of discrete control m. – Dm max – Dm is the maximum available amount of discrete controlm. – V kj (t) ∈ is the voltage of bus j at time t(t k ≤ t ≤ tk + Tp ) of the nominal system at time t k . kj (t) is the minimum voltage at bus j desired at – Vmin time tk ≤ t ≤ tk + Tp . kj – Vmax (t) is the maximum voltage at bus j desired at time tk ≤ t ≤ tk + Tp . – VCkjmn (t) is the trajectory sensitivity of the voltage at bus j at time tk ≤ t ≤ tk + Tp with respect to the continuous control m added at time t k + (n − 1)Ts . – VDkjmn (t) is the trajectory sensitivity of the voltage at bus j at time tk ≤ t ≤ tk + Tp with respect to the discrete control m added at time t k + (n − 1)Ts . The objective of the optimization is to minimize the voltage deviation and cumulative cost of continuous and discrete controls as shown in Equation (16). Equation (17) constraints the amount of the continuous control m to be added at time t k + (n − 1)Ts . Equation (18) is the control step constraints on discrete actuators. Equation (19) constraints the total amount of continuous control m to be added over [t k , tk + (N − k)Ts ]. Equation (20) constraints the total amount of discrete control m to be added over [t k , tk + (N − k)Ts ]. Equation (21) constraints the voltage fluctuation at time t ∈ [t k , tk +Tp ]. The number of candidate control locations and their upper limits are determined through a prior planning step (see for example [26]). The total number of control variables in the optimization is the number of candidate control locations times the number of control steps. The optimization problem is solved in Matlab, and it does converge to a global minimum. (3) At time tk , the solution of the optimization problem (16)k k , ΔDmn . (23) computes a sequence of controls ΔC mn k k k Add only the first control ΔC m1 , Sm1 ΔDm1 at time tk and obtain the system state x(t k+1 ) at time tk+1 = tk + T s . (4) Increase k to k + 1 and repeat steps (1)-(3) until k = N − 1.
IV. I MPLEMENTATION The functional structure of implementing the MPC based coordinated voltage control is shown in Figure 3. Line flow, bus voltage information, switch status measured by phase
5
measurement units (PMUs) and collected by Phasor Data Concentrators (PDCs) are sent to a control center through communication channels. These measurements plus a network model are used by the state estimator (SE) for filtering out the noise and making best use of the measured data. The results from the state estimator are used for power flow analysis. A power flow solution is then used by an on-line dynamic security assessment program to initialize the state variables of the dynamic models. Further, it uses system models and disturbance information to perform the contingency analysis to evaluate the security margin of the power system. If a contingency is identified where the system will become unstable, MPC based computation will get triggered at the time an identified critical contingency occurs. A final step is to implement the real time control computed to improve the security of the power system. The steps of the MPC computation in the k th iteration include: • Estimate static variables such as voltage magnitudes and angles at time tk as well as the dynamic variables x(tk ) such as generator angles, velocities and real and reactive load recovery. The values of the static variables is provided by the state-estimator. As far as the dynamic variables are concerned, they can be classified into shortterm dynamic variables (such as generator angles and velocities) and long-term dynamic variables (such as real and reactive load recovery). The values of the long-term dynamic variables can be directly measured and hence are known, whereas the short-term dynamic variables are in quasi steady-state (QSS) with respect to the long-term voltage/frequency stability phenomenon investigated in this proposal. Thus the values of the short-term dynamic variables can be obtained by solving an equilibrium equation of the form: 0 (= x˙ s ) = fs (xs , xl , y, u),
• •
•
(24)
where xs is the short-term dynamic variable vector (to be computed by solving (24)), x l is long-term dynamic variable vector (which is measured and hence known), y is the static variable vector (which is provided by the state-estimator and hence known), and u is the input variable vector (which is of course known). Then in equation (24), the number of unknowns (dimension of xs ) is the same as the number of equations (dimension of fs ), and so the short-term dynamic variables can be computed by solving (24). Run time-domain simulation to compute the system trajectory given the current state. Obtain trajectory sensitivities of voltage with respect to the control variables as a by-product of the time-domain simulation performed in the previous step. Solve the quadratic programming optimization problem and implement the first step of the control. V. A PPLICATION
The proposed method is illustrated using the modified WECC 9-bus system and New England 39-bus system. The
PMU measurements State estimator Power flow analysis
System model
Disturbance Recorder
On-line dynamic security program System secure? Yes
Fig. 3.
Control signal to power system
MPC based coordinated voltage controller
Yes Disturbance happens?
No
No End
Structure of implementing a MPC based Voltage stabilization
exponential recovery load model is used in both cases. The parameters 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 based on the following considerations. Any voltage instability following a contingency must be stabilized in a certain time duration (typically the time in which voltage will decrease by 15%). This is the prediction horizon T p . The control should be exercised on a time horizon T c , which is shorter than the prediction horizon, typically the time in which voltage will decrease by 10% (if no control is applied). A discrete-time control must be applied within this duration T c at a samplerate high enough to adequately react to the changing voltage trajectory, as well as to allow accurate enough predictions of the voltage trajectory based on the linearization of the trajectory-sensitivity. This dictates the sampling duration T s . The number of sampling point N is then determined as the ratio of Tc and the sampling duration T s . 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 each sampling point. For SVCs, the maximum control amount is 0.1 p.u.. The maximum number of under load tap changer steps is 3. And the maximum load shedding at one sampling point is 10%. The step size of ULTCs is 0.006 p.u.. The step size of load shedding is 5%. A. Modified WECC 3-Generator Test System 1) System description: Fig. 4 is a representation of the modified WECC 3-generator 9-bus system. Transformer banks with under load tap changers are connected to bus 6 and bus 8 to regulate the voltages of load buses 10 and 11. A fourthorder generator model is used. The state variables include rotor angle δ, rotor speed ω, q-axis transient voltage e q , and daxis transient voltage e d . Generator 1 is equipped with an automatic Voltage Regulator (AVR). The continuously acting regulator and exciter model [27] are employed in the study. It is represented by a four-dimensional state equation. The loads at buses 5, 10 and 11 are represented by the exponential recovery dynamic model. Thus, each load is described by a two-dimensional state equation. Therefore, the total dimension of the state space is 22. The voltage control mechanisms include the followings:
6
7
8
Voltage with coordinated control
9
1.1
Gen 2
Bus 5 Bus 7 Bus 9
Gen 3
3
2
1.05
Voltage magnitude (p.u.)
11
5
6
10 4
1
1
0.95
0.9
Gen 1
0.85
Fig. 4.
Modified WECC 3-generator 9-bus test system
0.8
77
Voltage without any control Bus 5
1
Fig. 6.
Bus 7
.)u .p ( ed uit ne ag m gea lto V
• •
100
150
200
250
300
350
400
Time (second)
Voltage behavior of the modified WECC with MPC control
TABLE I T HE RESULTING CONTROL STRATEGY FOR THE WECC SYSTEM
0.95
0.9
0.85
0.8
0
•
50
Bus 9
0.75
Fig. 5.
