# power_system_restoration_with_transient_stability__7ff416ec.pdf

Power System Restoration With Transient Stability

Hassan Hijazi and Terrence W.K. Mak and Pascal Van Hentenryck NICTA and Australian National University 7 London Circuit, Canberra, ACT 2601, Australia {Hassan.Hijazi,Terrence.Mak,Pascal.Vanhentenryck}@nicta.com.au

We address the problem of power system restoration after a signiﬁcant blackout. Prior work focus on optimization methods for ﬁnding high-quality restoration plans. Optimal solutions consist in a sequence of grid repairs and corresponding steady states. However, such approaches lack formal guarantees on the transient stability of restoration actions, a key property to avoid additional grid damage and cascading failures. In this paper, we show how to integrate transient stability in the optimization procedure by capturing the rotor dynamics of power generators. Our approach reasons about the differential equations describing the dynamics and their underlying transient states. The key contribution lies in modeling and solving optimization problems that return stable generators dispatch minimizing the difference with respect to steady states solutions. Computational efﬁciency is increased using preprocessing procedures along with traditional reduction techniques. Experimental results on existing benchmarks conﬁrm the feasibility of the new approach.

Introduction Power system restoration is a key ingredient in deﬁning the self-healing property of the future smart grid. Restoration plans are generated automatically using smart optimization procedures, reducing the cost and the inconvenience caused by blackouts. A restoration plan consists in a set of repair actions, such as connecting or disconnecting a line. The primary objective is to isolate faulty equipments while maximizing the percentage of load recovery. The problem has a time dimension related to the ordering of the repair actions. This ordering has a direct inﬂuence on the load recovery over time. It is, for instance, natural to favor early load recoveries. This captures the essence behind the Restoration Ordering Problem which is at the core of this paper. Traditionally, this problem is addressed with simplifying assumptions, e.g., ignoring ramping rates of generators but, most importantly, disregarding transient stability during line switching. Incorporating operational limits on transient states is computationally challenging, as they are deﬁned using partial differential equations (PDEs). Existing discrete optimization tools cannot deal with such non-algebraic expressions. In our context, this refers to the ability to reason

Copyright c 2015, Association for the Advancement of Artiﬁcial Intelligence (www.aaai.org). All rights reserved.

about the power system transient states over a short period of time (e.g., a few seconds) after the line closing. In this work, we focus on capturing the dynamics of generator rotor angles as a measure of transient stability related to switching operations in transmission networks. Our contributions are based on a few key ideas:

1. We use rotor angle stability, a traditional metric to assess the transient stability of a system, and the classical model of generator dynamics based on the Swing equations (Kundur, Balu, and Lauby 1994).

2. We capture transient states through a discretization of the dynamics model, including a nonlinear formulation of the power ﬂow equations on these transient states.

3. For each restoration step, we deﬁne an optimization model that ﬁnds a transient-stable generator dispatch minimizing the difference with respect to the steady state solution. This ensures that the size of the blackouts increase as little as possible while ensuring transient stability.

More generally, this paper can be viewed as a novel, hierarchical approach to the optimization of complex hybrid dynamic systems. Given a static version of the models, the key idea relies in adjusting optimal steady-state solutions taking into account the system dynamics. The overall computational tractability is thus improved.

Related Work

Our work is closely related to the transient-stable optimal power ﬂow problem, which was ﬁrst proposed by Gan et al. (Gan, Thomas, and Zimmerman 2000). The problem was later extended to multi-contingency settings in (Yuan, Kubokawa, and Sasaki 2003). Both approaches utilize the swing equation to reason on transient stability. Our work is also related to techniques improving stability during transmission loop closures in normal operating conditions, e.g., techniques on reducing rotor shaft impacts and standing phase angles (Martins et al. 2008; Ye and Liu 2013; Ketabi, Ranjbar, and Feuillet 2002; Hazarika and Sinha 1999). To the best of our knowledge, this work represents the ﬁrst attempt to incorporate transient stability into full power restoration schemes and to optimize generator dispatches and load pickups as additional degrees of freedom.

Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence

Background In this section, we introduce our notations and discuss the Restoration Ordering Problem (Van Hentenryck, Coffrin, and Bent 2011; Coffrin and Van Hentenryck 2014b).

Nomenclature N Set of buses in the power grid N(n) Set of buses connected to bus n L N N Set of lines G N Set of generators in the power grid G(n) Set of generators at bus n D N Set of demands in the power grid D(n) Set of demands at bus n R Set of damaged components to repair Sij = pij + iqij Power ﬂow for line < i, j > yij = gij + ibij Line admittance for line < i, j > Y = G + i B The Y-bus admittance matrix Vi = vi θi Voltage magnitude and phase angle at bus i θij, vij Shorthand for θi θj and vi vj x, x Upper and Lower bound on x

The Restoration Ordering Problem This section introduces the main notations used in the paper and formalizes the Restoration Ordering Problem (ROP) (Van Hentenryck, Coffrin, and Bent 2011; Coffrin and Van Hentenryck 2014b). The ROP takes as inputs the topology of the network PN and the set R of damaged components to be repaired. Its main goal is to ﬁnd the best ordering of the repairs in R in order to recover the loads in the network as quickly as possible. The intuition underlying the ROP is depicted in Figure 1. The ROP only reasons about steady states, each of which corresponds to a restoration action. Moreover, at each step, the ROP searches for a steady state that maximizes the served load, which can be expressed as a power ﬂow sub-problem. In other words, the ROP searches over sequences of |R| steady states and aims at ﬁnding the sequence maximizing the served load over time. This paper focuses on transmission line restorations for simplicity (i.e., R L) but the ROP can be generalized easily to other components (Van Hentenryck, Coffrin, and Bent 2011). Model 1 shows a simpliﬁed Mixed-Integer Nonlinear Program for the ROP. Its two main set of decision variables are the variables zrl and lri: Binary variable zrl speciﬁes whether line l is operational at step r and variable lri represents the percentage of served load at bus i for step r. The remaining variables are the traditional power ﬂow variables. There are |R| copies of these variables (and the associated constraints), one for each restoration step. The objective function (O.1) maximizes the load restored over time. Constraints (C.1.1 C.1.2) ensures that exactly only one line is repaired at each step. Constraints (C.2.1 C.2.2) enforce power conservation on nodes while constraints (C.2.3 C.2.8) implement the AC steady-state power ﬂow equations and the thermal limits on transmission lines (i.e., constraints (C.2.5) and (C.2.8)). Constraints (C.2.3 C.2.5) are conditional to the appropriate line being operational.

Model 1 AC Restoration Ordering Problem

Inputs: PN = N, L, G, D - Power network R L - Damaged items pl i, ql i - Active/reactive power loads on bus i Variables: (0 r |R|) zrl {0, 1}, l R - Operating state of line l at step r lri [0, 1], i D - load percentage of load i at step r θri ( , ), i N - bus angle at step r vri (vi, vi), i N - bus voltage magnitude at step r pg ri (0, pg i ), i G - Active injection of generator i qg ri (qg i , qg i ) , i G - Reactive injection of generator i prij ( Sij, Sij) - Active ﬂow from i to j qrij ( Sij, Sij) - Reactive ﬂow from i to j Maximize |R| X

pl ilri (O.1)

Subject to: (0 r |R|) X

l R zrl = r, (C.1.1)

z(r 1)l zrl, l R, r = 0 (C.1.2) i N: X

j G(i) pg rj X

j G(i) pl jlrj = X

j N(i) prij (C.2.1)

j G(i) qg rj X

j G(i) ql jlrj = X

j N(i) qrij (C.2.2)

i, j : l R between buses i and j prij = zrl(gijv2 ri vrivrj(gij cos(θrij) bij sin(θrij))) (C.2.3) qrij = zrl( bijv2 ri vrivrj(gij sin(θrij) bij cos(θrij))) (C.2.4) p2 rij + q2 rij Sijzrij (C.2.5) i, j : l L R between buses i and j prij = gijv2 ri vrivrj(gij cos(θrij) bij sin(θrij)) (C.2.6) qrij = bijv2 ri vrivrj(gij sin(θrij) bij cos(θrij)) (C.2.7) p2 rij + q2 rij Sij (C.2.8)

