Energy Conversion and Management 76 (2013) 20–25

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Energy Conversion and Management

journal homepage: www.elsevier .com/locate /enconman

Second-order cone programming for solving unit commitment strategyof thermal generators

0196-8904/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2013.07.019

⇑ Corresponding author. Tel.: +86 27 87543609.E-mail address: zhangsq_hust@163.com (S. Zhang).

Xiaohui Yuan a, Hao Tian a, Shuangquan Zhang b,⇑, Bin Ji a, Yanhong Hou a

a School of Hydropower and Information Engineering, Huazhong University of Science and Technology, 430074 Wuhan, Chinab School of Energy and Power Engineering, Huazhong University of Science and Technology, 430074 Wuhan, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 February 2013Accepted 13 July 2013

Keywords:Mixed-integer nonlinear programmingSecond-order cone programmingSeparable convex functionUnit commitment

The short-term unit commitment (UC) problem of hydrothermal generation systems is a mixed-integernonlinear programming (MINLP), which is difficult to solve efficiently, especially for large-scale instances.The perspective relaxation (PR) is an effective approach to constructing tight approximations to MINLPwith semi-continuous variables. In this paper, the PR of UC problem is formulated as a mixed integersecond-order cone programming (SOCP) model because the quadratic polynomial cost function of theUC problem is SOCP-representable. The proposed model is implemented by using the commercialoptimization software IBM CPLEX 12.4. Extensive numerical studies have been conducted to verify theadvantages of our proposed method. Instances of the test system vary from 10 to 1000 units. Our resultsindicate that the proposed method performs better than the existing methods in terms of production costsavings and faster computational times, especially for large systems.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Unit commitment (UC) is a very significant optimization task,which plays an important role in operation planning of power sys-tems. The unit commitment problem (UCP) in power systems re-fers to the optimization problem of determining the startup andshutdown schedules of generating units over a scheduling periodso that the total production cost is minimized while satisfying var-ious constraints [1]. The UCP can be considered as two linked pro-cesses of optimization decision, namely the unit-scheduled processwhich determines the on/off status of generating units in each per-iod of planning horizon, and the economic load dispatch process.Mathematically, the UCP has commonly been formulated as a non-linear, large scale, mixed-integer combinatorial optimization prob-lem with constraints. Furthermore, the number of combinations of0–1 integer variables grows exponentially as the size of the prob-lem gets larger. Therefore, the UCP is known as one of the most dif-ficult problems in power systems of optimization [2,3].

Many methods have been developed to solve the UCP in thepast decades. The major methods include dynamic programming(DP) [4,5], branch-and-bound methods (BBM) [6], priority listmethod (PL) [7,8], integer and mixed integer linear programming(MILP) [9,10], and Langrangian relaxation (LR) [11,12]. Amongthese methods, PL is simple and fast, but the quality of its final

solution cannot be guaranteed. DP is flexible but the disadvantagelies in the ‘‘curse of dimensionality’’. BBM is another popular meth-od, but typically it runs very slowly when the number of units in-creases. As MILP adopts linear programming techniques to checkan integer solution, it handles UCP’s nonlinear characteristicsapproximately. The main problem with the LR method lies in thedifficulty in obtaining feasible solutions. Due to the non-convexityof the UCP, optimality of the dual problem does not guarantee fea-sibility of the primal UCP. Furthermore, the solution’s quality of LRdepends on the method to update Lagrange multipliers. In short,these methods have only been applied to small UCP instancesand typically they require strong assumptions that limit the solu-tion space.