0
50
100
150
200
250
Time (second)
300
350
400
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 connecting 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-ground fault at bus 5 at t = 1.0 second, which is cleared at t = 1.2 seconds by tripping of the line between bus 4 and bus 5. Based on the time domain simulation, the voltage performance is not satisfactory as shown in Fig. 5. At t = 1.0 second, the voltages begin to drop dramatically due to the fault. At t = 1.2 seconds, the voltages start to recover since the fault is cleared. However, the voltages begin to oscillate. Fifteen seconds later, voltages begin to decline gradually. The dynamic load models result in slightly recovery of load consumption, which deteriorates the voltage condition. Assume that the post-transient load bus voltages must be above 0.95 p.u. Therefore, some control actions are required to satisfy the voltage performance requirement. 3) Simulation result: In this example, we have chosen prediction horizon T p to be 60 seconds (the time in which voltage drops by nearly 15% at bus 5). T c has been chosen to be 50 seconds. We found that a sample duration of T s = 10 seconds works well for this example, and so we have the number of control steps: N = TTcs = 50 10 = 5. The model predictive control approach determines a coordinated control strategy 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. For generator buses, we set the maximum voltage magnitude to be 1.08 p.u., which is slightly higher than load buses. These
Time(second) ULTC between buses 6 and 10 (step) ULTC between buses 8 and 11 (step) Load shedding at bus 5(%) Load shedding at bus 10 (%) Load shedding at bus 11(%) SVC capacitor change at bus 5 (p.u.) SVC capacitor change at bus 7 (p.u.) SVC capacitor change at bus 8 (p.u.)
20 3 3 0 0 0 0.1
30 3 3 0 0 0 0.1
40 3 0 0 0 0 0.1
50 3 0 0 0 0 0
60 0 0 0 0 0 0
0.1
0.1
0.022
0.025
0.001
0.1
0.059
0
0
0
settings are practical. Fig. 6 shows the bus voltages after MPC based control was implemented. From the figure, we can see that all the bus voltages were restored to be above 0.95 p.u. The control strategy is shown in Table I. The first row has the time information of the 5 control sampling points, i.e. 20 seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds. Each column corresponding to the control sampling point has the information of the control actions. For example, at time 20 second, both under load tap changers increase their tap ratios by 3 steps, which is 0.018 p.u.. No load shedding has been taken. All the existing three SVCs increase their susceptance output by 0.1 p.u.. B. Modified New England 9-Generator 39-Bus Test System 1) System description: Fig. 7 shows the modified New England 9-generator 39-bus system. There are totally 41 buses and 9 generators. Two transformer banks with under load tap changers are added between bus 8 and bus 40, bus 4 and bus 41. A fourth-order generator model is used. The exception is that a third-order model is used for the generator at bus 39. In addition, all generators excluding those at bus 34 and bus 37 have automatic voltage regulators (AVRs), which are represented by fourth-order models. The loads are represented by the exponential recovery dynamic models. The control variables 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
30
Voltage behavior with SVC control
G
1.05
37
Bus 34
25
26
28
29
Bus 21 Bus 20
1
27
Voltage magnitude (p.u.)
2
38
1
18
3
G
G
17
39
21
16 15 41
G 14
4
31 G
0.8
20
10
34
32 G
G
0
20
40
22
33 G
60
80
100
120
Time (second)
35 G
Fig. 9. control
Voltage behavior of the modified New England system with SVC
TABLE II T HE CONTROL STRATEGY FOR THE MODIFIED N EW E NGLAND SYSTEM
Modified New England 10-generator 39-bus test system Voltage behavior without control 1.05 Bus Bus Bus Bus
1
Voltage magnitude (p.u.)
0.9
0.85
19
12 11
8
0.95
23
6
7
Fig. 7.
36
13
5 9
40
24
Bus 16
34 21 20 16
0.95
0.9
Time(second) SVC at bus 1 (p.u.) SVC at bus 6 (p.u.) SVC at bus 14 (p.u.) SVC at bus 28 (p.u.) ULTC between buses 8 and 40 (steps) ULTC between buses 4 and bus 41 (steps) Load shedding at bus 15 (%) Load shedding at bus 16 (%)
20 0 0 0 0 3
35 0 0 0.0806 0 3
50 0 0.1 0.1 0.1 3
65 0 0 0.0274 0 3
75 0 0.0544 0.092 0 3
3
3
3
3
3
5
10
10
10
0
0
0
0
0
0
0.85
0.8
0
20
40
60
80
100
120
Time (second)
Fig. 8. Voltage behavior of the modified New England system without MPC control
2) Fault scenario: The contingency considered here is a three-phase-to-ground fault at bus 21 at t = 1.0 second, which is cleared at t = 1.2 seconds by the tripping of the transmission line between bus 21 and bus 22. Bus voltages drop dramatically when the fault occurs as shown in Fig. 8. After the fault is cleared at 1.2 seconds, the voltages recover greatly whereas some oscillations follow. About 20 seconds later, the oscillations are damped out, but the voltages start to decline slowly because of the exponential recovery of the loads. Around 2 minutes later, the voltages collapse. 3) Simulation result with only SVC control: In this test case, there are three types of voltage control options. They are ULTCs, SVCs and load shedding. This subsection studies the effect of SVCs on the restoration of the voltage behavior. There are four SVCs, which locate at bus 1, bus 6, bus 14 and bus 28. The upper limit of these SVCs is 0.3 p.u.. The control strategy is to switch all the available capacity of SVCs at 20 seconds. The voltage behavior is presented in Fig. 9. From this 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 chosen prediction horizon T p to be 90 seconds (the time in which
voltage drops by nearly 12% at bus 20). T c has been chosen to be 75 seconds. We found that a sample duration of T s = 15 seconds works well for this example, and so we have the number of control steps: N = TTsc = 75 15 = 5. The control action determined by the MPC based algorithm starts around 20 seconds to recover voltage. The system response with MPC in 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 are at the maximum change steps at each sampling point. Load shedding is also used to stabilize the system. The table shows a coordinated control strategy between under load tap changer, static var compensators as well as load shedding. VI. C ONCLUSION AND DISCUSSION This paper presents a coordinated voltage control strategy. The design is based on a modified MPC method with a decreasing control horizon. Trajectory sensitivity is used to evaluate the effect of controls on the voltage improvement. The iterative optimization process of MPC helps ensure that errors introduced due to trajectory sensitivity linearizations and any model inaccuracies are minimized. The coordination of static var compensators, under load tap changers and load shedding is achieved by solving a quadratic mixed integer optimization formulation. The test cases indicated that the proposed MPCbased coordinated control strategy can effectively improve the system performance. The proposed methodology also works
8 Voltage behavior with coordinated controls 1.05
Bus Bus Bus Bus
Voltage magnitude (p.u.)