The ROP is computationally challenging. On the one hand, it suffers from the combinatorial explosion of ordering problems. On the other hand, the non-convex power ﬂow equations represent a major computational challenge in cold-start contexts (Stott, Jardim, and Alsac 2009; Overbye, Cheng, and Sun 2004). The ﬁrst difﬁculty is typically addressed using greedy algorithms, large neighborhood search, and/or randomized adaptive decompositions (Bent and Van Hentenryck 2007; Van Hentenryck, Coffrin, and Bent 2011). The second difﬁculty is avoided by convexifying the nonlinear power ﬂow equations, e.g., using linear approximations (Van Hentenryck, Coffrin, and Bent 2011; Coffrin and Van Hentenryck 2014a; 2014b), or quadratic relaxations (Coffrin et al. 2014; Hijazi, Coffrin, and Van Hentenryck 2013).

Transient Stability for Restoration Planning

Prior work on power restoration has largely focused on computing optimal restoration plans maximizing load recovery over steady states. However, in practice, these restorations

Figure 1: A diagram illustrating a sequence of steady states generated by Restoration Ordering Problem

may not always be feasible as there could be violations of operational limits, safety limits, and stability limits (Adibi and Kafka 1991). The ROP only captures such operational limits as algebraic constraints over steady states, e.g., thermal limits imposed by Constraint C.2.5 in Model 1. It does not capture limits on transient states. Hence, it is possible to derive restoration actions that are not transient stable (Kheradmandi and Ehsan 2004), thus increasing the size of the blackout. Adding constraints on the steady states (Mak et al. 2014) can reduce this possibility. This paper goes one step further by incorporating the system dynamics, which we now present the system dynamics before describing the overall approach.

Rotor Dynamics and Swing Equations The general form for computing power system dynamic response (Kundur, Balu, and Lauby 1994; Yuan, Kubokawa, and Sasaki 2003) after a disturbance is usually written as follows:

x = f(x, y) 0 = g(x, y)

where g( ) represents a set of algebraic equations describing the power transmission network and f( ) represents a set of ﬁrst order differential equations describing the internal dynamics of network equipments. Vector x captures the short-term dynamic variables and y is a vector of algebraic state variables. Given an initial condition for variables x and y, we can compute a set of transient states over time by using equations f( ) and g( ). To incorporate rotor angle stability, we ﬁrst introduce the classical model of generator dynamics corresponding to the Swing equations (Kundur, Balu, and Lauby 1994) (for implementing f( )):

dt = pm i pe i Diωi

where Hi, δi, Di, ωi, and ω0 respectively denote the inertia constant, the rotor angle, the damping coefﬁcient, the current angular velocity, and the nominal angular velocity of a generator i (which is assumed constant for all generators at 50Hz/60Hz). On the right hand side of the equation, pm i and pe i are the mechanical and electrical powers acting on the rotor of generator i. In the Swing equations, δi and ωi are short-term dynamic variables (i.e., the x-vector above) and pm i and pe i are algebraic state variables (i.e., the y-vector above). In steady states, the mechanical power is assumed to be equal to the electrical power and the rotor angle of all generators remains constant. Once a disturbance occurs, the

electrical power changes and there is a difference between mechanical and electrical power. Hence, the rotor angle of generators changes according to the Swing equation. Figure 2 shows a the rotor swings after the closure of a transmission line for a 3-bus example. The y-axis shows the rotor angle (in degrees) of the generators and the x-axis shows the time in seconds after the line closing.

