SOLVING A FUZZY UNIT COMMITMENT MODEL WITH TABU SEARCHALGORITHM

A. H. Mantawy Youssef L. Abdel-Magid Member , IEEE Senior Member, IEEE amantawy@kfupm.edu.sa amagid@kfupm.edu.sa

Electrical Engineering DepartmentKing Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi ArabiaAbstract

This paper presents a new algorithm based on integrating tabu search and fuzzy logic methods to solve theunit commitment problem. The uncertainties in the load demand and the spinning reserve constraints areformulated in a fuzzy logic frame. The tabu search is used to solve the combinatorial part of the unitcommitment problem, while the nonlinear part of the problem is solved via a quadratic programmingroutine. A simple tabu search algorithm based on the short-term memory procedures has been implemented.Numerical results show the superiority of the solutions obtained compared to the classical methods and thetabu search method as individual.

1. INTRODUCTIONThe Unit Commitment Problem (UCP) is the problem

of selecting the power generating units to be in serviceduring a scheduling period and for how long. Thecommitted units must meet the system load and reserverequirements at minimum operating cost, subject to avariety of constraints. The Economic Dispatch Problem(EDP) is to optimally allocate the load demand amongthe running units while satisfying the power balanceequations and units operating limits [1-18].

The exact solution of the UCP can be obtained by acomplete enumeration of all feasible combinations ofgenerating units, which could be a very huge number,while the economic dispatch problem is solved for eachfeasible combination. However, the high dimension ofthe possible solution space is the real difficulty insolving the problem.

Artificial intelligence techniques have come to be themost widely used tool for solving many optimizationproblems. These methods (e.g., simulated annealing,fuzzy logic, genetic algorithms, and tabu search) seemto be promising and are still evolving.

Tabu Search (TS) is an efficient optimizationprocedure that has been successfully applied to anumber of combinatorial optimization problems [11,19-22]. It has the ability to avoid entrapment in localminima. TS employs a flexible memory system (incontrast to memoryless systems, as SA and GA, andrigid memory systems as in branch-and-bound).Specific attention is given to the short-term memorycomponent of TS, which has provided solutionssuperior to the best obtained by other methods forvariety of problems [19].

Fuzzy Logic (FL), which may be viewed as anextension of classical logical systems, provides aneffective conceptual framework for dealing with theproblem of knowledge representation in an environmentof uncertainty and imprecision. The FL is used torealize the expected error in the forecasted load demandand the soft limits of the spinning reserve requirements[25-28].

In this paper we propose a new hybrid algorithm(TSFL) for solving the UCP. In the proposed algorithmwe consider the load demand uncertainties and thereserve constraints as soft limits in a FL frame. The TSalgorithm is then used to solve the combinatorialoptimization problem of the UCP.

Several examples are solved to test the proposedalgorithm. A comparison of results with other methodsin the literature [5,6,11] is presented.

In the next section, a mathematical formulation ofthe problem is introduced. In Section 3, the proposedTSFL algorithm is described. Sections 4 and 5 presentthe detailed implementation of the TS and FLcomponents. In Section 6, the computational resultsalong with a comparison to previously published workare presented. Section 7 outlines the conclusions.

2. PROBLEM STATEMENTIn the UCP under consideration, one is interested in a

solution that minimizes the total operating cost of thegenerating units during the scheduling time horizonwhile several constraints are satisfied [1-11].

2.1 The Objective functionThe overall objective function of the UCP of N

generating units for a scheduling time horizonT , (e.g.,24 HRs), is:

F (U F (P ) V S )T it it it it iti 1

N

t 1

T

= +== $ (1)

Where Uit : is status of unit i at hour t (ON=1, OFF=0).

Vit : is start-up/shut-down status of unit i at hour t.

Pit: is the output power from unit i at time t

The production cost, F (Pit)it , of a committed unit i, is

conventionally taken in a quadratic form:

F (P ) A P B P Cit it i2it i it i= + + $/HR (2)

Where, A i,Bi,C i : are the cost function parameters of

unit i.The start-up cost,Sit , is a function of the down time ofunit i [6]:

S So [1 D exp( Toff / Tdown)] Eit i i i i i= - - + $ (3) Where,Soi : is unit i cold start-up cost, and

D ,Ei i : are start-up cost coefficients for unit i.

2.2 The ConstraintsThe constraints that have been taken into

consideration in this work, may be classified into twomain groups:(i) System Constraints:a- Load demand constraints:

U P PDit it ti 1

N

== ;"t (4)

Where PD t : is the system peak demand at hour t (MW).b- Spinning Reserve

Spinning reserve,Rt, is the total amount ofgeneration capacity available from all unitssynchronized (spinning) on the system minus thepresent load demand.