1
34 21 20 16
0.95
0.9
0.85
0.8
0
20
40
60
80
100
120
Time (Second)
Fig. 10. Voltage behavior of the modified New England system with MPCbased coordinated voltage control
for controls such as generator voltage setting points, shunt capacitors. The proposed approach is applicable to industrial-size systems. The control computation at each control step requires (i) estimation of static and dynamic variables, (ii) time-domain simulation to predict system trajectory starting from newly estimated state under the controls applied in the past steps, (iii) trajectory-sensitivity computation, (iv) quadratic mixed-integer programming solution. The most time-consuming component, dominating the other components, is time-domain simulation. Currently there already exist on-line dynamic security assessment (DSA) programs, e.g., the on-line version of DSA of Power Tech. It runs stability study for 3000 contingencies for a 12000 bus system based on a time domain simulation in a 10 minute cycle using around 10 servers. We believe therefore that it should be possible to design controls based on the proposed method for on-line real-time system protection against a single contingency. R EFERENCES [1] C. W. Taylor, Power system voltage stability. McGraw Hill: EPRI Power System Engineering Series, 1994. [2] T. V. Cutsem and C. Vournas, Voltage stability of electric power systems. Boston/Dordrecht/London: Kluwer academic publishers, 1998. [3] S. Corsi, M. Pozzi, C. Sabelli, and A. Serrani, “The coordinated automatic voltage control of the Italian transmission grid-part I: reasons of the choice and overview of the consolidated hierarchical system,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1723–1732, 2004. [4] ——, “The coordinated automatic voltage control of the Italian transmission grid-part II: control apparatuses and field performance of the consolidated hierarchical system,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1733–1741, 2004. [5] H. Vu, P. Pruvot, C. Launay, and Y. Harmand, “An improved voltage control on large-scale power system,” IEEE Transactions on Power Systems, vol. 11, no. 3, pp. 1295–1303, 1996. [6] H. Liu, L. Jin, J. D. McCalley, R. Kumar, and V. Ajjarapu, “Linear complexity search algorithm to locate shunt and series compensation for enhancing voltage stability,” in Proceedings of the 37th Annual North American Power Symposium, 2005, pp. 344–350. [7] J. L. Sancha, J. L. Fernandez, A. Cortes, and J. T. Abarca, “Secondary voltage control: analysis, solutions and simulation results for the Spanish transmission system,” IEEE Transactions on Power Systems, vol. 11, no. 2, pp. 630–638, 1996. [8] J. V. Hecke, N. Janssens, J. Deuse, and F. Promel, “Coordinated voltage control experience in Belgium,” CIGRE Task Force 38-111, Tech. Rep., 2000.
[9] G. Taranto, N. Martins, A. C. B. Martins, D. M. Falcao, and M. G. D. Santos, “Benefits of applying secondary voltage control scheme to the Brazilian system,” in Proceedings of Power Engineering Society Summer Meeting, 2000, pp. 937–942. [10] J. Paserba, D. Leonard, N. Miller, S. Naumann, M. Lauby, and F. Sener, “Coordination of a distribution level continuously controlled compensation device with existing substation equipment for long term var management,” IEEE Transactions on Power Delivery, vol. 20, no. 2, pp. 1034–1040, 1994. [11] J. Y. Park, S. R. Nam, and J. K. Park, “Control of a ULTC considering the dispatch schedule of capacitors in a distribution system,” IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 755–761, 2007. [12] G. Kim and K. Lee, “Coordination control of ULTC transformer and STATCOM based on an artificial neural network,” IEEE Transactions on Power Systems, vol. 2, no. 2, pp. 580–586, 2005. [13] M. Larsson, D. J. Hill, and G. Olsson, “Emergency voltage control using search and predictive control,” International Journal of Power and Energy Systems, vol. 24, no. 2, pp. 121–130, 2002. [14] M. Larsson and D. Karlsson, “Coordinated system protection scheme against voltage collapse using heuristic search and predictive control,” IEEE Transactions on Power Systems, vol. 18, no. 3, pp. 1001–1006, 2003. [15] J. Y. Wen and Q. H. Wu, “Optimal coordinated voltage control for power system voltage stability,” IEEE Transactions on Power Systems, vol. 19, no. 2, pp. 1115–1122, 2004. [16] M. Zima, P. Korba, and G. Andersson, “Power systems voltage emergency control approach using trajectory sensitivities,” in Proceedings of 2003 IEEE Conference on Control Applications, Istanbul, Turkey, 2003, pp. 189–194. [17] M. Zima and G. Andersson, “Stability assessment and emergency control method using trajectory sensitivies,” in Proceedings of 2003 IEEE Bologna Power Tech Conference, Bologna, 2003, p. 7. [18] I. A. Hiskens and B. Gong, “MPC-based load shedding for voltage stability enhancement,” in 44th IEEE Conference on Decision and Control, Seville,Spain, 2005, pp. 4463–4468. [19] M. Zima and G. Andersson, “Model predictive control employing trajectory sensitivities for power systems applications,” in Proceedings of the 44-nd IEEE Conference on Decision and Control, Seville,Spain, 2005, pp. 4452–4456. [20] J. Rawlings, “Tutorial overview of model predictive control,” IEEE Control Systems Magazine, vol. 20, no. 3, pp. 38–52, 2000. [21] H. K. Khalil, Nonlinear System (Third Edition). New Jeasey: Prentice Hall, 2002. [22] I. A. Hiskens and M. A. Pai, “Trajectory sensitivity analysis of hybrid systems,” IEEE Transactions on Circuits and Systems, vol. 47, no. 2, pp. 204–220, 2000. [23] F. Milano, “An open source power system analysis toolbox,” IEEE Transactions on Power Systems, vol. 20, no. 3, pp. 1199–1206, 2005. [24] D. Karlsson and D. Hill, “Modeling and identification of nonlinear dynamic loads in power systems,” IEEE Transactions on Power Systems, vol. 9, no. 1, pp. 157–166, 1994. [25] D. Hill, “Nonlinear dynamic load models with recovery for voltage stability studies,” IEEE Transactions on Power Systems, vol. 8, no. 1, pp. 166–176, 1993. [26] H. Liu, L. Jin, J. D. McCalley, R. Kumar, and V. Ajjarapu, “Planning minimum reactive compensation to mitigate voltage instability,” in Proceedings of 2006 IEEE Power Engineering Society General Meeting, Montreal, Canada, 2006, pp. 4452–4456. [27] I. C. Report, “Computer representation of excitation systems,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-87, no. 6, pp. 1460–1464, 1968. Licheng Jin received the bachelor’s degree in 2000 and master’s degree of science in 2003 from Zhejiang University, Hangzhou, China. She is currently working towards the Ph.D. from Department of Electrical and Computer Engineering, 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 from the Indian Institute of Technology at Kanpur, India, in 1987, and the M.S. and the Ph.D. degree in Electrical and Computer Engineering from the University of Texas at Austin, in 1989 and 1991, respectively. From 1991-2002 he was on the faculty of University of Kentucky, and since 2002 he has been on the faculty of the Iowa State University. He has held visiting positions at Univ. of Maryland at College Park, NASA Ames, Applied Research Lab. at Penn. State, Argonne National Lab.—West, and United Technology Research Center. He is a Fellow of the IEEE.