Figure 2: Rotor swings during transmission loop closures

To implement the Swing equations in optimization tools, we perform a trapezoidal discretization over a given time horizon (1 t T) (Gan, Thomas, and Zimmerman 2000). This leads to the following discretized equations, i G, 1 t T 1,

2 (ωit+1 + ωit) = 0 (1)

2 (ait+1 + ait) = 0 (2)

2Hi (pe i1 pe it+1 Diωit) = 0 (3)

with the following initial conditions at t = 1,

ωi1 = 0, ai1 = 0 (4)

where δit, ωit, ait, and pe it are the rotor angle, the angular velocity, the angular acceleration, and the electrical power of generator i at time step t, 1 t T. is a constant representing the time interval between every time step. To increase rotor stability, (Gan, Thomas, and Zimmerman 2000) suggest to bound at all time steps the difference between the rotor angle of a generator δit and a reference angle δr t representing the Center Of Inertia (COI):

δr t = P i G Hiδit P i G Hi , 1 t T (5)

The stability constraints then become:

i G, 1 t T : δ δit δr t δ (6)

where δ represents the bound constant to be determined.

AC Power Flow Equations Over Time The set of algebraic constraints g( ) include the AC power ﬂow equations characterizing power transmission and passive network equipments (Liu and Thorp 2000). Two admittance matrices deﬁne the line properties and the load characteristics, before and after a line switch, resp. Y o and Y c. The AC power ﬂow equations are then speciﬁed as follows, i N, 2 t T, X

j N [Vit Vjt(Gc ij cos(θijt) + Bc ij sin(θijt))] = X

j G(i) pe jt (7)

j N [Vit Vjt(Gc ij sin(θijt) Bc ij cos(θijt))] = X

j G(i) qe jt (8)

with the following initial conditions for t = 1: X

j N [Vi1Vj1(Go ij cos(θij1) + Bo ij sin(θij1))] = X

j G(i) pe j1 (9)

j N [Vi1Vj1(Go ij sin(θij1) Bo ij cos(θij1))] = X

j G(i) qe j1 (10)

In the classical generator model, every generator i is deﬁned as an internal bus connected to a terminal node with transient reactance Xi and constant voltage V g i . The rotor angles δit are then treated as bus phase angles. Note that we ignore the reactive power ﬂow constraints for transient states (i.e., when time t > 1) (Liu and Thorp 2000). For all i G, 1 t T,

pe it = V g i Vit

Xi sin(θit δit) (11)

with the following extra condition on time step t = 1,

qe i1 = V 2 i1 Xi V g i Vi1

Xi cos(θi1 δi1) (12)

The last part corresponds to boundary conditions representing safety/operating limits. For the initial state at t = 1, for all i G we have,

0 pe i1 pe i, qe i qe i1 qe i, V g i V g i V g i (13)

For all time steps t, we have,

i N :V i Vit V i (14)

i G : θ δit θit θ (15)

i, j L : θ θit θjt θ (16)

Let us emphasize that this formulation allows voltage magnitudes to ﬂuctuate at every transient time step t. This may naturally lead to other stability issues. In order to avoid drastic ﬂuctuations, we add the following constraints bounding their magnitude for all time steps. i N, 1 t T 1,

Vit V Vit+1 Vit + V . (17)

A Non-Linear Program for Stability Enhancement Given the set of constraints (1)-(17) deﬁning the transient behavior of the system, we now introduce Model 2 for stability enhancement after a line-switching operation. An optimal solution in Model 2 guarantees a transient-stable switching with generator dispatches that minimize the distance to the original steady-state solution.

Model 2 Transient Stable Line Closing Model

Inputs: PN - Power network Y o, Y c - Admittance matrix before and after line closing p T i , q T i - ith generator active/reactive target dispatch, Xi, Hi, Di - transient reactance, inertia, and damping constant fq = ω0

2π , , T - Grid frequency, integration step, time horizon δ, V - Maximum rotor swing and voltage ﬂuctuations Variables: ( t : 1 t T) i N, θit - Terminal bus angle Vit - Bus voltage i G, V g i - Internal bus voltage pe it, qe it - Generator active and reactive injection δit, ωit, ait - Generator rotor angle, velocity, and acceleration δr t - Rotor angle reference index Minimize X

i G [(pe i1 p T i )2 + (qe i1 q T i )2]- Distance from target dispatch

Subject to: Equations (1) - (17)