Aside from the above methods, meta-heuristic approaches, suchas artificial neural network (ANN) [13], genetic algorithm (GA)[14–16], evolutionary programming (EP) [17], memetic algorithm(MA) [18], Tabu search (TS) [19], simulated annealing (SA) [20],particle swarm optimization (PSO) [21–24], greedy random adap-tive search procedure (GRASP) [25], immune algorithm (IA) [26]and differential evolution (DE) [27,28], have also been used tosolve UCP. These meta-heuristic optimization methods attractmuch attention, because of their ability to search not only localoptimal solution but also global optimal solution, and because theycan easily deal with various complicated nonlinear constraints.However, these meta-heuristic methods require a considerableamount of computational time to find the near-global minimumespecially for a large-scale UCP. To reduce the search space in thelarge-scale UCP, hybrid methods combined with meta-heuristic

X. Yuan et al. / Energy Conversion and Management 76 (2013) 20–25 21

approaches and other optimization methods have been used tosolve UCP, such as GA combined with LR (LRGA) [29], MA com-bined with LR (LRMA) [18], PSO combined with LR (LRPSO) [30],hybrid PSO and Nelder–Mead [31], hybrid binary PSO and realcoded GA) [32]. These hybrid methods tend to be more efficientand effective than the individual method alone. However, they stillhave the drawbacks of the single method we mentioned earlier.Thus, it’s still of great significance to improve current optimizationtechniques and to explore new algorithm solutions to solving UCP.

Second order cone programming (SOCP) problems are convexoptimization problems in which a linear function is minimizedover the intersection of a polyhedron and the Cartesian productof second order or Lorentz cones [33]. Linear programming, convexquadratic programming and quadratically constrainted convexquadratic programming can all be formulated as SOCP problems.SOCP has been used to model a broad range of applications fromengineering to business, which are capable of dealing with robustoptimization and combinatorial optimization. SOCP problems canbe solved in polynomial time by interior point method. Thus, ow-ing to its broad applicability and its computational tractability,SOCP deserves to be studied in its own right.

In this paper, we exploit the structure of the UC problem in or-der to derive a strong second order conic formulation. We applythe conic strengthening to this problem. As the conic reformulationtechnique is based only on the objective function of the problem, itcan easily be applied to solve the mixed 0–1 optimization UC prob-lem with quadratic cost objective functions. So, we put forwardsecond order cone programming to solve the UC problem, whichcan be solved directly and does not need to be divided firstly intotwo or more sub-problems. Moreover, the SCOP algorithm mayconverge in polynomial time with a very high precision in solutioneven for large-scale UC problems. In this paper, we constructstrong second order conic reformulations of the UC problem basedon the convex hull description of mixed integer set defined by non-linear inequalities. The effectiveness of the proposed method istested on many instances with the number of units ranging from10 to 1000. In terms of the quality of solution and computationtime, our numerical results demonstrate the superiority of the pro-posed method when compared with other optimization methodsreported in the literatures.

The rest of this paper is organized as follows. Section 2 gives themathematical formulation of the UCP. Section 3 introduces theconic strengthening formulation which is applied to the UC Prob-lem. Section 4 gives the numerical studies and Section 5 makesthe conclusion. Finally, acknowledgements are given.

2. Formulation of the unit commitment problem

2.1. Notation

First, we introduce the notations for UC problem used in thispaper.

CSCi

cold startup cost of unit i HSCi hot startup cost of unit i N number of thermal generators Pt

i

generation power of unit i at time t

Pi min

minimum generation power of unit i Pi max maximum generation power of unit i Pt

D

system load demand at time t

PtR

spinning reserve at time t

SCti

startup cost of unit i at time t

T

number of time intervals

Ti;cold

cold startup time of unit i

Tti;on

time period that unit i had been continuously up tillperiod t

Tti;off

time period that unit i had been continuously downtill period t

Ti;down

minimum down time of unit i Ti;up minimum up time of unit i ut

i

on/off status of unit i at time t (on = 1 and off = 0)

2.2. Objective function

The objective of UCP is to find the generation scheduling overthe scheduled-time horizon so that the total production cost canbe minimized while satisfying certain types of constraints. The to-tal production cost in the entire scheduling periods is the sum ofthe operating cost and the startup cost of all the units. Thus, theoverall objective function of the UC problem can be given by

