a decomposition-based practical approach to transient stability-constrained unit commitment

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 30, NO. 3, MAY 2015 1455

A Decomposition-Based Practical Approach toTransient Stability-Constrained Unit CommitmentYan Xu, Member, IEEE, Zhao Yang Dong, Senior Member, IEEE, Rui Zhang, Student Member, IEEE,

Yusheng Xue, Member, IEEE, and David J. Hill, Fellow, IEEE

Abstract—Traditional security-constrained unit commitment(SCUC) considers only static security criteria, which may howevernot ensure the ability of the system to survive dynamic transitionbefore reaching a viable operating equilibrium following a largedisturbance, such as transient stability. This paper proposes atractable mathematical model for transient stability-constrainedunit commitment (TSCUC) and a practical solution approach.The problem is modeled without explicit differential-algebraicequations, reducing the problem size to one very similar to a con-ventional SCUC. The whole problem is decomposed into a masterproblem for UC and a range of subproblems for steady-statesecurity evaluation and transient stability assessment (TSA).Additional constraints including Benders cut and so-named sta-bilization cut are generated for eliminating the security/stabilityviolations. The extended equal-area criterion (EEAC) is used forfast TSA and analytically deriving the stabilization cut, whereinmultiple contingencies having common instability mode can besimultaneously stabilized by one cut. The proposed approach isdemonstrated on the New England 10-machine system and theIEEE 50-machine system, reporting very high computationalefficiency and high-quality solutions.Index Terms—Benders decomposition, extended equal-area

criterion, mixed-integer programming, transient stability-con-strained unit commitment.

NOMENCLATURE

Constants:Generation cost coefficients of unit .Generation cost function of unit .Start-up and shut-down cost of unit in period .

Manuscript received February 12, 2014; revised May 13, 2014 and July08, 2014; accepted August 14, 2014. Date of publication August 29, 2014;date of current version April 16, 2015. This work was supported in part bythe Australian Research Council (ARC) through a Linkage Project (Grantno. LP120100302), in part by the University of Newcastle through a FacultyStrategic Pilot Grant, and in part by the State Key Laboratory of China forAlternate Electrical Power Systems with Renewable Energy Sources throughan Open Grant. Paper no. TPWRS-00214-2014.Y. Xu and R. Zhang are with the Centre for Intelligent Electricity Net-

works, University of Newcastle, Newcastle, NSW 2308, Australia (e-mail:[email protected]; [email protected]).Z. Y. Dong is with the School of Electrical and Information Engineering, Uni-

versity of Sydney, Sydney, NSW 2006, Australia (e-mail: [email protected]).Y. Xue is with State Grid Electric Power Research Institute, Nanjing, China

(e-mail: [email protected]).D. J. Hill is with the Department of Electrical and Electronic Engineering,

University of Hong Kong, Pokfulam, Hong Kong, and also with the School ofElectrical and Information Engineering, University of Sydney, Sydney, NSW2006, Australia (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2014.2350476

System load demand in period .Load demand of bus in period .Machine inertia of unit .Number of units and dispatching time periods.System spinning reserve requirement in period .

Ramping down and up limit of unit .Shut-down and start-up ramping limit of unit .Minimum up and down time of unit .

Desired transient stability margin.Steady-state variables:

Real flow on line in period t for contingency .Binary variable for on/off status of unit inperiod .Active power output of unit in period .Spinning reserve of unit in period .ON and OFF time of unit in period .

Dynamic variables:Transient stability margin for contingency inperiod .Rotor angle of unit at transient time .Angle speed of unit at transient time .Mechanical power of unit at transient time .Electrical power of unit at transient time .

I. INTRODUCTION

A. Background and Motivation

A S an effective tool to clear day-ahead electricity markets,an unit commitment (UC) program aims to determine the

generators' operating states including their turn on/off statusesand power outputs with the objective of minimizing the totalproduction cost while meeting prevailing operational limits [1].Conventionally, the prevailing constraints include power bal-ance, unit minimum on/off time limits, ramping up/down limitsand system spinning reserve requirements, etc. State-of-the-artmethods for solving a UC are Lagrangian relaxation (LR) [1]and mixed-integer programming (MIP) [2].In recent years, due to rapid load growth and unmatched

infrastructure investments, power systems are being pushedto operate near their security limits, as a consequence, thereis a strong need to consider the security constraints in the

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1456 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 30, NO. 3, MAY 2015

UC, yielding security-constrained UC (SCUC) problems [3].Traditionally, SCUC aims to satisfy the static security criteriaincluding network power flow limits and voltage limits atsteady states (before and/or after a contingency) [3]. Presently,SCUC can be effectively solved by decomposition-basedmethods, such as Benders decomposition (BD) [3]. As a nec-essary extension of UC, SCUC has become an essential toolto balance the economy and security requirements in day-headelectricity market operations.While the static security requirements can be soundly sat-

isfied in SCUC, today's power systems is also facing signifi-cant risk of dynamic insecurity, which means a power systemfailing to survive the dynamic transition before reaching a vi-able steady-state operating equilibrium after a contingency. Ac-cording to previous investigations [25], [26], lack of adequatedynamic performance is one key driven force for cascading fail-ures and wide-spread blackouts around the world. As one essen-tial dynamic performance criterion, transient stability refers tothe ability of the power system to maintain synchronism afterbeing subjected to a large disturbance [4]. The loss of transientstability can be extremely fast (e.g., 100 200 ms followingthe fault clearance) and can trigger generator and load shed-ding, leading to system-wide failures and/or blackouts. Whileconventional SCUC only considers the static security, it maybe unable to meet the transient stability requirement. To fur-ther enhance the system security and reduce the risk of black-outs, it is sensible and imperative to include transient stability inthe day-ahead generation dispatch stage, which yields the tran-sient stability-constrained UC (TSCUC) problem studied in thispaper.