Simpliﬁcation and Kron Reduction

In Model 2, the number of variables is O(|N|T). The model dimension can be reduced by constructing a smaller equivalent network and computing the corresponding admittance matrices. Since a line closing only affects a subset of connected complements in the network, we disregard islands that are not related to the current switching. An admittance preserving reduction known as Kron reduction (Ward 1949) is then performed. This technique removes all buses that are not under study by translating their properties into new lines. In our case, since we are only interested in rotor dynamics, Kron reduction is used to remove all non-generator buses. This results in two admittance matrices Y o and Y c representing the reduced network before and after line closing. The number of variables in Model 2 is then O(|G| T), where G represents the number of generators on the island corresponding to the line closings. Figure 3 shows an example of Kron reduction on a 6-bus network with three generators (G1 to G3), three loads (L1 to L3), and ﬁve transmission lines. After performing the Kron reduction, only three buses (with generators) are left. Three virtual transmission lines are constructed and loads L1 to L3 are re-adjusted to maintain an equivalent transfer admittance and power ﬂows.

Figure 3: An example network illustrating Kron reduction

Figure 4: Flow chart for the stability enhancement routine

Kron reduction is inspired by Kirchhoff s Current Law (KCL) (Kundur, Balu, and Lauby 1994). Given an admittance matrix Y , a vector of voltages V , and a vector of current injections I, the KCL law (in matrix form) states that:

Since the current injections will be zero for non-generator buses, we can use Gaussian elimination to delete corresponding rows from the system of equations. This will allow us to generate a reduced Y-bus matrix representing the network. Following the Gaussian elimination, we obtain the following set of equations allowing to remove a non-generator bus n from the network.

i, j N {n} : Y ij = Yij Yin Ynj

Stability Enhancement Routine Figure 4 presents our overall approach to power restoration with transient stability. Step 1 solves the ROP to obtain a restoration plan deﬁned as a sequence of steady states. Since this solution may be computed using approximation techniques, it may be necessary to convert the potentially non-AC feasible steady states into AC feasible solutions. A standard load ﬂow procedure can be used in the conversion process. Step 2 extracts the sub-sequence of steady states corresponding to line switchings within an energized island. One can safely ignore lines connecting two islands as this is covered by standard approaches (e.g., monitoring periodic oscillations of the phase angle variables or using a synchronization check relay)(Kundur, Balu, and Lauby 1994). Steps 3 and 4 remove islands which are not related to the line-switching operations and perform a Kron reduction (as

discussed in the last section) in order to reduce the size of Model 2. Finally, our optimization model is run in Step 5 to ﬁnd generator dispatches that ensure transient stability during the switching operations.

Experimental Results We now present experimental results to demonstrate the beneﬁts of the proposed approach. The experimental results study whether the steady states returned by the ROP are transient-stable under various increasingly strict rotor angle limits and for various reactance values. They also study the magnitude of changes in generator dispatches to ensure transient stability, i.e., how close the transientstable dispatches pe i1 and qe i1 and the target dispatches p T i and q T i are. Our model is implemented in AMPL (Fourer, Gay, and Kernighan 2002; AMPL 2014), and uses Ipopt 3.11 (W achter and Biegler 2006) as a nonlinear solver. We have tested our model on 5 MATPOWER benchmarks (Zimmerman, Murillo-S andnchez, and Thomas 2011). Note that the ROP was solved using the utilization heuristic (Coffrin, Van Hentenryck, and Bent 2012). Table 1 presents experimental results for various generator reactances and maximum rotor swing limits δ. The table presents aggregate results obtained by summing up all the objective values after solving Model 2 for all of the restoration steps (after ﬁltering and simpliﬁcation). Hence, each value provides a measure on the degree of dispatch (measured in per-unit (Kundur, Balu, and Lauby 1994)) required to obtain transient stability. In other words, larger values indicate large adjustments in dispatch. The table also indicates, in parentheses, the number of restoration steps in which Model 2 is not solved by IPOPT within one hour. Note that, in general, a rotor angle limit of 90 degrees is considered acceptable, the goal of the current experiment is to push the limits on this bound as far as possible. The results indicate that the steady-state dispatches are in general transient-stable except for the 39-bus and 57-bus benchmarks. Fortunately, in general, it is possible to ensure transient stability with minor changes in generator dispatches. The changes in dispatch increases with stricter limits on rotor angles. In extreme cases, where the bound is set to 1 degree, IPOPT fails to ﬁnd transient stable dispatches for the 14-bus, 39-bus, and 57-bus benchmarks. This indicates that adjusting generator dispatch may not be sufﬁcient for an arbitrary degree constraint, but those limits are extremely strict. Overall, the results show that the proposed approach produces transient-stable restoration plans with minimal adjustments in generator dispatches. These adjustments are however necessary to guarantee transient stability, highlighting the beneﬁts of the proposed approach.