ðUCP0ÞminXT

t¼1

XN

i¼1

½fiðPti Þ þ SCt

i � ð1Þ

Generally, the fuel cost, fiðpti Þ per unit is a function of the gener-

ator power output. The most frequently-used cost function is in theform of

fiðpti Þ ¼ ai þ bipt

i þ ciðpti Þ

2 ð2Þ

where ai, bi and ci represent the cost coefficients of unit i.The generator startup cost is incurred when a generator is put

into operation. This cost depends on the time when the unit isoff. It is given by

SCti ¼

HSCi if Ti;down 6 Tti;off 6 Ti;down þ Ti;cold

CSCi if Tti;off > Ti;down þ Ti;cold

(ð3Þ

2.3. Constraints

� System power balanceThe power generated from all committed units must be enoughto meet the system power demand.

XN

i¼1

uti p

ti ¼ Pt

D ð4Þ

� System spinning reserve requirementSpinning reserve requirements are necessary in the operation ofa power system due partly to possible outages of equipments.The reserve is considered to be a pre-specified amount or a gi-ven percentage of the forecasted demand.

XN

i¼1

uti pi max P Pt

D þ PtR ð5Þ

� Generation power limitsEach unit has a generation range, which is represented as

uti pi min 6 pt

i 6 uti pi max ð6Þ

� Minimum up time of unitA unit must be on for certain hours before shuting it down.

Tti;on P Ti;up ð7Þ

� Minimum down time of unitA unit must be off for certain hours before bringing it online.

22 X. Yuan et al. / Energy Conversion and Management 76 (2013) 20–25

Tti;off P Ti;down ð8Þ

� Unit initial statusThe initial status at the start of the scheduling period must be

taken into account.

3. Second order conic reformulation of the UC problem

3.1. Overview of second order cone programming

SOCP is a polynomial-time generalization of convex quadrati-cally constrained programming [34]. The standard form of SOCPis much similar to that of linear programming.

We consider the second order cone programming as

min f T x

s:t: kAixþ bik 6 cTi xþ di; i; . . . ;N

ð9Þ

where x 2 Rn is the optimization variable, and f 2 Rn, Ai 2 Rðni�1Þ�n,bi 2 Rni�1, ci 2 Rn and di 2 R are the problem parameters. The normappearing in the constraints is the standard Euclidean norm, i.e.,kuk ¼ ðuT uÞ1=2.

The constraint in the form kAixþ bik 6 cTi xþ di is called a sec-

ond order cone constraint, namely the term second order cone pro-gramming. The SOCP (9) is a convex programming problem sincethe objective is a convex function and the constraints define a con-vex set.

Second order cone constraints can be used to represent severalcommon convex constraints. For example, when ni ¼ 1 fori = 1, . . . ,N, the SOCP reduces to the linear programming (LP)

min f T x

s:t: 0 6 cTi xþ di; i; . . . ;N

ð10Þ

Another interesting special case arises when ci ¼ 0, so the sec-ond order cone constraint reduces to kAixþ bik 6 di, which isequivalent (assuming di P 0) to the (convex) quadratic constraintkAixþ bik2

6 d2i . Thus, when all ci vanish, the SOCP reduces to a

quadratically constrained linear programming (QCLP). The (con-vex) quadratic programming (QP), quadratically constrained qua-dratic programming (QCQP) and many other nonlinear convexoptimization problems can be reformulated as SOCP as well.

The set of points that satisfy (10) forms a convex set, and thereare efficient and robust algorithms to solve optimization problemswhich have second order cone constraints.