B. Literature StudyAlthough very important, very limited work on TSCUC has

been reported due to the tremendous complexity of the problem.On one hand, SCUC is a large-scale, multi-stage, mixed non-linear integer programming problem. On the other hand, tran-sient stability study usually calls for a vast number of differen-tial-algebraic equations (DAEs) which are intractable for cur-rent programming algorithms.To the best of our knowledge, [5] is the first and sole work

on this topic to date. In [5], the authors propose an augmentLR method combined with variable duplication technique tosolve the TSCUC model. The whole problem is divided into abasic UC subproblem and a transient stability-constrained op-timal power flow (TSCOPF) subproblem. The first subproblemis solved by dynamic programming and the second by a re-duce-space interior point method.Although effective, the method reported in [5] can suffer from

several practical limitations. First, the transient stability con-straint is formulated as DAEs, and time-domain numerical dis-cretization is applied to convert the differential equations intothe algebraic form. It is clear that this would result in a dramaticexplosion of the problem size proportional to the number of in-tegration time steps times the number of generators. The lengthof the whole integration period is usually arbitrarily selected.Although a reduced-space interior method is used to solve theTSCOPF model, the overall problem dimension and computa-tional burden remain extensive, making the parallel computinga necessity for implementing the method. Second, the transientstability is constrained by a rotor angle limit index, i.e., the max-imum rotor angle deviation of each generator against the centerof inertia (COI) during the transient period being bounded by apre-defined threshold. However, it has been widely shown that

this threshold is usually system dependent and not easy to de-fine: when it is set to a small value, the operation tends to beconservative and less economic; while if it is too relaxed, thetransient stability may not be ensured [9], [16]. Meanwhile, itprovides very little information about the system stability de-gree, which is an important metric for system operators. Fur-thermore, this method only considers a single contingency case,whereas in practice, it is usually necessary to stabilize multiplecontingencies [13].

C. Contributions of This PaperThis paper proposes 1) a new TSCUC model which dramat-

ically reduces the problem size and 2) a practical approach toefficiently solve the problem. The model does not explicitlycontain any DAEs, yielding a tractable problem size. Followingthe state-of-the-art SCUC solution strategy, the proposedapproach is based on a decomposition framework, wherethe master problem consists of solving a basic UC model todetermine the unit status and the generation output, and theslave subproblems consist of feasibility checks including bothnetwork steady-state security evaluation (NSE) and transientstability assessment (TSA), and generating additional con-straints for the master problem to retrieve the security/stability.For solving the master problem, the MIP method is used.

Compared with LR, MIP is advantageous in that high-perfor-mance commercial solvers can be employed and higher-qualitysolutions can be usually obtained [2], [3], [27]. For the slavesubproblems, to deal with the transient stability constraints, ahybrid TSA method called extended equal-area criteria (EEAC)[8] is utilized. EEAC, also called single machine equivalent(SIME) [9], is a quantitative TSA method, which can not onlymeasure the stability degree (margin) but also quantify thegeneration shifting to stabilize the system. Based on EEAC,the transient stability can be quantitatively constrained and thestability control can be analytically derived and formulated aslinear constraints (named stabilization cuts in this paper). Inthis way, the TSCUC can be solved in a process very similar toa standard SCUC problem. Meanwhile, contingencies that havecommon instability mode can be simultaneously stabilized(i.e., one stabilization cut to stabilize multiple contingencies),reducing further the dimension of the problem. Furthermore,while major computing effort is on the TSA stage, EEAC-basedearlier termination for time-domain simulation (TDS) can beused to speed-up the solution significantly.

II. PROPOSED MODEL

A. Objective FunctionThe objective of TSCUC is to determine a day-ahead UC for

minimizing the total production cost:

(1)

where is the generation cost function:

(2)

The decision variables are unit on/off status and active powerdispatch.

B. Operational ConstraintsThe following operational constraints are considered:

XU et al.: A DECOMPOSITION-BASED PRACTICAL APPROACH TO TRANSIENT STABILITY-CONSTRAINED UNIT COMMITMENT 1457

a) Power balance:

(3)

b) Generation limits:

(4)

c) Spinning reserve limits:

(5)

d) Ramping limits:

(6)

e) Minimum up and down time limits:

(7)

C. Steady-State Security ConstraintsSince this paper mainly focuses on the transient stability cri-

terion, for the steady-state security, a DC network is used andthe transmission flow limits are considered:

(8)

where denotes the base case, and denotesa contingency case, is the power transfer distributionfactor of bus to line for contingency at period , is theload demand of bus .It should be indicated that AC network constraints containing

bus voltage limits [10] can also be used for this method.