Coordinated Dynamic Voltage Stabilization based on Model Predictive Control Licheng Jin, Member, IEEE, and Ratnesh Kumar, Fellow, IEEE
Abstract—Keeping voltages of all buses within acceptable bounds is very important for power system operations. This paper presents an approach for optimal coordination of static var compensators (SVCs), transformer under load tap changers (ULTCs) and load shedding to improve voltage performance following large disturbances. The approach is based on model predictive control (MPC) with a decreasing control horizon. The MPC formulation leads to a mixed integer quadratic programming (MIQP) problem, since both continuous and discrete controls are considered. Trajectory sensitivities are used to evaluate the effect of controllers on voltage performance. The iterative optimization process of MPC helps ensure that errors introduced due to any model inaccuracies and approximations are minimized. The MPC based coordination of these controllers is applied to a modified WECC system to enhance the voltage performance and to a modified 39-bus New England system to prevent voltage collapse. Index Terms—Coordinated voltage control, model predictive control, trajectory sensitivity, power system
I. I NTRODUCTION Voltage instability takes the form of a dramatic drop in bus voltages in a transmission system, which may result in system collapse. Nowadays, voltage stability has become a major concern in power system planning and operation [1], [2]. The deregulation of power industry has created an economical incentive to operate power systems closer to their limits. Voltage instability can occur under certain severe disturbances. In practice there exist various choices for exercising voltage control, i.e., reactive power compensation devices, generator reactive power control, transformer tap changer control and load shedding. As shown in Section V, in certain applications, a single type of control does not stabilize the system whereas coordination of multiple types of controls is able to stabilize the system. Therefore, it is imperative that coordinated voltage controls be in place to mitigate the catastrophic effects such as large scale shutdowns and collapses caused by such disturbances. There exists prior work on coordinated voltage control. One approach is based on static analysis. A hierarchical voltage control system for the Italian transmission grid is introduced in [3], [4]. The power plants adjust their reactive The work was supported in part by the National Science Foundation under Grants NSF-ECCS-0424048, NSF-ECCS-0601570, NSF-ECCS-0801763, and NSF-CCF-0811541 L. Jin is with the Department of Electrical and Computer Engineering of Iowa State University, Ames, IA 50011, USA and also with California ISO, Folsom, CA, 95630 USA (email:[email protected]) R. Kumar is with the the Department of Electrical and Computer Engineering of Iowa State University, Ames, IA 50011, USA (email:[email protected])
power outputs based on voltages measured at pivot buses. Hierarchical control schemes are also studied in other countries as shown in [5], [6], [7], [8] and [9]. [10] discusses the coordination of distribution-level under load tap changers (ULTCs), mechanically-switched capacitors and static compensators (STATCOMs) to improve voltage profile and to reduce the mechanical switching operations within a substation. [11] proposes a coordinated control method for ULTCs and capacitors in distribution systems to reduce power loss and to improve voltage profiles during a day. [12] presents an artificial neural network based coordination control scheme for ULTCs and STATCOMs to minimize the amount of transformer tap changes and STATCOM outputs while maintaining acceptable voltage magnitudes at substation buses. Some work has also been done to design a coordinated voltage control strategy by considering dynamic response of a power system. [13] presents a method of coordination of load shedding, capacitor switching and tap changers using model preventive control. The prediction of states is based on the numerical simulation of nonlinear differential algebraic equations (DAEs) together with Euler state prediction. A tree search 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 the predictive control. The prediction of states is based on the linearization of nonlinear DAEs. [15] presents an optimal coordinated voltage control using model predictive control. The controls used include: shunt capacitors, load shedding, tap changers and generator voltage setting points. The prediction of voltage trajectory is based on the Euler state prediction. The optimization problem is solved by a pseudo gradient evolutionary programming (PGEP) technique. In [16] and [17], authors present a method to compute a voltage emergency control strategy based on model predictive control. The prediction of the output trajectories is based on trajectory sensitivity. However, in these two papers, the authors employ a simplified model predictive control, which computes the control actions only at the initial time and implements it over the entire control horizon. A voltage stabilization control strategy is also proposed in [18] based on load shedding, where the objective function is to minimize the amount of load shedding required to restore the voltages. It shows load shedding is an effective voltage control under emergency condition. [19] presents a MPC based voltage control design. The controls are reference voltage of automatic voltage regulators and load shedding. In this paper, we design a coordinated control of SVCs, ULTCs and load shedding to improve voltage performance following disturbances. Given the locations and capabilities
2
Magnitude
of SVCs, ULTCs and interruptible load, the control design problem is to determine the control sequences and the control amounts to satisfy voltage performance requirements. The comparison of our work and the prior works is summarized as follows: •
•
•
•
•
Trajectory sensitivity is used to compute a 1st-order (linear) approximation of the effect of control without having to linearize the system model. Further at each control step, the trajectory sensitivity is updated (based on a prediction of system trajectory starting from an estimate of the current state under the control applied in the past steps). This way of computing the effect of control provides a better approximation as compared to [13], [14], [15], where either system linearization or numerical simulation of DAEs was used. Optimization minimizes costs of control as well as voltage-deviations. In contrast [18], only considers the amount of controls to restore the voltage. Optimization at each control step is a quadratic programming problem, and hence can be efficiently solved. In contrast [15] uses a pseudo gradient evolutionary programming. [13], [14] use a tree search method. In contrast to [16] and [17], where the control action is calculated only at the initial time, and remains the same over the entire control horizon, a sequence of control inputs is determined and only the first of them is applied in our case. [13], [14], [15], [18], [19] are all based on a traditional MPC with constant control horizon. However, our paper is based on a modified MPC with decreasing control horizon which facilitates the convergence of the optimization and reduces the computation time. II. M ETHODOLOGY
A. Model Predictive Control Model predictive control is a class of algorithms that compute a sequence of control variable adjustments in order to optimize the future behavior of a plant (system). MPC was originally developed to meet the specialized control needs of petroleum refineries. Now it has been used in a wide variety of application areas including chemicals, food processing, automotive, aerospace, metallurgy, and power plants. An introduction to the basic concepts and formulations of MPC can be found in [20]. The principle of MPC is graphically depicted in Fig. 1. Here x represents the state variable that needs to be controlled to a specific range. The available control is represented by variable u. At a current time tk , MPC solves an optimization problem over a finite prediction horizon [t k , tk + Tp ] with respect to a predetermined objective function such that the predicted state variable xˆ(tk + Tp ) can optimally stay close to a reference trajectory. 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-plant mismatch and the prediction horizon is infinite, one could apply the control strategy found at current time t k for all
Predicted state
State trajectory until current time, xk
Manipulated input
Input until current time, uk
tk
t k + Ts
tk + Tp
tk + Tc
Time
Control horizon Tc Prediction horizon Tp
Fig. 1.