A rotated second order cone constraint is of the form

x26 yz with y P 0; z P 0 ð11Þ

Rotated second order cone constraints (11) are equivalent tosecond order cone constraints kAixþ bik 6 cT

i xþ di since

kð2x; y� zÞTk 6 yþ z() x26 yz; y P 0; z P 0 ð12Þ

3.2. Strengthening the continuous relaxation

In this section we describe how to reformulate the UC problemby using conic quadratic constraints. By introducing auxiliary vari-ables ct

i 2 Rþ, we can bring the nonlinear objective of the problem(UCP0) into the constraints and linearize the objective of the for-mulation as

ðUCP1Þ

minXT

t¼1

XN

i¼1

½cicti þ bipt

i þ aiuti þ SCt

i �

s:t: ðpti Þ

26 ct

iut

i 2 f0;1gi ¼ 1;2; :::N; t ¼ 1;2; :::Tð4Þ; ð5Þ; ð6Þ; ð7Þ; ð8Þ

ð13Þ

Problem (UCP1) is not necessarily easier to solve than (UCP0).On the contrary, solvers can usually deal with nonlinearity in theobjective more easily than nonlinearity in the constraints. Problem(UCP1) is an intermediate formulation that will enable us to derivea strong conic formulation.

To achieve our purpose it suffices to concentrate on the mixed0–1 set

C ¼ ðu; p; cÞ 2 ð0;1Þ � Rþ � Rþ : p26 c;

u � pmin 6 p 6 u � pmax

( )ð14Þ

Observe that constraints of C are of the form (6) and (13) andconsider that solutions of C satisfy p ¼ au. It is easy to find thatfor a > 0 each point on the curve defined asL ¼ fðu; p; cÞ 2 R3 : 0 < u < 1; p ¼ au; p2 ¼ cg is an extreme pointof the continuous relaxation of C.

Next we will reformulate C so that L is eliminated from its con-tinuous relaxation.

We propose to strengthen (13) as

p26 cu ð15Þ

For 0 6 u 6 1, inequality (15) implies (13). Also it is clear that(15) is valid for C as it reduces to (13) for u 2 f0;1g.

Thus, we may replace (13) with (15). Next, consider thestrengthened continuous relaxation of C:

Cs ¼ðu;p; cÞ 2 R3 : p2

6 c � u;u � pmin 6 p 6 u � pmax;

0 6 u 6 1;0 6 c

( )ð16Þ

Although (15) is nonlinear, Cs is a convex set. Indeed, it can berevealed that Cs is the smallest convex relaxation of C. Accordingto [34,35], the convex hull of C, conv(C), equals Cs.

3.3. Conic quadratic representation of UC problem

Here we use the nonlinear constraint (13) explicitly by reformu-lating it via conic quadratic constraints.

According to (12), constraint (15) p26 c � u can be written as a

conic quadratic (second order cone) constraint

kð2p; ðc� uÞÞk 6 cþ u or p2 þ ðc� uÞ2

46ðcþ uÞ2

4ð17Þ

Therefore, the reformulation of problem (UCP0) as a SOCP isactually quite simple for the quadratic cost function case, as when

uti > 0, constraint ðpt

i Þ2=ut

i 6 cti can be transformed algebraically

into the equivalent form ðpti Þ

2 þ cti�ut2

i4 6

ðctiþut

iÞ2

4 , leading to the SOCPof the UC problem:

ðUC � SOCPÞ

minXT

t¼1

XN

i¼1

½cicti þ bipt

i þ aiuti þ SCt

i �

s:t:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpt

i Þ2 þ ðct

i � uti Þ

2=4

q6

ctiþut

i2

cti P 0

uti 2 f0;1g

i ¼ 1;2; :::N; t ¼ 1;2; :::Tð4Þ; ð5Þ; ð6Þ; ð7Þ; ð8Þ

ð18Þ

The second order conic programming reformulation of theabove UC problem (UC-SOCP) can be directly solved by solverssuch as CPLEX and MOSEK. This can shorten the solution time asit does not require the process of solving mixed-integer nonlinearprogramming.