U Pmax (PD R )it ii 1

N

t t= + ; "t (5)

(ii) Unit constraints:The constraints on the generating units area- Generation limits U Pmin P PmaxUit i it i it "i,t (6)Where,Pmini ,Pmaxi is minimum and maximum

generation limit (MW) of unit i, respectively.b- Minimum up/down time

Toff Tdown

Ton Tupi i

i i

;"i (7)

WhereTupi ,Tdowni are unit i minimum up/downtime.

Toni ,Toffi are time periods during which uniti is continuously ON/OFF.

c- Unit initial statusd- Crew constraintse- Unit availability; e.g. , must run, unavailable,

available, or fixed output (MW).f- Unit derating

3. THE PROPOSED ALGORITHM

3.1 OverviewIn the proposed algorithm we consider the load

demand uncertainties and the reserve constraints as softlimits in a FL frame. The fuzzy load demand iscalculated based on the error statistics and loadmembership function [25]. A penalty factor is thendetermined, as function of both the load demand andreserve membership functions to guide the search in theTS algorithm.

The major steps of the TSFL algorithm aresummarized as follow: (One) Apply FL rules to calculate the fuzzy load demand. (Two) Generate randomly an initial feasible solution andlet it be the current and best solutions. For the k thiteration apply the following steps:

(Three) Generate randomly a set of trial solutions asneighbors to the current solution.

(Four) Calculate the objective function of the set of trialsolutions by solving the EDP.

(Five) Use the FL approach to calculate the penalty factorto be added to the objective function as reflection tothe amount of reserve in the trial solution as follow: Calculate the amount of spinning reserve in the

trial solution. Apply FL rules to calculate the reserve

membership function. Estimate the value of the penalty factor according

the output of the load and reserve membershipfunctions.

(Six) Apply the TS test to accept one of the set of trialsolutions.

(Seven) Update the current and best solutions. (Eight) Check for stopping criteria. If satisfied stop,

otherwise go to Step (c).

3.2 Stopping CriteriaThere are several possible stopping conditions for the

search. In our implementation, we stop the search if oneof the following two conditions is satisfied in the ordergiven:

The number of iterations performed since the bestsolution last changed is greater than a prespecifiedmaximum number of iterations, or

Maximum allowable number of iterations isreached.

4. TS IMPLEMENTATION IN THE TSFLALGORTIHM

In the TS part of the TSFL algorithm the short-term memory procedures are implemented. In this workthe tabu List (TL) is implemented using our approachin [11]. In this approach the TL is recorded as adecimal number equivalent to the solution vector foreach generating unit (which is 0-1 values). Hence theTL is a two dimensional array of dimension NxZ, whereN is the number of generating units and Z is the TLsize. By using this approach we record all informationof the trial solution in minimum memory.

The tabu test can be described as follows: Sort the set of trial solutions (neighbors) in an

ascending order according to their objectivefunctions.

Apply the acceptance test in order until one ofthese solutions is accepted.

Accept the trial solution and set the currentsolution equal to the accepted trial solution:

* If it is not in the TL, or * In the TL but satisfy the aspiration level

criteria. Otherwise apply the test to the next solution.

5. FL IMPLEMENTATION IN THE TSFLALGORITHM

In the proposed algorithm FL is used to deals withthe uncertainties in the forecasted load demand and theprespecified spinning reserve requirements. Theimplemented fuzzy logic system consists of two inputs:the load demand and the spinning reserve, and twooutputs: a fuzzy load demand and a penalty factor.

5.1 Membership function for the load demandThe fuzzy set of input for the load demand is divided

into six fuzzy values (LN, MN, HN, LP, MP, HP). Themembership function for load forecast error is taken asfollow [25]:

committed units in the 24 hours. Table (3) gives thehourly crisp and fuzzy load demands, and thecorresponding economic dispatch costs, start-up costs,and total operating cost.