Principle of MPC
times t ≥ tk . However, due to the disturbances, modelplant mismatch and finite prediction horizon, the true system behavior is different from the predicted behavior. In order to incorporate the feedback information about the true system state, the computed optimal control is implemented only until the next measurement instant (t k + Ts ), at which point the entire computation is repeated. In a MPC, the optimization problem to be solved at time tk can be formulated as follows: tk +Tp minuˆ F (ˆ x(τ ), uˆ(τ ))dτ (1) tk
subject to x ˆ˙ (τ ) = f (ˆ x(τ ), uˆ(τ )), x ˆ(tk ) = x(tk )
(2)
ˆ(τ ) ≤ umax , ∀τ ∈ [tk , tk + Tc ] umin ≤ u
(3)
u ˆ(τ ) = u ˆ(tk + Tc ), ∀τ ∈ [tk + Tc , tk + Tp ]
(4)
ˆ(τ ) ≤ xmax (τ ), ∀τ ∈ [tk , tk + Tp ] xmin (τ ) ≤ x
(5)
Here, Tc and Tp are the control and prediction horizon with Tc ≤ Tp . x ˆ denotes the estimated state and uˆ represents “estimated” control (The true state may be different and the true control matches the estimated control only during the first sampling period). Equation (1) represents the cost function of the MPC optimization. Equation (2) represents the dynamic system model with initial state x(tk ). Equations (3) and (4) represent the constraints on the control input during the prediction horizon. Equation (5) indicates the state operation requirement during the prediction horizon. B. Trajectory Sensitivity Consider a differential algebraic equation (DAE) of a system, x˙
= f (x, y, u),
0
= g(x, y, u)
x(0) = x0
(6) (7)
where x is a vector of state variables, y is a vector of algebraic variables, and u is a vector of control variables. Trajectory sensitivity considers the influence of small variations in the
3 Magnitude
control u (and any other variable of interest) on the solution of the equations (6) and (7). Let u 0 be a nominal value of u, and assume that the nominal system in (8) and (9) has a unique solution x(t, x 0 , u0 ) over [t0 , t1 ]. x˙ = 0 =
f (x, y, u0 ), g(x, y, u0 )
x(0) = x0
Step n
Δx = x Δu nk kl n
Step 1
(8) (9)
x(t, x0 , u0 ) + xu (t)(u − u0 ) + h.o.t. (10) y(t, x0 , u0 ) + yu (t)(u − u0 ) + h.o.t. (11)
0 ,u) is called the trajectory sensitivities Here xu (t) = ∂x(t,x ∂u of state variables with respect to variable u and y u (t) = ∂y(t,x0 ,u) is the trajectory sensitivities of algebraic variables ∂u with respect to variable u. The evolution of trajectory sensitivities can be obtained by differentiating Equations (6) and (7) with respect to the control variables u and is expressed as:
x˙ u (t) = fx (t)xu (t) + fy (t)yu (t) + fu (t) 0 = gx (t)xu (t) + gy (t)yu (t) + gu (t)
(12) (13)
Detailed information about trajectory sensitivity theory can be found in [21]. The trajectory sensitivity can be solved numerically. [22] provides a methodology for the computation of trajectory sensitivity. When time domain simulation of a power system is based on trapezoidal numerical integration, the calculation of trajectory sensitivity requires solving a set of linear equations, thus costing a little time. In our work, we extended the Power System Analysis Tool [23] (a MATLAB based tool) to do trajectory sensitivity calculation and the MPC optimization. Fig. 2 illustrates the application of trajectory sensitivity in evaluating the effect of controls on system behavior. The trajectory xk of the nominal system represents the behavior under the control u k . When the control is increased by Δu k1 at time tk , the change in predicted system behavior based on sensitivity analysis at time t l , can be approximated as l k l Δxkl 1 = xuk Δu1 . Here xuk is the trajectory sensitivity of 1 1 the state variable at time tl with respect to the control at time tk . Similarly if we increase the control by Δu kn at time tk + (n − 1)Ts , the change in the state variable at time t l is l represented by Δx kl Δukn . Here, xluk is the trajectory n = xuk n n sensitivity of the state variable at time t l with respect to the control at time tk + (n − 1)Ts . III. P ROBLEM 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 of differential algebraic equations (DAE) as in Equations (6) and (7). Here x is a vector of states including state variables in generator dynamic models, AVR models and dynamic load models such as, rotor angles and angular speeds of generators, outputs of AVRs, and active power recovery and reactive
Nominal system trajectory
Δx1kl = xul k Δu1k
xk
1
Δunk Δu1k
Then the system in Equations (6) and (7) has a unique solution x(t, x0 , u) over [t0 , t1 ] that is related to x(t, x0 , u0 ) as: x(t, x0 , u) = y(t, x0 , u) =
l u nk
uk tk
Fig. 2.
tk + Ts
t k + (n − 1)Ts
tl
Time
Application of trajectory sensitivity in system behavior prediction
power recovery of dynamic load models. y is a vector of algebraic variables such as bus voltage magnitudes and phase angles. The vector u includes all the control variables. In our setting, it represents the position of under load tap changers, the susceptance output of SVCs and the amount of load shedding. Whenever the occurrence of a certain pre-identified contingency is detected and the system performance is not satisfactory, for instance, voltages are out of the their limits, an optimal coordinated control strategy is identified based on a decreasing horizon MPC algorithm Let Tp be the prediction horizon, T c be the control horizon, Ts be the control sampling interval, and N = TTcs be the total number of control steps. The procedure to determine the control strategy at the k th sampling instant is as follows: (1) At time tk (i.e. the (k+1)th sampling instant), an estimate of the current state x(t k ) is obtained. The nominal power system evolves according to Equations (14) and (15). x˙ = f (x, y, uc , ud ),
x(0) = x0
(14)
0 = g(x, y, uc , ud ) (15) m=Mc 0 i + k−1 is the continuous Here, uc = {Cm i=0 ΔCm1 }m=1 control variable (e.g. amounts of SVC currently in use). 0 Cm is the amounts of continuous variables that exist k−1 i at time 0. i=0 ΔCm1 is the amounts of the continuous variablethat were added over time [0, t k − Ts ]. k−1 i 0 i c +Md ud = {Dm + i=0 Sm1 ΔDm1 }m=M is the discrete m=Mc +1 0 control amount. D m is the amounts of discrete variables k−1 i i that exist at time 0. i=0 Sm1 ΔDm1 is the amount of the discrete control that were added over time [0, t k −Ts ]. i Here Sm1 is the step size of the discrete actuator m at i sampling point t i , and ΔDm1 is the number of steps of the discrete actuator at time t i . Time domain simulation is used to obtain the trajectory of the nominal system (14) and (15), starting from the state x(tk ) at time tk to the end of prediction horizon t k + Tp . At the same time, the trajectory sensitivities of bus voltages with respect to the continuous and discrete controls to be added at instants t k +(n−1)Ts , n = 1 . . . N −k are obtained and denoted as V Ckjmn (t), VDkjmn (t) (see below for the explanation of notation). (2) At time tk , solve the quadratic integer programming optimization problem over the prediction horizon [t k , tk +Tp ] and a control horizon [t k , tk + (N − k)Ts ] as stated in (16)-(23).
4
k k Minimize (with respect to ΔC mn and ΔDmn ) tk +Tp (V k (t) − Vref ) R(V k (t) − Vref )dt tk
+ +
m=M c n=N −k
k Wmn ΔCmn
m=1 n=1 m=M c +Md n=N −k m=Mc +1
k k Wmn Smn ΔDmn
(16)
n=1
Subject to min k max ΔCm ≤ ΔCmn ≤ ΔCm ,
(17)
min k max ΔDm ≤ ΔDmn ≤ ΔDm ,
(18)
min 0 Cm ≤ Cm +
k−1
i ΔCm1 +
i=0 min 0 Dm ≤ Dm +
k−1
N −k n=1
i i Sm1 ΔDm1 +
N −k
≤V
kj
k k max Smn ΔDmn ≤ Dm
n=1
i=0
kj (t) Vmin
k max ΔCmn ≤ Cm (19)
(t) +
Mc N −k
(20)
k VCkjmn (t)ΔCmn
m=1 n=1
+
M −k c +Md N m=Mc +1 n=1
k k kj VDkjmn (t)Smn ΔDmn ≤ Vmax (t)
k ΔCmn ≥ 0, m = 1, · · · , Mc
(21) (22)
k ΔDmn 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 ∈ [t k , tk + TP ] as predicted at the sampling instant t k . – 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 variables, 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 changer plus the number of load shedding candidate locations. – N is the total number of control steps. k is the amount of continuous actuator m to be – ΔCmn added at time tk + (n − 1)Ts in iteration k. k – ΔDmn is the number of steps of discrete actuator m to be added at time t k + (n − 1)Ts in iteration k. It is an integer. k is the step size of discrete actuator m at time – Smn tk + (n − 1)Ts in iteration k. min – ΔCm ∈ is the minimum amount of continuous control m to be added at control sampling points, typically 0. max ∈ is the maximum amount of continuous – ΔCm control m to be added at control sampling points.