The UC problem includes many constraints. However, it isworth mentioning that thermal units are subject to the minimumup-time and down-time constraints in (7) and (8), which are

X. Yuan et al. / Energy Conversion and Management 76 (2013) 20–25 23

difficult to handle directly. In [36], Rajan and Takriti claim that thefollowing constraints (19) and (20), along with the following logi-cal constraints (21) and variable-bound constraints (22), define theconvex hull of all feasible solutions in the minimum up/down-timepolytope:

Xt

j¼t�Ti;upþ1

v ji 6 ut

i ; t 2 ½Ti;up þ 1; T� ð19Þ

Xt

j¼t�Ti;downþ1

wji 6 1� ut

i ; t 2 ½Ti;down þ 1; T� ð20Þ

In addition to constraints (19) and (20), constraint (21) isneeded to ensure that v t

i and wti take the appropriate values when

a generator is either turned on or turned off:

ut�1i � ut

i þ v ti �wt

i ¼ 0 ð21Þ

0 6 v ti 6 1

0 6 wti 6 1

ð22Þ

where v ti and wt

i denote startup and shutdown status at time t ofgenerator i, respectively.

Note that if all the variables uti are known, then v t

i and wti vari-

ables can be easily determined. The unit commitment variable uti

must be modeled as binary. v ti and wt

i can be treated, in practice,as continuous variables. Once ut

i is defined binary, v ti and wt

i willtake 0/1 values. Using binary variables to indicate startup or shut-down status can make it easier to generate strong valid inequali-ties. As a result, the minimum up/down-time constraints (7) and(8) in UC-SOCP drops and we replace (7) and (8) with constraints(19-22).

Finally, the mixed-integer SOCP formulation of the UC mathe-matical model is obtained as follows:

minXT

t¼1

XN

i¼1

½cicti þ bipt

i þ aiuti þ SCt

i �

s:t:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpt

i Þ2 þ ðct

i � uti Þ

2=4

q6

cti þ ut

i

2ct

i P 0XN

i¼1

pti u

ti ¼ Pt

D

XN

i¼1

uti pi max P Pt

D þ PtR

uti pi min 6 pt

i 6 uti pi max

ut�1i � ut

i þ v ti �wt

i ¼ 0Xt

j¼t�Ti;upþ1

v ti 6 ut

i k 2 ½Ti;up þ 1; T�

Xt

j¼t�Ti;downþ1

wti 6 1� ut

i k 2 ½Ti;down þ 1; T�

0 6 v ti 6 1

0 6 wti 6 1

uti 2 f0;1g

i ¼ 1;2; :::N; t ¼ 1;2; :::T

ð23Þ

4. Numerical studies

In order to verify the feasibility and effectiveness of the UC-SOCP model for solving the UC problem, the UC-SOCP model istested on problem instances generated from a basic system of 10

units [17]. The scheduling time horizon T is chosen by dividingone day into 24 intervals, each with one hour. The spinning reserverequirement is set to be 10% of the total load demand. For systemsof 20, 40, 60, 80, 100, 200, 500 and 1000 units, the basic 10-unitsystem is duplicated and the total load demands are adjusted pro-portionally to its size. The demand and generating unit data of thetest system are given in Tables 1 and 2, respectively.

Due to the growing demand for solving conic quadratic integerprogramming, commercial optimization packages, such as CPLEXand MOSEK, have already offered branch-and-bound solvers forconic quadratic integer programming. Therefore, we solve theUC-SOCP model by using CPLEX (Version 12.4) second-order conicMIP solver, on a laptop with an Intel Core(TM) 2Duo 2.50 GHz CPUand 4 GB memory. In CPLEX, an optimality parameter can be spec-ified to decide whether to find the optimal solution or to quicklyobtain a suboptimal solution. In this numerical study, we set theoptimality gap tolerance of the relative mixed integer to be0.001. Under the chosen parameters, we run CPLEX to solve prob-lem instances of the UC-SOCP model, which ranges from 10 to 100units. The total production costs and the corresponding executiontime of CPU are summarized in Table 3.