D. Transient Stability ConstraintsHistorically, transient stability constraints have beenmodeled

as a large set of DAEs [4], [5], [11], [12]. Including them in aprogramming model can result in a prohibitive problem size andcomputational difficulties as discussed in Section I.To alleviate the computational complexity, we express the

transient stability constraints by enforcing a positive stabilitymargin for each contingency:

(9)

is calculated through a rigorous time-domain simulation-based TSA procedure which will be introduced later. Basically,a larger stability margin would result in a more conservativeoperating condition and therefore higher operating costs. Hence,it is an engineering practice to limit the stability margin with asmall threshold according to practical needs.

E. Model FeaturesIt is important to note that, instead of modeling the transient

stability constraints as a large set of DAEs, which can result inan enormous problem dimension (i.e., for each contingency, thenumber of stability constraints equals the number of generatorstimes the number of integration time steps), we strive to simplifythe stability constraints in the TSCUCmodel while retaining theinherent instability mechanism and essential accuracy. This canreduce drastically the dimension of the programming problem.Even more importantly, it provides the opportunity for decom-posing the TSCUC problem in a BD manner, wherein high so-lution efficiency can be gained.

III. EXTENDED EQUAL-AREA CRITERION

A. BasicsThe original EEAC was firstly proposed by Xue et al. in

[8]. Its improved version—IEEAC [17] (also known as SIME[9]) is a hybrid TSA approach combining a full TDS processand the well-known equal-area criterion (EAC). The principleof IEEAC is to transform the multi-machine trajectories toan equivalent one-machine-infinite-bus (OMIB) trajectory,and apply the EAC to the equivalent OMIB. In such a way, itprovides a good engineering approximation of investigating thestability characteristics of the original multi-machine system.Specifically, IEEAC or SIME drives a full TDS engine to

obtain the multi-machine trajectories (wherein complex systemmodel is not a limitation), and then separate them into two exclu-sive clusters: one composed of critical machines (CMs) whichare responsible for the loss of synchronism, and the other com-posed of non-critical machines (NMs) which correspond to theremaining machines. The two clusters of CMs and NMs are rep-resented as two corresponding equivalent machine trajectories[8], [9]:

(10)

(11)

where subscripts and denote the CMs and NMs, respec-tively; and are respectively the inertia coefficient ofCMs and NMs, calculated as

(12)

The multi-machine equivalent OMIB trajectory is then con-structed as follows [8],[9]:

(13)(14)

(15)

(16)(17)


Fig. 1. Multi-machine rotor angle trajectories (left window) and the corre-sponding OMIB representation (right window)—an unstable case forthe New England 10-machine system.

Fig. 2. Multi-machine rotor angle trajectories (left window) and the corre-sponding OMIB representation (right window)—a stable case for theNew England 10-machine system.

where and here denote the rotor angle and angular speed ofthe OMIB, respectively; and denote the mechanical andelectrical power of the OMIB, respectively.Then, the EAC is applied to the OMIB plane for quan-

tifying the transient stability degree and extracting stability in-formation of the original multi-machine system. Illustrations ofmulti-machine rotor angle trajectories and correspondingOMIB

representations are shown in Figs. 1 and 2. The figuresare obtained from the simulation results in Section V.

B. Transient Stability Assessment Based on EEACIEEAC or SIME can offer the following TSA results [8],[9]:1) CMs and NMs, which determines an “instability mode”.2) The “time to instability” , which indicates the time that

system loses synchronism. At this time, the curve ofcrosses —see Fig. 1, that is

(18)

3) The “time to first-swing stability” , which indicates thetime that system can be declared as first-swing stable. Atthis time, the curve stops its excursion and return backbefore crossing —see Fig. 2, that is

(19)

Note that and can be used to terminate the TDSmuchearlier, saving a great deal of computation time.

4) Transient stability margin , which quantifies the degreeof system stability. It is calculated by the decelerating area

minus the accelerating area of the OMIBplane:

(20)

means the system is unstable; otherwise stable.

Based on IEEAC, TSA can be realized much more efficientlywith abundant system stability information while maintainingnecessary accuracy. Currently, there are mature commercialIEEAC software tools available, such as one called FASTEST[17]. FASTEST incorporates a powerful full TDS program androbust IEEAC algorithm to accurately and reliably calculatethe stability margin and other TSA parameters. It has beenwidely implemented in China and many other countries foryears reporting satisfactory engineering performance [17].

C. Transient Stability Control Based on EEACIEEAC provides significant information for transient stability

control (TSC). In the literature, it has been used for efficientlysolving preventive TSC and TSCOPF problems [12]–[16].For preventive actions, stabilizing an unstable system con-

sists of modifying the pre-contingency conditions until the sta-bility margin becomes zero (or positive). This can be achievedby increasing the decelerating area and/or decreasing theaccelerating area of the OMIB representation. Inpractice, this can be realized by decreasing the OMIB mechan-ical power , that is

(21)

where denotes the pre-contingency state, and arerespectively the changes in the total power of CMs and NMs:

(22)

To maintain the power balance, the following conditionshould be satisfied:

(23)

Substituting (23) into (21), we have

(24)

Equations (21)–(24) reveal that by shifting real power outputof CMs to NMs, the transient stability can be restored [9],[13]–[15].Numerous examples have reported a quasi-linear relation-

ship between changes of stability margin and OMIBmechanicalpower at pre-contingency state [9], [13]–[15], [16], that is

(25)

where is the approximate linear sensitivity of the stabilitymargin with respect to generation change.In practice, the sensitivity value around the operating point

can be numerically estimated via two successive IEEAC runs:

(26)

With , the required generation shifting for TSC can be an-alytically calculated. Specifically, to control an unstable case,whose stability margin is , if the desired stabilitymargin is , the required increment in stability margin is

. Combining (24)–(26), the required generationshifting between CMs and NMs can be calculated as

(27)


Fig. 3. Proposed decomposition strategy for TSCUC.