Conclusion Automated power restoration is a key component in future smart grids. Finding the optimal restoration plan is only one step towards system recovery. Guaranteeing transient stability of all switching operations is equally critical. In this paper, we propose an algorithmic procedure for validating

Table 1: Generator total dispatch adjustments on 5 Matpower benchmarks (T = 400, = 0.005)

6 Bus 14 Bus 30 Bus

Maximum Rotor Swing Maximum Rotor Swing Maximum Rotor Swing Gen. Reactance 90 40 10 1 90 40 10 1 90 40 10 1 0.02 0.00002 0.00002 0.00002 0.17967 0.00000 0.00000 0.00000 6.53902 0.00004 0.00003 0.00004 2.91547 0.06 0.00003 0.00003 0.00003 0.10593 0.00000 0.00000 0.00000 8.67193 0.00004 0.00003 0.00004 3.07084 0.10 0.00003 0.00002 0.00002 0.00843 0.00000 0.00000 0.00000 8.18311 0.00004 0.00004 0.00004 3.76922 0.14 0.00002 0.00003 0.00002 0.30335 0.00000 0.00000 0.00000 6.09923 (1) 0.00004 0.00005 0.00004 4.17696 0.20 0.00003 0.00003 0.00003 1.01248 0.00000 0.00000 0.00000 3.66457 (3) 0.00004 0.00004 0.00004 4.07680

39 Bus 57 Bus

Maximum Rotor Swing Maximum Rotor Swing Gen. Reactance 90 40 10 1 90 40 10 1 0.02 0.00001 0.00001 20.52951 7.95879 (6) 0.00000 0.00000 0.66260 131.39311 0.06 0.00001 0.00001 81.80927 (1) 0.00002 (7) 0.00000 0.00000 1.36766 1.12417 (21) 0.10 0.42052 0.28436 (1) 78.24781 48.13747 (5) 0.23766 0.23766 39.33088 1.14312 (21) 0.14 0.49691 0.49698 60.35684 41.99449 (5) 0.68299 0.67222 (1) 73.84672 1.26118 (21) 0.20 0.00002 0.00002 34.26957 69.35875 (3) 0.83704 0.83744 120.01411 (1) 1.41655 (21)

and enhancing transient stability of such restoration plans. Stability is measured using the nonlinear dynamic swing equations corresponding to rotor angles of power generators. Computational scalability is then improved using preprocessing and reduction routines. Numerical experiments are carried on state-of-the-art benchmarks, assessing the scalability of the approach and validating the stability results.

Acknowledgement The authors would like to thank David Hill and Ian Hiskens for many interesting discussions on power restoration. NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

References Adibi, M., and Kafka, R. 1991. Power system restoration issues. IEEE Computer Applications in Power 4(2):19 24. 2014. AMPL. AMPL Optimization: http://www.ampl.com. Bent, R., and Van Hentenryck, P. 2007. Randomized adaptive spatial decoupling for large-scale vehicle routing with time windows. In AAAI 07, 173 178. Coffrin, C., and Van Hentenryck, P. 2014a. A linearprogramming approximation of AC power ﬂows. Informs Journal on Computing 26(4):718 734. Coffrin, C., and Van Hentenryck, P. 2014b. Transmission system restoration: Co-optimization of repairs, load pickups, and generation dispatch. In Proceedings of the 18th Power Systems Computation Conference (PSCC 14). Coffrin, C.; Hijazi, H.; Lehmann, K.; and Van Hentenryck, P. 2014. Primal and dual bounds for optimal transmission switching. In Proceedings of the 18th Power Systems Computation Conference (PSCC 14). Coffrin, C.; Van Hentenryck, P.; and Bent, R. 2012. Last Mile restoration for multiple interdependent infrastructures. In AAAI 12, 455 463.