To verify the quality of the solutions found by the UC-SOCP, wecompare the performance of the UC-SOCP to that of other ap-proaches in terms of the total production cost and the CPU execu-tion time. Other approaches include greedy random adaptivesearch procedure (GRASP)[25], Lagrangian relaxation (LR) [11], pri-ority list (PL) [8], genetic algorithm (GA) and integer-coded GA(ICGA) [15], evolutionary programming (EP) [17], memetic algo-rithm (MA) [18], Lagrangian relaxation and memetic algorithm(LRMA) [18], improved binary particle swarm optimization (IBPSO)[22], binary differential evolution (BDE) [27] and particle swarmoptimization combined with the Lagrangian relaxation method(LRPSO) [30]. Table 4 provides a comparison between the total pro-duction cost of the SOCP method and that of other methods. Wecan see clearly from Table 4 that the total production costs ob-tained by the SOCP method in test cases are smaller than thoseof the above methods. Besides, the superiority of the SOCP methodis the most salient in larger systems. On the one hand, when com-pared to the BDE solution (which obtains the lowest values amongthe listed methods), the UC-SOCP saves $3847, $3913, $6642,$5536 in the test systems of 40, 60, 80, 100 units, respectively.On the other hand, the difference between the best solutions andthe UC-SOCP solution for the smaller test systems of 10 and 20units is in the order of 0.098% and 0.016%, partially because theoptimality gap tolerance of the relative mixed integer in CPLEX isset only at 0.001 for the UC-SOCP solution.

To observe the rate of increase in the CPU execution time of theUC-SOCP, Fig. 1 is plotted to show the CPU execution time of theUC-SOCP method versus the problem size of the UC problem aswell as its quadratic polynomial fitting curve. From Fig. 1, we cannotice that the rate of the increase is roughly the problem size ofa quadratic polynomial function.

To get a fair comparison of computational efforts, the CPU exe-cution time of each method must be converted into a common basefor comparison. In this paper, we normalize the CPU executiontime of each method, that is, by comparing the factor of whichthe execution time of a method increases when the number ofunits increases (from N = 10 to N = 20, then to N = 40, etc.). Insteadof just comparing the execution time of a method, the normalizedexecution time is chosen as an indicator for the computational ef-fort of a method for the sole reason that these methods do not usethe same computing platforms. Fig. 2 illustrates the normalizedexecution time of the SOCP and that of other methods.

From Fig. 2, we can learn that the SOCP method runs faster thanthe others. Although they cannot be compared directly due to dif-ferent computers which have been used, the computational time

Table 1Hourly load demand.

Hour 1 2 3 4 5 6 7 8 9 10 11 12

Load (MW) 700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500Hour 13 14 15 16 17 18 19 20 21 22 23 24Load (MW) 1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800

Table 2Unit data of the 10-unit system.

Unit 1 2 3 4 5

Pimax (MW) 455 455 130 130 162Pimin (MW) 150 150 20 20 25ai 1000 970 700 680 450bi 16.19 17.26 16.60 16.50 19.7ci 0.00048 0.00031 0.002 0.00211 0.00398Min up (h) 8 8 5 5 6Min down (h) 8 8 5 5 6Hot start cost ($) 4500 5000 550 560 900Cold start cost ($) 9000 10000 1100 1120 1800Cold start time (h) 5 5 4 4 4Initial States (h) 8 8 -5 -5 -6Unit 6 7 8 9 10Pimax (MW) 80 85 55 55 55Pimin (MW) 20 25 10 10 10ai 370 480 660 665 670bi 22.26 27.74 25.92 27.27 27.79ci 0.00712 0.00079 0.00413 0.00222 0.00173Min up (h) 3 3 1 1 1Min down (h) 3 3 1 1 1Hot start cost ($) 260 260 30 30 30Cold start cost ($) 520 520 60 60 60Cold start time (h) 2 2 0 0 0Initial states (h) �3 �3 �1 �1 �1

Table 3Simulation results of the total cost and execution time.