Meanwhile, IEEAC reveals that multiple contingencieshaving common instability mode, i.e., common CMs, can be si-multaneously stabilized. This is to apply the most constrainingpower shifting, imposed by the severest contingency, to thesecommon CMs only [9], [13]–[15]. Hence, the computationalefforts for multi-contingency can be remarkably reduced.

IV. PROPOSED APPROACH

A. Decomposition StrategyDecomposition-based methods, such as BD, have become a

mature approach to SCUC problems. Typically, it decomposesthe whole problem into a master problem and a range of slavesubproblems. The master problem is essentially a UC problem,which determines the commitment and dispatch of the unitswithout the network security constraints. Given the UC resultsfrom the master problem, the slave subproblems check the se-curity constraints. If violation exists, Benders cuts (or similari-ties) are generated and added to the master problem. The wholeproblem is iteratively solved between the master and slave prob-lems until no violation exits. The major advantage of the BDmethod is that it can decompose the large problem into a seriesof smaller and tractable problems, diminishing significantly thecomplexity of the whole problem and enabling parallel com-puting to reduce the total execution time.This paper adopts a similar decomposition strategy to effi-

ciently solve the TSCUC problem. The description of the pro-posed decomposition strategy is shown in Fig. 3.Let denote the UC status I and generation dispatch P, and y

denote the system state variables. The TSCUC problem can berewritten as the following standard BD form:

(28)(29)(30)(31)

where (28) corresponds to the cost function (1), (29) corre-sponds to the operational constraints (3)–(7) as well as the ad-ditional constraints generated from the subproblem, (30) corre-sponds to the network steady-state security constraints (8), and(31) corresponds to the transient stability constraints (9).

B. Master ProblemThe master problem is to solve the UC model, i.e., (28), (29),

determining the commitment and generation dispatch .

At the initial iteration, (29) only comprises the operationalconstraints (3)–(7), where the network steady-state security con-straints and transient stability constraints are disregarded. Afterthe subproblems are solved, additional constraints are generated(if necessary) and added to (29) in subsequent iterations to mit-igate the security and stability violations.As already mentioned, there are generally two mature

methods for solving a UC problem: LR and MIP. A comprehen-sive comparison and discussion on the two methods for UC canbe found in [3]. In this paper, we apply the MIP method sincehigh-performance commercial MIP packages are availabletoday, such as CPLEX [6] and GUROBI [7]. Actually, mostISOs in the US are switching to MIP method for UC calculation[27]. However, it should be indicated that the LR method canalso be adopted for the proposed approach here.To apply the MIP, the nonlinear UC model should be refor-

mulated into a solvable form [i.e., mixed integer linear program-ming (MILP)] for standard MIP solvers. In this paper, a compu-tationally efficient MILP formulation reported in [2] is adopted.It has been shown that this MILP formulation requires fewer bi-nary variables and constraints than other reported models andcan significantly saving the computational efforts.

C. SubproblemThe subproblem evaluates the hourly network steady-state se-

curity and the transient stability of a master UC solution andgenerates additional constraints (if necessary): Benders cut andstabilization cut.1) Network Steady-State Security Evaluation (NSE): The

NSE involves both the base case and contingency cases. Foreach case, a linear programming (LP) model is built [3]:

(32)(33)

where is the vector of ones, is the slack vector used to checkthe violation of line flow constraints, and is the Lagrangianmultiplier vector of inequality constraints in (33).means the violation occurs, and the Benders cut is generated as

(34)

mathematically represents the marginal decrement orincrement of the objective function (32) when is adjusted.In the next iteration, (34) will be added to (29) of the masterproblem to eliminate the steady-state security violation.2) Transient Stability Assessment (TSA): For each contin-

gency case, the EEAC-based TSA is performed. The TSA re-sults include those presented in Section III-B. The left side ofconstraint (31) is checked: if the stability margin is negative, thestabilization cut, i.e., required generation shifting between CMsand NMs, is generated.In order to simultaneously stabilize multiple contingencies

with least number of additional constraints, the contingenciescan be grouped according to their resulting instability modes.The contingencies having a common instability mode are di-vided into one group. Each contingency group is then repre-sented by the severest contingency which is the one with thesmallest stability margin in that group. For each representativecontingency, the stabilization cut is generated as

(35)


Fig. 4. Computation flowchart.

where denotes the generation output of unit obtainedfrom the master problem.The stabilization cut (35) conveys information about how the

generation dispatch should be modified to preventively retainthe transient stability.It is important to note that the derived stabilization cut is in

the same mathematical form as the Benders cut, namely, both ofthem are the linear constraints. Hence, they can be seamlesslyintegrated together to be added to the master problem. In such away, the programming problem size can be limited to as smallas a traditional SCUC.