min ∈ is the minimum number of steps of – ΔDm discrete control m to be added at control sampling points, typically 0. max ∈ is the maximum number of steps of – ΔDm discrete control m to be added at control sampling points. i is the amount of control m implemented at – ΔCm1 the control sampling point t i , i = 0, ..., k − 1. min ∈ is the minimum amount of continuous – Cm control m that must be used, typically 0. max – Cm ∈ is the maximum available amount of continuous control m. min is the minimum amount of discrete control m. – Dm max – Dm is the maximum available amount of discrete controlm. – V kj (t) ∈ is the voltage of bus j at time t(t k ≤ t ≤ tk + Tp ) of the nominal system at time t k . kj (t) is the minimum voltage at bus j desired at – Vmin time tk ≤ t ≤ tk + Tp . kj – Vmax (t) is the maximum voltage at bus j desired at time tk ≤ t ≤ tk + Tp . – VCkjmn (t) is the trajectory sensitivity of the voltage at bus j at time tk ≤ t ≤ tk + Tp with respect to the continuous control m added at time t k + (n − 1)Ts . – VDkjmn (t) is the trajectory sensitivity of the voltage at bus j at time tk ≤ t ≤ tk + Tp with respect to the discrete control m added at time t k + (n − 1)Ts . The objective of the optimization is to minimize the voltage deviation and cumulative cost of continuous and discrete controls as shown in Equation (16). Equation (17) constraints the amount of the continuous control m to be added at time t k + (n − 1)Ts . Equation (18) is the control step constraints on discrete actuators. Equation (19) constraints the total amount of continuous control m to be added over [t k , tk + (N − k)Ts ]. Equation (20) constraints the total amount of discrete control m to be added over [t k , tk + (N − k)Ts ]. Equation (21) constraints the voltage fluctuation at time t ∈ [t k , tk +Tp ]. The number of candidate control locations and their upper limits are determined through a prior planning step (see for example [26]). The total number of control variables in the optimization is the number of candidate control locations times the number of control steps. The optimization problem is solved in Matlab, and it does converge to a global minimum. (3) At time tk , the solution of the optimization problem (16)k k , ΔDmn . (23) computes a sequence of controls ΔC mn k k k Add only the first control ΔC m1 , Sm1 ΔDm1 at time tk and obtain the system state x(t k+1 ) at time tk+1 = tk + T s . (4) Increase k to k + 1 and repeat steps (1)-(3) until k = N − 1.
IV. I MPLEMENTATION The functional structure of implementing the MPC based coordinated voltage control is shown in Figure 3. Line flow, bus voltage information, switch status measured by phase
5
measurement units (PMUs) and collected by Phasor Data Concentrators (PDCs) are sent to a control center through communication channels. These measurements plus a network model are used by the state estimator (SE) for filtering out the noise and making best use of the measured data. The results from the state estimator are used for power flow analysis. A power flow solution is then used by an on-line dynamic security assessment program to initialize the state variables of the dynamic models. Further, it uses system models and disturbance information to perform the contingency analysis to evaluate the security margin of the power system. If a contingency is identified where the system will become unstable, MPC based computation will get triggered at the time an identified critical contingency occurs. A final step is to implement the real time control computed to improve the security of the power system. The steps of the MPC computation in the k th iteration include: • Estimate static variables such as voltage magnitudes and angles at time tk as well as the dynamic variables x(tk ) such as generator angles, velocities and real and reactive load recovery. The values of the static variables is provided by the state-estimator. As far as the dynamic variables are concerned, they can be classified into shortterm dynamic variables (such as generator angles and velocities) and long-term dynamic variables (such as real and reactive load recovery). The values of the long-term dynamic variables can be directly measured and hence are known, whereas the short-term dynamic variables are in quasi steady-state (QSS) with respect to the long-term voltage/frequency stability phenomenon investigated in this proposal. Thus the values of the short-term dynamic variables can be obtained by solving an equilibrium equation of the form: 0 (= x˙ s ) = fs (xs , xl , y, u),
• •
•
(24)
where xs is the short-term dynamic variable vector (to be computed by solving (24)), x l is long-term dynamic variable vector (which is measured and hence known), y is the static variable vector (which is provided by the state-estimator and hence known), and u is the input variable vector (which is of course known). Then in equation (24), the number of unknowns (dimension of xs ) is the same as the number of equations (dimension of fs ), and so the short-term dynamic variables can be computed by solving (24). Run time-domain simulation to compute the system trajectory given the current state. Obtain trajectory sensitivities of voltage with respect to the control variables as a by-product of the time-domain simulation performed in the previous step. Solve the quadratic programming optimization problem and implement the first step of the control. V. A PPLICATION
The proposed method is illustrated using the modified WECC 9-bus system and New England 39-bus system. The
PMU measurements State estimator Power flow analysis
System model
Disturbance Recorder
On-line dynamic security program System secure? Yes
Fig. 3.
Control signal to power system
MPC based coordinated voltage controller
Yes Disturbance happens?
No
No End
Structure of implementing a MPC based Voltage stabilization
exponential recovery load model is used in both cases. The parameters 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 based on the following considerations. Any voltage instability following a contingency must be stabilized in a certain time duration (typically the time in which voltage will decrease by 15%). This is the prediction horizon T p . The control should be exercised on a time horizon T c , which is shorter than the prediction horizon, typically the time in which voltage will decrease by 10% (if no control is applied). A discrete-time control must be applied within this duration T c at a samplerate high enough to adequately react to the changing voltage trajectory, as well as to allow accurate enough predictions of the voltage trajectory based on the linearization of the trajectory-sensitivity. This dictates the sampling duration T s . The number of sampling point N is then determined as the ratio of Tc and the sampling duration T s . 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 each sampling point. For SVCs, the maximum control amount is 0.1 p.u.. The maximum number of under load tap changer steps is 3. And the maximum load shedding at one sampling point is 10%. The step size of ULTCs is 0.006 p.u.. The step size of load shedding is 5%. A. Modified WECC 3-Generator Test System 1) System description: Fig. 4 is a representation of the modified WECC 3-generator 9-bus system. Transformer banks with under load tap changers are connected to bus 6 and bus 8 to regulate the voltages of load buses 10 and 11. A fourthorder generator model is used. The state variables include rotor angle δ, rotor speed ω, q-axis transient voltage e q , and daxis transient voltage e d . Generator 1 is equipped with an automatic Voltage Regulator (AVR). The continuously acting regulator and exciter model [27] are employed in the study. It is represented by a four-dimensional state equation. The loads at buses 5, 10 and 11 are represented by the exponential recovery dynamic model. Thus, each load is described by a two-dimensional state equation. Therefore, the total dimension of the state space is 22. The voltage control mechanisms include the followings:
6
7
8
Voltage with coordinated control
9
1.1
Gen 2
Bus 5 Bus 7 Bus 9
Gen 3
3
2
1.05
Voltage magnitude (p.u.)