Results Number of units

10 20 40 60 80 100

Total cost ($) 564531 1124713 2244369 3363758 4484357 5603728Computing time (s) 0.78 2.13 5.34 10.67 18.13 28.97

Fig. 1. The CPU execution time of the UC-SOCP method.

Fig. 2. Normalized execution time of different methods.

24 X. Yuan et al. / Energy Conversion and Management 76 (2013) 20–25

shows that the SOCP is able to find close-to-optimal solutions inmuch shorter time than the other methods. Moreover, the CPUtime of the SOCP method does not increase exponentially with re-spect to the system size of UCP, which is favorable for a large-scaleUCP implementation.

At the same time, the UC-SOCP model is tested on very large in-stances of 200, 500 and 1000 units with a scheduling time horizon

Table 4Comparison of the total production costs ($) with other methods.

Method Number of units

10 20 40 60 80 100

LR 565825 1130660 2258503 3394066 4526022 5657277GRASP 565825 1128160 2259340 3383184 4525934 5668870PL 564950 1123938 2248645 3371178 4492909 5615530GA 565825 1126243 2251911 3376625 4504933 5627437ICGA 566404 1127244 2254123 3378108 4498943 5630838EP 564551 1125494 2249093 3371611 4498479 5623885MA 565827 1127254 2252937 3388676 4501449 5640543LRMA 566686 1128192 2249589 3370595 4494214 5616314LRPSO 565869 1128072 2251116 3376407 4496717 5623607IBPSO 563977 1125216 2248581 3367865 4491083 5610293BDE 563977 1124538 2248216 3367671 4490999 5609264SOCP 564531 1124713 2244369 3363758 4484357 5603728

Table 5Simulation results for large test systems.

Results Number of units

200 500 1000

Total cost ($) 11199983 27993083 55979918Computing time (s) 49.38 147.08 504.56

of 24 h. As to the very large-scale systems up to 1000 units, com-putation results of the UC-SOCP model are shown in Table 5.

From Table 5, we can see that the CPU time appears to be a sys-tem size of the quadratic polynomial function, which is very prac-tical for large-scale implementations.

In summary, the total production costs of the UC-SOCP tend tobe less than those of other methods, and the UC-SOCP performs

X. Yuan et al. / Energy Conversion and Management 76 (2013) 20–25 25

much better than other methods in terms of the CPU executiontime, especially on a large-scale UCP. As a result, we conclude thatthe UC-SOCP is a better approach to solving UCP.

5. Conclusion

The perspective relaxation is an useful tool in obtaining tighterand lower bounds in nonlinear programming with semicontinuousvariables. So the UC problem can take advantage of this relaxationtechnique. In addition, the nonlinear inequalities of the UC prob-lem in the perspective relaxation can be casted as second ordercone constraints, a transformation that have greatly improvedthe solvability of the UC problem. Based on SOCP, we have pro-posed a new way of constructing the perspective relaxation ofthe UC problem. In this way, the mixed integer nonlinear unit com-mitment problem can be transformed into SOCP form that can beefficiently implemented by using the optimization software. Theextensive computation studies show that the proposed SOCPmethod dominates the existing methods in the solution qualityas well as the solution time, especially for the large-scale UC prob-lem. Moreover, the CPU time of the proposed SOCP method in-creases quadratically only with the system size, which isfavorable for large-scale implementations.

Acknowledgements

The authors sincerely thank Prof. Zuo-Jun ‘‘Max’’ Shen, Dr. SarahDrewes and Dr. Chen Chen for their contributions to the originalversion of this manuscript. At the same time, they had like to givehearted thanks for the financial supports from National NaturalScience Foundation of China under Grant No. 40971018.

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