D. Solution ProcedureThe general computation flowchart of the TSCUC is pre-

sented in Fig. 4 and the detailed steps are as follows.Step 1) Solve the master problem (28), (29) using MILP.Step 2) Given the master UC solution, perform the hourly

NSE and hourly TSA for each considered contin-gency.

Step 3) If all the contingencies are both steady-state secureand transient stable, stop; otherwise, go the nextstep.

Step 4) For the steady-state security block, for each inse-cure contingency, generate the Benders cut (34) bysolving (32), (33); for the transient stability block,for each representative unstable contingency, gen-erate the stabilization cut (35) following the proce-dure presented in Section III-C.

Step 5) Add the generated Benders cut and stabilization cut(if any) to the master problem, and go back to Step1).

V. NUMERICAL RESULTS

A. Implementation of the Proposed ApproachThe simulation is conducted on an ordinary 64-bit PC with

3.10-GHz CPU and 4.0 GB of RAM. Tim-domain simulationis performed using the commercial power system simulationpackage PSS/E [20], and the IEEAC algorithm is realizedin the MATLAB platform (note that, commercial packagesfor IEEAC combined with TDS can also be used here, e.g.,FASTEST [17] which has been put into practice in manycountries). An interface developed in our previous work [21] isused to connect PSS/E and MATLAB. The optimization model

Fig. 5. Implementation of the proposed approach.

Fig. 6. One-line diagram of the New England 10-machine system.

is written in MATLAB language and solved by invoking thecommercial solver GUROBI [7] which has a built-in interfacewith MATLAB. The implementation of the proposed approachis schematically described in Fig. 5.

B. New England 10-Machine SystemThe proposed approach is first tested on the New England

10-machine system. This test system consists of 10 machines,39 buses, and 46 branches (see Fig. 6). The system network dataand machine dynamic parameters are obtained from [18], and a10-unit UC data is obtained from [19]. The UC data is modifiedto accommodate the base loading level of this system. Specif-ically, the total load demand, generation limits, and rampinglimits are increased by 3 times. Then, the total load is pro-portionally distributed to the load buses according to the baseloading level.1) SCUC Results: For comparison purposes, the SCUC

model is first solved without including the transient stabilityconstraints. The total generation cost is $1 590 910.6 andthe detailed unit status and generation dispatch are given inTable I and Fig. 7, respectively.2) Single Contingency Case: A single contingency (called

C1) is considered first. The contingency is a three-phase short-circuit at bus 21 with the duration time of 0.12 s. Under thiscontingency, it is found that the SCUC solution, although viable


TABLE ISCUC/TSCUC RESULTS (NEW ENGLAND 10-MACHINE SYSTEM)

Fig. 7. SCUC generation dispatch.

for steady-state security, fails to maintain the transient stabilityin 17 out of 24 h. The transient stability margin and CMs arealso given in Table I.To illustrate, the system multi-machine rotor angle trajecto-

ries and the corresponding OMIB representations of the5th and 1st dispatch hour are respectively shown in Fig. 1 andFig. 2—see Section III-A. In Fig. 1, it can be seen that, there isone CM which is unit G31 and the system losses synchronismat 0.63 s. Consequently, the TDS can be early terminated at thattime. In Fig. 2, the “time to first-swing stability” is 0.4 s, whichmeans the TDS can be early terminated at that time.The TSCUC model considering C1 is then solved using the

proposed approach. The desired stability margin is set to asmall positive value 1.0 to avoid over-stabilization. Note how-ever, that this threshold can also be set larger depending on prac-tical needs. After the 1st iteration, hour 7, 10, and 11 remain un-stable (but their stability margin have been increased to 8.8,8.1, and 9.8, respectively) and the other unstable hours be-

come stable with positive stability margin close to the margintarget. The 2nd iteration is then executed and after this itera-tion, all hours have become transient stable with respect to C1.For the obtained TSCUC solution, the total generation cost is

$1 615 187.1, which increases only 1.53% over the SCUC so-lution. The increased cost accounts for preventively stabilizing

Fig. 8. TSCUC generation dispatch—single contingency (C1).

Fig. 9. TSCUC generation dispatch—multiple contingencies (C1 and C2).

the system over the 24 h for C1. The difference in unit status be-tween TSCUC and SCUC solutions is highlighted in Table I, andthe single-contingency TSCUC generation dispatch is shown inFig. 8.Comparing the results of TSCUC and the SCUC, it can be

seen that the generation output of unit G31 has been remarkablydecreased, and to compensate its generation shifting, the on/offstatus and generation dispatch of other units are modified.3) Multi-Contingency Case: The proposed approach is tested

for multi-contingency stabilization, which is not treated in pre-vious work [5]. Another three-phase short-circuit applied at bus30 with the duration time of 0.09 s is considered (called C2). ForC2, 19 out of 24 hours are unstable, which are the hour 5–23.For common unstable hour 5–12 with C1, C2 has the same com-position of CMs as C1, but C2 is more severe, i.e., has smallerstability margin value, in the hour 5, 7–9, and 12.To stabilize C1 and C2 simultaneously, a representative con-

tingency is selected (given the same instability mode) as the onewith smaller stability margin for each unstable hour. Then, themulti-contingency stability constraint is imposed for the repre-sentative contingency only. In this way, the computational ef-forts can be effectively reduced.After 2 iterations, a viable TSCUC solution has been ob-

tained, which can ensure the transient stability for both contin-gencies for each hour. The total generation cost of themulti-con-tingency TSCUC solution is $1 616 903.5, which increases only0.11% and 1.64% over the single-contingency TSCUC and theSCUC solution, respectively. The difference in unit status be-tween the multi- and single-contingency TSCUC solutions ison the unit G34 at hour 8. The multi-contingency TSCUC gen-eration dispatch is shown in Fig. 9.4) Computation Efficiency Analysis: Compared with the

existing method for TSCUC [5], it is manifest that the com-putation efficiency of the proposed approach is substantiallyhigher: it does not solve a time-consuming high-dimensionalprogramming problem such as TSCOPF; rather, it decomposes