Fourer, R.; Gay, D. M.; and Kernighan, B. W. 2002. AMPL: A Modeling Language for Mathematical Programming. Cengage Learning. Gan, D.; Thomas, R.; and Zimmerman, R. 2000. Stabilityconstrained optimal power ﬂow. IEEE Transactions on Power Systems 15(2):535 540. Hazarika, D., and Sinha, A. 1999. An algorithm for standing phase angle reduction for power system restoration. IEEE Transactions on Power Systems 14(4):1213 1218. Hijazi, H.; Coffrin, C.; and Van Hentenryck, P. 2013. Convex quadratic relaxations of nonlinear programs in power systems. In Optimization Online: http://www.optimizationonline.org/DB HTML/2013/09/4057.html. Ketabi, A.; Ranjbar, A.; and Feuillet, R. 2002. New approach to standing phase angle reduction for power system restoration. European transactions on electrical power 12(4):301 307. Kheradmandi, M., and Ehsan, M. 2004. Dynamic analysis of transmission line connection in restoration of interconnected power systems. In Electrical and Computer Engineering, 2004. Canadian Conference on, volume 3, 1621 1624. Kundur, P.; Balu, N. J.; and Lauby, M. G. 1994. Power system stability and control. Mc Graw-hill New York. Liu, C.-W., and Thorp, J. 2000. New methods for computing power system dynamic response for real-time transient stability prediction. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 47(3):324 337. Mak, T. W. K.; Coffrin, C.; Van Hentenryck, P.; Hiskens, I. A.; and Hill, D. 2014. Power system restoration planning with standing phase angle and voltage difference constraints. In Proceedings of the 18th Power Systems Computation Conference (PSCC 14). Martins, N.; De Oliveira, E.; Moreira, W.; Pereira, J. L. R.; and Fontoura, R. 2008. Redispatch to reduce rotor shaft impacts upon transmission loop closure. IEEE Transactions on Power Systems 23(2):592 600.

Overbye, T. J.; Cheng, X.; and Sun, Y. 2004. A comparison of the AC and DC power ﬂow models for LMP calculations. In Proceedings of the 37th Annual Hawaii International Conference on System Sciences. Stott, B.; Jardim, J.; and Alsac, O. 2009. DC power ﬂow revisited. IEEE Transactions on Power Systems 24(3):1290 1300. Van Hentenryck, P.; Coffrin, C.; and Bent, R. 2011. Vehicle routing for the last mile of power system restoration. In Proceedings of the 17th Power Systems Computation Conference (PSCC 11). W achter, A., and Biegler, L. T. 2006. On the implementation of an interior-point ﬁlter line-search algorithm for largescale nonlinear programming. Mathematical Programming 106(1):25 57. Ward, J. B. 1949. Equivalent circuits for power-ﬂow studies. Transactions of the American Institute of Electrical Engineers 68(1):373 382. Ye, H., and Liu, Y. 2013. A new method for standing phase angle reduction in system restoration by incorporating load pickup as a control means. International Journal of Electrical Power & Energy Systems 53:664 674. Yuan, Y.; Kubokawa, J.; and Sasaki, H. 2003. A solution of optimal power ﬂow with multicontingency transient stability constraints. Power Systems, IEEE Transactions on 18(3):1094 1102. Zimmerman, R.; Murillo-S andnchez, C.; and Thomas, R. 2011. MATPOWER: Steady-state operations, planning, and analysis tools for power systems research and education. IEEE Transactions on Power Systems 26(1):12 19.