11
5
6
10 4
1
1
0.95
0.9
Gen 1
0.85
Fig. 4.
Modified WECC 3-generator 9-bus test system
0.8
77
Voltage without any control Bus 5
1
Fig. 6.
Bus 7
.)u .p ( ed uit ne ag m gea lto V
• •
100
150
200
250
300
350
400
Time (second)
Voltage behavior of the modified WECC with MPC control
TABLE I T HE RESULTING CONTROL STRATEGY FOR THE WECC SYSTEM
0.95
0.9
0.85
0.8
0
•
50
Bus 9
0.75
Fig. 5.
0
50
100
150
200
250
Time (second)
300
350
400
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 connecting 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-ground fault at bus 5 at t = 1.0 second, which is cleared at t = 1.2 seconds by tripping of the line between bus 4 and bus 5. Based on the time domain simulation, the voltage performance is not satisfactory as shown in Fig. 5. At t = 1.0 second, the voltages begin to drop dramatically due to the fault. At t = 1.2 seconds, the voltages start to recover since the fault is cleared. However, the voltages begin to oscillate. Fifteen seconds later, voltages begin to decline gradually. The dynamic load models result in slightly recovery of load consumption, which deteriorates the voltage condition. Assume that the post-transient load bus voltages must be above 0.95 p.u. Therefore, some control actions are required to satisfy the voltage performance requirement. 3) Simulation result: In this example, we have chosen prediction horizon T p to be 60 seconds (the time in which voltage drops by nearly 15% at bus 5). T c has been chosen to be 50 seconds. We found that a sample duration of T s = 10 seconds works well for this example, and so we have the number of control steps: N = TTcs = 50 10 = 5. The model predictive control approach determines a coordinated control strategy 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. For generator buses, we set the maximum voltage magnitude to be 1.08 p.u., which is slightly higher than load buses. These
Time(second) ULTC between buses 6 and 10 (step) ULTC between buses 8 and 11 (step) Load shedding at bus 5(%) Load shedding at bus 10 (%) Load shedding at bus 11(%) SVC capacitor change at bus 5 (p.u.) SVC capacitor change at bus 7 (p.u.) SVC capacitor change at bus 8 (p.u.)
20 3 3 0 0 0 0.1
30 3 3 0 0 0 0.1
40 3 0 0 0 0 0.1
50 3 0 0 0 0 0
60 0 0 0 0 0 0
0.1
0.1
0.022
0.025
0.001
0.1
0.059
0
0
0
settings are practical. Fig. 6 shows the bus voltages after MPC based control was implemented. From the figure, we can see that all the bus voltages were restored to be above 0.95 p.u. The control strategy is shown in Table I. The first row has the time information of the 5 control sampling points, i.e. 20 seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds. Each column corresponding to the control sampling point has the information of the control actions. For example, at time 20 second, both under load tap changers increase their tap ratios by 3 steps, which is 0.018 p.u.. No load shedding has been taken. All the existing three SVCs increase their susceptance output by 0.1 p.u.. B. Modified New England 9-Generator 39-Bus Test System 1) System description: Fig. 7 shows the modified New England 9-generator 39-bus system. There are totally 41 buses and 9 generators. Two transformer banks with under load tap changers are added between bus 8 and bus 40, bus 4 and bus 41. A fourth-order generator model is used. The exception is that a third-order model is used for the generator at bus 39. In addition, all generators excluding those at bus 34 and bus 37 have automatic voltage regulators (AVRs), which are represented by fourth-order models. The loads are represented by the exponential recovery dynamic models. The control variables 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
30
Voltage behavior with SVC control
G
1.05
37
Bus 34
25
26
28
29
Bus 21 Bus 20
1
27
Voltage magnitude (p.u.)
2
38
1
18
3
G
G
17
39
21
16 15 41
G 14
4
31 G
0.8
20
10
34
32 G
G
0
20
40
22
33 G
60
80
100
120
Time (second)
35 G
Fig. 9. control
Voltage behavior of the modified New England system with SVC
TABLE II T HE CONTROL STRATEGY FOR THE MODIFIED N EW E NGLAND SYSTEM
Modified New England 10-generator 39-bus test system Voltage behavior without control 1.05 Bus Bus Bus Bus
1
Voltage magnitude (p.u.)
0.9
0.85
19
12 11
8
0.95
23
6
7
Fig. 7.
36
13
5 9
40
24
Bus 16
34 21 20 16
0.95
0.9
Time(second) SVC at bus 1 (p.u.) SVC at bus 6 (p.u.) SVC at bus 14 (p.u.) SVC at bus 28 (p.u.) ULTC between buses 8 and 40 (steps) ULTC between buses 4 and bus 41 (steps) Load shedding at bus 15 (%) Load shedding at bus 16 (%)
20 0 0 0 0 3
35 0 0 0.0806 0 3
50 0 0.1 0.1 0.1 3
65 0 0 0.0274 0 3
75 0 0.0544 0.092 0 3
3
3
3
3
3
5
10
10
10
0
0
0
0
0
0
0.85
0.8
0
20
40
60
80
100
120
Time (second)
Fig. 8. Voltage behavior of the modified New England system without MPC control
2) Fault scenario: The contingency considered here is a three-phase-to-ground fault at bus 21 at t = 1.0 second, which is cleared at t = 1.2 seconds by the tripping of the transmission line between bus 21 and bus 22. Bus voltages drop dramatically when the fault occurs as shown in Fig. 8. After the fault is cleared at 1.2 seconds, the voltages recover greatly whereas some oscillations follow. About 20 seconds later, the oscillations are damped out, but the voltages start to decline slowly because of the exponential recovery of the loads. Around 2 minutes later, the voltages collapse. 3) Simulation result with only SVC control: In this test case, there are three types of voltage control options. They are ULTCs, SVCs and load shedding. This subsection studies the effect of SVCs on the restoration of the voltage behavior. There are four SVCs, which locate at bus 1, bus 6, bus 14 and bus 28. The upper limit of these SVCs is 0.3 p.u.. The control strategy is to switch all the available capacity of SVCs at 20 seconds. The voltage behavior is presented in Fig. 9. From this 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 chosen prediction horizon T p to be 90 seconds (the time in which
voltage drops by nearly 12% at bus 20). T c has been chosen to be 75 seconds. We found that a sample duration of T s = 15 seconds works well for this example, and so we have the number of control steps: N = TTsc = 75 15 = 5. The control action determined by the MPC based algorithm starts around 20 seconds to recover voltage. The system response with MPC in 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 are at the maximum change steps at each sampling point. Load shedding is also used to stabilize the system. The table shows a coordinated control strategy between under load tap changer, static var compensators as well as load shedding. VI. C ONCLUSION AND DISCUSSION This paper presents a coordinated voltage control strategy. The design is based on a modified MPC method with a decreasing control horizon. Trajectory sensitivity is used to evaluate the effect of controls on the voltage improvement. The iterative optimization process of MPC helps ensure that errors introduced due to trajectory sensitivity linearizations and any model inaccuracies are minimized. The coordination of static var compensators, under load tap changers and load shedding is achieved by solving a quadratic mixed integer optimization formulation. The test cases indicated that the proposed MPCbased coordinated control strategy can effectively improve the system performance. The proposed methodology also works
8 Voltage behavior with coordinated controls 1.05
Bus Bus Bus Bus
Voltage magnitude (p.u.)