TABLE IICPU TIME (NEW ENGLAND 10-MACHINE SYSTEM)

the whole large-scale problem into small-scale, tractable sub-problems which are efficiently solved using commercial solversand fast TDS tools.Given the clear computational structure, the total CPU time

required for a single contingency calculation can be roughly es-timated as follows:

(36)

where respectively denote the CPU time forUC solution, NSE and TDS for a contingency, denotes thetotal iteration number, denote the number of unstable hourfor the th iteration, and denotes all the other elapsed timeduring the whole computation process, including the time forreading and exporting data files, interfacing between differentsoftware tools, etc. In particular, the summation item in (36)represents that an unstable contingency requires an additionalTDS to calculate the sensitivity value [see (26)] for deriving thestabilization cut. Table II lists the CPU time for each task of theproposed approach.For the master problem, theMILP solver GUROBI only costs

0.24 s. For each contingency, the NSE and generating BD cut re-quires only 0.01 s. The major computational burden lies in theTSA phase, which has been significantly alleviated thanks toEEAC-based early termination: for a contingency, it only costabout 0.7 s 0.9 s using PSS/E package. For this test system, thetotal CPU times are 62.8 s and 89.5 s for single- and multi-con-tingency TSCUC calculations, respectively. Given the decom-position structure, the proposed approach is also ideal for par-allel implementing, e.g., paralleling 24 h or paralleling contin-gencies. Such a parallel computing platform tailored for PSS/Eis developed in [21].

C. IEEE 50-Machine System

The IEEE 50-machine system is used to further demonstratethe high solution speed of the proposed approach. This system isderived from a representative model of a realistic power systemin North America [23]. It consists of 50 machines, 145 buses,and 453 branches. Fig. 10 shows a portion of the one-line dia-gram of the high-voltage lines of this test system.The system data is obtained from [24], and is extended

for a TSCUC study. The size of this system is similar to theIEEE 118-bus and 300-bus systems used in [5], which have54 machines/186 branches and 69 machines/411 branches,respectively. Therefore it is sensible to use the reported CPUtime in [5] to benchmark the proposed approach.A 3-phase short-circuit fault at the branch between bus 6

and 10 is studied here. Using the proposed approach, a viableTSCUC solution is obtained after 3 iterations. Fig. 11 showsthe system trajectories (in different colors to distinguish CMsand NMs) for a specific dispatch hour from the SCUC solution,

Fig. 10. Portion of one-line diagram of the IEEE 50-machine system.

Fig. 11. Multi-machine rotor angle trajectories (upper window) and the corre-sponding OMIB representation (lower window)—an unstable case forthe IEEE 50-machine system.

where it can be seen that the system losses synchronism afterthe fault and its stability margin is 18.6.Fig. 12 shows the system trajectories for the same dispatch

hour from the TSCUC solution, where it can be seen that thesystem is able to maintain synchronism after the fault and itsstability margin becomes to 0.72. Note that the trajectories ofCMs and NMs are respectively plotted in red and blue colors inthe upper windows of these two figures.Since the major computational burden lies at the TDS stage, it

is important to note that, for the unstable case (Fig. 10), the insta-bility condition is reached at 1.09 s; for the stable case (Fig. 11),the first-swing stability condition is met at 1.36 s. Hence, thereis no need to run the TDS for the whole simulation time (e.g., 5


Fig. 12. Multi-machine rotor angle trajectories (upper window) and the corre-sponding OMIB representation (lower window)—an stable case for theIEEE 50-machine system.

TABLE IIICPU TIME (IEEE 50-MACHINE SYSTEM)

TABLE IVTOTAL CPU TIME FOR TSCUC COMPUTATIONS

or 10 s); rather, the TDS can be early terminated at these times,saving a great deal of computation time.The CPU time for each task of the TSCUC calculation is

given in Table III. For this test system, the total CPU time isaround 319 s. To compare, the CPU time of the existing ap-proach on the two similar systems are listed in Table IV. It canbe seen that the proposed approach is around 140 times fasteron the similar test system.