Table (1) Comparison with TSA in [11]

Example TSA[11] TSFL % Saving1 538390 537686 0.13Total

Cost ($) 2 59512 59385 0.21

Table (2) Power Sharing (MW) of Example 1

HR

Unit Number** 2 3 4 6 7 8 9 10

1 400.0 0.00 0.00 185.0 0.00350.2 0.00 89.702 395.3 0.00 0.00 181.0 0.00338.3 0.00 85.193 355.3 0.00 0.00 168.6 0.00300.9 0.00 75.004 333.1 0.00 0.00 161.7 0.00280.1 0.00 75.005 400.0 0.00 0.00 185.0 0.00350.2 0.00 89.706 400.0 0.00 295.6200.0 0.00375.0 0.00129.37 400.0 0.00 342.9200.0 0.00375.0507.0145.08 400.0295.5396.6200.0 0.00375.0569.9162.89 400.0468.0420.0200.0 0.00375.0768.0218.910400.0444.6420.0200.0 358.0375.0741.0211.211400.0486.3420.0200.0 404.8375.0788.9224.812400.0514.1420.0200.0 436.0375.0820.8233.913400.0479.3420.0200.0 397.0375.0780.9222.614400.0388.9420.0200.0 295.6375.0677.1193.215400.0310.0410.8200.0 250.0375.0586.5167.516400.0266.6368.2200.0 250.0375.0536.6153.417400.0317.3417.9200.0 250.0375.0594.8169.818400.0458.5420.0200.0 373.6375.0757.0215.819400.0486.3420.0200.0 404.8375.0788.9224.820400.0375.0420.0200.0 280.0375.0661.2188.621400.0 0.00 404.8200.0 0.00375.0579.5165.522400.0 0.00 0.00 200.0 0.00375.0675.00.0023400.0 0.00 0.00 191.6 0.00370.1338.20.0024377.6 0.00 0.00 175.5 0.0321.7275.00.00

* Units 1 & 5 are OFF at all hours.

Table (3) Load, Capacities (MW), and Hourly Costs ($) ofExample 1.

Total-COST

ST-COST

ED-COST

FuzzyLoad

CrispLoad

HR

9,670.0 - 9,670.0 1,002.61 1,025 1 9,446.6 - 9,446.6 1,026.08 1,000 2 8,560.9 - 8,560.9 880.54 900 3 8,123.1 - 8,123.1 872.01 850 4 9,670.0 - 9,670.0 1,002.73 1,025 5 15,140.0 1,706.0 13,434.1 1,435.06 1,400 6 21,876.8 2,659.1 19,217.7 1,929.65 1,970 7 26,500.6 2,685.1 23,815.5 2,351.39 2,400 8 28,253.9 - 28,253.9 2,794.37 2,850 9 34,709.3 3,007.6 31,701.7 3,213.98 3,150 10

33,219.8 - 33,219.8 3,234.62 3,300 11 34,242.1 - 34,242.1 3,337.41 3,400 12 32,965.5 - 32,965.5 3,206.45 3,275 13 29,706.3 - 29,706.3 2,892.06 2,950 14 27,259.7 - 27,259.7 2,653.29 2,700 15 25,819.8 - 25,819.8 2,501.50 2,550 16 27,501.6 - 27,501.6 2,673.12 2,725 17 32,205.7 - 32,205.7 3,139.54 3,200 18 33,219.8 - 33,219.8 3,370.87 3,300 19 29,212.5 - 29,212.5 2,847.30 2,900 20 20,698.4 - 20,698.4 2,085.57 2,125 21 15,878.2 - 15,878.2 1,686.43 1,650 22 12,572.8 - 12,572.8 1,277.54 1,300 23 11,232.0 - 11,232.0 1,126.86 1,150 24

7. CONCLUSIONSIn this paper we proposed a new hybrid algorithm for

the UCP. The algorithm integrates the main features oftwo of the most commonly used artificial intelligencemethods, TS and FL. The UCP is formulated in a FLframe to deal with the uncertainties in the load demandand the soft limit constraint of the spinning reserve.The TS algorithm is used to solve the combinatorialoptimization part of the UCP while quadraticprogramming algorithm is used to solve the nonlinearprogramming part of the problem.

The TS algorithm is implemented via a simple short-term algorithm to simplify and speed up the calculation[11].

Two examples from the literature were solved forcomparison purposes with other methods. The obtainedresults are superior to those reported in [5,6] using LRand IP. Moreover the obtained results (using theproposed algorithm) are better than those obtainedusing the TS algorithm [11].

A basic advantage of the proposed algorithm is thehigh quality of solutions compared to those obtained byLR, IP and TS. Moreover, the algorithm is capable ofhandling practical issues such as the uncertainties inthe UCP. Further work in this area may be in theapplication of parallel processing techniques, thusreducing the computation time or exploring widersolution space.

ACKNOWLEDGMENTThe authors acknowledge the support of King Fahd

University of Petroleum and Minerals.

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