1
34 21 20 16
0.95
0.9
0.85
0.8
0
20
40
60
80
100
120
Time (Second)
Fig. 10. Voltage behavior of the modified New England system with MPCbased coordinated voltage control
for controls such as generator voltage setting points, shunt capacitors. The proposed approach is applicable to industrial-size systems. The control computation at each control step requires (i) estimation of static and dynamic variables, (ii) time-domain simulation to predict system trajectory starting from newly estimated state under the controls applied in the past steps, (iii) trajectory-sensitivity computation, (iv) quadratic mixed-integer programming solution. The most time-consuming component, dominating the other components, is time-domain simulation. Currently there already exist on-line dynamic security assessment (DSA) programs, e.g., the on-line version of DSA of Power Tech. It runs stability study for 3000 contingencies for a 12000 bus system based on a time domain simulation in a 10 minute cycle using around 10 servers. We believe therefore that it should be possible to design controls based on the proposed method for on-line real-time system protection against a single contingency. R EFERENCES [1] C. W. Taylor, Power system voltage stability. McGraw Hill: EPRI Power System Engineering Series, 1994. [2] T. V. Cutsem and C. Vournas, Voltage stability of electric power systems. Boston/Dordrecht/London: Kluwer academic publishers, 1998. [3] S. Corsi, M. Pozzi, C. Sabelli, and A. Serrani, “The coordinated automatic voltage control of the Italian transmission grid-part I: reasons of the choice and overview of the consolidated hierarchical system,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1723–1732, 2004. [4] ——, “The coordinated automatic voltage control of the Italian transmission grid-part II: control apparatuses and field performance of the consolidated hierarchical system,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1733–1741, 2004. [5] H. Vu, P. Pruvot, C. Launay, and Y. Harmand, “An improved voltage control on large-scale power system,” IEEE Transactions on Power Systems, vol. 11, no. 3, pp. 1295–1303, 1996. [6] H. Liu, L. Jin, J. D. McCalley, R. Kumar, and V. Ajjarapu, “Linear complexity search algorithm to locate shunt and series compensation for enhancing voltage stability,” in Proceedings of the 37th Annual North American Power Symposium, 2005, pp. 344–350. [7] J. L. Sancha, J. L. Fernandez, A. Cortes, and J. T. Abarca, “Secondary voltage control: analysis, solutions and simulation results for the Spanish transmission system,” IEEE Transactions on Power Systems, vol. 11, no. 2, pp. 630–638, 1996. [8] J. V. Hecke, N. Janssens, J. Deuse, and F. Promel, “Coordinated voltage control experience in Belgium,” CIGRE Task Force 38-111, Tech. Rep., 2000.
[9] G. Taranto, N. Martins, A. C. B. Martins, D. M. Falcao, and M. G. D. Santos, “Benefits of applying secondary voltage control scheme to the Brazilian system,” in Proceedings of Power Engineering Society Summer Meeting, 2000, pp. 937–942. [10] J. Paserba, D. Leonard, N. Miller, S. Naumann, M. Lauby, and F. Sener, “Coordination of a distribution level continuously controlled compensation device with existing substation equipment for long term var management,” IEEE Transactions on Power Delivery, vol. 20, no. 2, pp. 1034–1040, 1994. [11] J. Y. Park, S. R. Nam, and J. K. Park, “Control of a ULTC considering the dispatch schedule of capacitors in a distribution system,” IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 755–761, 2007. [12] G. Kim and K. Lee, “Coordination control of ULTC transformer and STATCOM based on an artificial neural network,” IEEE Transactions on Power Systems, vol. 2, no. 2, pp. 580–586, 2005. [13] M. Larsson, D. J. Hill, and G. Olsson, “Emergency voltage control using search and predictive control,” International Journal of Power and Energy Systems, vol. 24, no. 2, pp. 121–130, 2002. [14] M. Larsson and D. Karlsson, “Coordinated system protection scheme against voltage collapse using heuristic search and predictive control,” IEEE Transactions on Power Systems, vol. 18, no. 3, pp. 1001–1006, 2003. [15] J. Y. Wen and Q. H. Wu, “Optimal coordinated voltage control for power system voltage stability,” IEEE Transactions on Power Systems, vol. 19, no. 2, pp. 1115–1122, 2004. [16] M. Zima, P. Korba, and G. Andersson, “Power systems voltage emergency control approach using trajectory sensitivities,” in Proceedings of 2003 IEEE Conference on Control Applications, Istanbul, Turkey, 2003, pp. 189–194. [17] M. Zima and G. Andersson, “Stability assessment and emergency control method using trajectory sensitivies,” in Proceedings of 2003 IEEE Bologna Power Tech Conference, Bologna, 2003, p. 7. [18] I. A. Hiskens and B. Gong, “MPC-based load shedding for voltage stability enhancement,” in 44th IEEE Conference on Decision and Control, Seville,Spain, 2005, pp. 4463–4468. [19] M. Zima and G. Andersson, “Model predictive control employing trajectory sensitivities for power systems applications,” in Proceedings of the 44-nd IEEE Conference on Decision and Control, Seville,Spain, 2005, pp. 4452–4456. [20] J. Rawlings, “Tutorial overview of model predictive control,” IEEE Control Systems Magazine, vol. 20, no. 3, pp. 38–52, 2000. [21] H. K. Khalil, Nonlinear System (Third Edition). New Jeasey: Prentice Hall, 2002. [22] I. A. Hiskens and M. A. Pai, “Trajectory sensitivity analysis of hybrid systems,” IEEE Transactions on Circuits and Systems, vol. 47, no. 2, pp. 204–220, 2000. [23] F. Milano, “An open source power system analysis toolbox,” IEEE Transactions on Power Systems, vol. 20, no. 3, pp. 1199–1206, 2005. [24] D. Karlsson and D. Hill, “Modeling and identification of nonlinear dynamic loads in power systems,” IEEE Transactions on Power Systems, vol. 9, no. 1, pp. 157–166, 1994. [25] D. Hill, “Nonlinear dynamic load models with recovery for voltage stability studies,” IEEE Transactions on Power Systems, vol. 8, no. 1, pp. 166–176, 1993. [26] H. Liu, L. Jin, J. D. McCalley, R. Kumar, and V. Ajjarapu, “Planning minimum reactive compensation to mitigate voltage instability,” in Proceedings of 2006 IEEE Power Engineering Society General Meeting, Montreal, Canada, 2006, pp. 4452–4456. [27] I. C. Report, “Computer representation of excitation systems,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-87, no. 6, pp. 1460–1464, 1968. Licheng Jin received the bachelor’s degree in 2000 and master’s degree of science in 2003 from Zhejiang University, Hangzhou, China. She is currently working towards the Ph.D. from Department of Electrical and Computer Engineering, 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 from the Indian Institute of Technology at Kanpur, India, in 1987, and the M.S. and the Ph.D. degree in Electrical and Computer Engineering from the University of Texas at Austin, in 1989 and 1991, respectively. From 1991-2002 he was on the faculty of University of Kentucky, and since 2002 he has been on the faculty of the Iowa State University. He has held visiting positions at Univ. of Maryland at College Park, NASA Ames, Applied Research Lab. at Penn. State, Argonne National Lab.—West, and United Technology Research Center. He is a Fellow of the IEEE.