VI. CONCLUSION AND FUTURE WORK

Lack of adequate dynamic performance is one key drivenforce for cascading failures and/or widespread blackouts. Inday-ahead generation dispatch stage, TSCUC is a reasonableand necessary extension of SCUC to satisfy transient stabilityrequirements for day-ahead generation dispatch. As studied inthe pioneer work [5], its difficulty lies in the high dimension,nonlinearity and DAE-based nature of the problem. This paperproposes a new, tractable TSCUC model which significantlyreduces the problem size to one very similar to a conventionalSCUC. A decomposition-based practical approach is thendeveloped. Similar to the BD strategy, the proposed approach

decomposes the whole problem into a master problem for UCand a range of subproblems for NSE and TSA. Benders cut andso-named stabilization cut are generated for eliminating thesecurity/stability violations. The EEAC is used for fast TSAand analytically deriving the stabilization cut. Compared withthe existing method [5], the proposed approach in this paper isadvantageous for much higher solution speed, more accurateand quantitative modeling of stability constraints, and highefficiency in stabilizing multiple contingencies simultaneously.Case studies on the New England 10-machine system and theIEEE 50-machine system have validated the approach.Given the strong and inherent coupling relationship between

the active power and the rotor angle stability, this paper onlyoptimize the active power quantities with the fully time-domainsimulation-based TSA check. Note that this is consistent withindustry practice, where DC OPF with AC feasibility check isadopted (see [27]). Future efforts can be devoted to co-optimizeboth active and reactive power in the TSCUC problem. This canbe achieved using the AC network-based UCmodeling reportedin [10]. In addition, future works can be done to integrate moredimensions into the problem, such as FACTS devices which canimpact system's dynamic performance.

REFERENCES[1] J. Wood and B. F. Wollenberg, Power Operation, Generation and Con-

trol. New York, NY, USA: Wiley, 1996.[2] M. Carrion and J. Arroyo, “A computationally efficient mixed-integer

linear formulation for the thermal unit commitment problem,” IEEETrans. Power Syst., vol. 21, no. 3, pp. 1371–1378, Aug. 2006.

[3] Y. Fu, Z. Li, and L. Wu, “Modeling and solution of the large-scalesecurity-constrained unit commitment,” IEEE Trans. Power Syst., vol.28, no. 4, pp. 3524–3533, Nov. 2013.

[4] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C.Canizares, N. Hatziargyriou, D. J. Hill, A. Stankovic, C. Taylor, T. VanCutsem, and V. Vittal, “Definition and classification of power systemstability,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 1387–1401,May 2004.

[5] Q. Jiang, B. Zhou, and M. Zhang, “Parallel augment Lagrangian re-laxation method for transient stability constrained unit commitment,”IEEE Trans. Power Syst., vol. 28, no. 2, pp. 1140–1148, May 2013.

[6] CPLEX Optimizer [Online]. Available: http://www.ibm.com/us/en/[7] GUROBI Optimizer [Online]. Available: http://www.gurobi.com/[8] Y. Xue, T. Van Cutsem, and M. Pavella, “A simple direct method for

fast transient stability assessment of large power systems,” IEEE Trans.Power Syst., vol. 3, no. 2, pp. 400–412, May 1988.

[9] M. Pavella, D. Ernst, and D. Ruiz-Vega, Transient Stability of PowerSystems: A Unified Approach to Assessment and Control. Norwell,MA, USA: Kluwer, 2000.

[10] Y. Fu, M. Shahidehpour, and Z. Li, “Security-constrained unit com-mitment with AC constraints,” IEEE Trans. Power Syst., vol. 20, no.3, pp. 1538–1550, Aug. 2005.

[11] D. Gan, R. J. Thomas, and R. D. Zimmerman, “Stability-constrainedoptimal power flow,” IEEE Trans. Power Syst., vol. 15, no. 2, pp.535–540, May 2000.

[12] Y. Xu, Z. Y. Dong, K. Meng, J. Zhao, and K. P. Wong, “A hybridmethod for transient stability constrained-optimal power flow compu-tation,” IEEE Trans. Power Syst., vol. 27, no. 4, pp. 1769–1777, Nov.2012.

[13] D. Ruiz-Vega and M. Pavella, “A comprehensive approach to tran-sient stability control. I: Near optimal preventive control,” IEEE Trans.Power Syst., vol. 18, no. 4, pp. 1446–1453, Nov. 2003.

[14] Y. Xue, W. Li, and D. J. Hill, “Optimization of transient stability con-trol Part-I: For cases with identical unstable modes,” Int. J. ControlAutom. Syst., vol. 3, no. 2, pp. 334–340, Jun. 2005.

[15] Y. Xue, W. Li, and D. J. Hill, “Optimization of transient stability con-trol Part-II: For cases with different unstable modes,” Int. J. ControlAutom. Syst., vol. 3, no. 2, pp. 341–345, Jun. 2005.

[16] A. Pizano-Martinez, C. R. Fuerte-Esquivel, and D. Ruiz-Vega, “A newpractical approach to transient stability-constrained optimal powerflow,” IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1686–1696, Aug.2011.


[17] Y. Xue, “Fast analysis of stability using EEAC and simulation tech-nologies,” in Proc. 1998 Int. Conf. Power Syst. Tech (POWERCON),Beijing, China, 1998.

[18] M. A. Pai, Energy Function Analysis for Power System Stability.Norwell, MA, USA: Kluwer, 1989.

[19] S. A. Kazarlis, A. G. Bakirtzis, and V. Petridis, “A genetic algorithmsolution to the unit commitment problem,” IEEE Trans. Power Syst.,vol. 11, no. 1, pp. 83–92, Feb. 1996.

[20] Siemens PTI, PSS/E 33.0 Program Application Guide, May 2011.[21] K. Meng, Z. Y. Dong, K. P. Wong, and Y. Xu et al., “Speed-up the

computing efficiency of PSS/E-based power system transient stabilitysimulations,” IET Gener., Transm., Distrib., vol. 4, no. 5, pp. 652–661,May 2010.

[22] Y. Xue, “Integrated extended equal area criterion-theory and applica-tion,” in Proc. 5th Symp. Specialists in Electric Operational and Ex-pansion Planning, Recife, Brazil, 1996.

[23] IEEE Committee Report, “Transient stability test systems for directstability methods,” IEEE Trans. Power Syst., vol. 7, no. 1, pp. 37–44,Feb. 1992.

[24] Power Systems Test Case Archive [Online]. Available: http://www.ee.washington.edu/research/pstca/

[25] IEEE PES Power System Dynamic Performance Committee, “Causesof the 2003 major grid blackouts in North America and Europe, recom-mended means to improve system dynamic performance,” IEEE Trans.Power Syst., vol. 20, no. 4, pp. 1922–1928, Nov. 2005.

[26] Y. V. Makarov, V. I. Reshetov, A. Stroev, and I. Voropai, “Blackoutprevention in the United States, Europe, and Russia,” Proc. IEEE, vol.93, no. 11, pp. 1942–1955, Nov. 2005.

[27] FERC Report, Recent ISO Software Enhancements and Future Soft-ware and Modeling Plans [Online]. Available: http://www.ferc.gov/in-dustries/electric/indus-act/rto/rto-iso-soft-2011.pdf

Yan Xu (S'10–M'13) received the B.E. and M.E. degrees from South ChinaUniversity of Technology, China, in 2008 and 2011, respectively, and the Ph.D.degree from the University of Newcastle, Australia, in 2013.He was with the Hong Kong Polytechnic University, Hong Kong, between

2009 and 2011. He is now a Research Fellow at the Center for Intelligent Elec-tricity Networks (CIEN), University of Newcastle, Australia. His research inter-ests include power system stability and control, power system planning, SmartGrid, and intelligent system applications to power engineering.

Zhao Yang Dong (M'99–SM'06) received the Ph.D. degree from the Universityof Sydney, Australia, in 1999.He is now Professor and Head of School of Electrical and Information Engi-

neering, University of Sydney, Australia. He was previously Ausgrid Chair andDirector of the Centre for Intelligent Electricity Networks (CIEN), The Uni-versity of Newcastle, Australia, and is now a conjoint professor there. He alsoheld academic and industrial positions with the Hong Kong Polytechnic Uni-versity and Transend Networks, Tasmania, Australia. His research interest in-cludes Smart Grid, power system planning, power system security, load mod-

eling, electricity market, and computational intelligence and its application inpower engineering.Prof. Dong is an editor of the IEEE TRANSACTIONS ON SMART GRID, the

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, IEEE POWER ENGINEERINGLETTERS, and IET Renewable Power Generation.

Rui Zhang (S'12) received the B.E. degree from the University of Queensland,Australia, and the Ph.D. degree from the University of Newcastle, Australia, in2009 and 2014, respectively.She is now a Research Associate at the Centre for Intelligent Electricity Net-

works (CIEN), University of Newcastle, Australia. She was with Mawan Elec-tric Power Company, Shenzhen, China, from 2010 to 2011. Her research inter-ests include power system operation, stability, and control.

Yusheng Xue (M'87) received the Ph.D. degree from the University of Liege,Belgium, in 1987.He is a member of Chinese Academy of Engineering (CAE), and a Standing

Committee member of Division of Energy and Mining Engineering, CAE. Hewas Chief Engineer at the Nanjing Automation Research Institute (NARI),China during 1993–2009. He is now the Honorary President of State GridElectric Power Research Institute (SGEPRI or NARI), China. His researchinterests are power system automation and control, power system dynamicsand stability, and power system computing methods.Prof. Xue is a member of the PSCC Council, and the Editor-in-Chief of Au-

tomation of Electric Power System since 1999, and a member of Editorial Boardof IET Generation, Transmission & Distribution.

David J. Hill (F'93) received the B.E. degree in electrical engineering and theB.Sc. degree in mathematics from the University of Queensland, Australia, in1972 and 1974, respectively, and the Ph.D. degree from the University of New-castle, Australia, in 1976.He is now Chair of Electrical Engineering at The University of Hong Kong

and part-time professor with the School of Electrical and Information Engi-neering, The University of Sydney, Australia. He was previously Ausgrid Chairat the University of Sydney and is a Principal Researcher in National ICT Aus-tralia. During 2005–2010, he was an Australian Research Council FederationFellow at the Australian National University. During 2006–2010, he was also aChief Investigator of the ARC Centre of Excellence for Mathematics and Sta-tistics of Complex Systems. Since 1994, he has held various positions at theUniversity of Sydney, Melbourne, California (Berkeley), Newcastle (Australia),Lund (Sweden) and City University and The University of Hong Kong. His re-search interests are in network systems, stability analysis, distributed controland applications to infrastructure type networks, with work now focused on fu-ture electricity networks.Dr. Hill is a Fellow of Engineers Australia, the Society for Industrial and Ap-

plied Mathematics (SIAM), USA, the Australian Academy of Science, and theAustralian Academy of Technological Sciences and Engineering (ATSE). He isa Foreign Member of the Royal Swedish Academy of Engineering Sciences.

a decomposition-based practical approach to transient stability-constrained unit commitment

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