research article a unit commitment model with implicit

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 912825, 11 pageshttp://dx.doi.org/10.1155/2013/912825

Research ArticleA Unit Commitment Model with Implicit Reserve ConstraintBased on an Improved Artificial Fish Swarm Algorithm

Wei Han, Hong-hua Wang, Xin-song Zhang, and Ling Chen

College of Energy & Electrical Engineering, Hohai University, Nanjing 211100, China

Correspondence should be addressed to Wei Han; [email protected]

Received 8 June 2013; Revised 29 September 2013; Accepted 1 November 2013

Academic Editor: Pedro Ribeiro

Copyright © 2013 Wei Han et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An implicit reserve constraint unit commitment (IRCUC) model is presented in this paper. Different from the traditional unitcommitment (UC) model, the constraint of spinning reserve is not given explicitly but implicitly in a trade-off between theproduction cost and the outage loss. An analytical method is applied to evaluate the reliability of UC solutions and to estimatethe outage loss. The stochastic failures of generating units and uncertainties of load demands are considered while assessing thereliability.The artificial fish swarm algorithm (AFSA) is employed to solve this proposedmodel. In addition to the regular operation,a mutation operator (MO) is designed to enhance the searching performance of the algorithm. The feasibility of the proposedmethod is demonstrated from 10 to 100 units system, and the testing results are compared with those obtained by genetic algorithm(GA), particle swarm optimization (PSO), and ant colony optimization (ACO) in terms of total production cost and computationaltime. The simulation results show that the proposed method is capable of obtaining higher quality solutions.

1. Introduction

During the operation of a power system, the balance ofactive power is often interrupted by some unpredictablefactors, such as a random load change, transmission andtransformation of power equipment, and failure of the gen-erating unit. In order to keep the balance of power system,a reasonable amount of spinning reserve needs to be setin advance. During the actual power generation dispatch,the system spinning reserve requirements are predeterminedby the knowledge of experienced power system operatorsand introduced as a set of operating constraints which givebetter performance of the power system [1, 2]. The basictask of the unit commitment (UC) model containing thespinning reserve constraint is to schedule the planning ofpower generation and the arranging output with minimumcost in order to satisfy the power load, spinning reserve, andoperational condition of units [3].

To deal with the problem of random failure of the unit,the reserve demand was defined as the largest capacity ofthe installed generating units within the system. Further-more, some practical systems for scheduling would assume

the spinning reserve to be a fixed percentage of powerloads.The selecting principle of deterministic reserve neitherconsidered the various random factors of the operatingsystem nor provided the cost of reserve. Therefore, thesystem state would be subjective to the experience of thedispatcher and in some cases the system would not operateefficiently from economy aspect. As more and more large-scale renewable energy sources are injected into the grid,the uncertainty of the power system will be significantlyincreased and the current calculation of the system spinningreserve would be more inefficient. Consequently, Anstineet al. [4] and Li and Zhou [5] put forth a principle ofprobabilistic reserve with the risk of UC model. Based onreliability constraints, Snyder et al. [6] proposed a UC modelusing the inequality constraints of reliability index to replacethe deterministic reserve constraints of the traditional UCmodel; a good performance was obtained. However, a lotof subjectivity is introduced by selecting the threshold ofprobabilistic reliability index and the deterministic reserve.

Based on the above discussion, the spinning reserve con-straint is removed from theUCmodel and an implicit reserveconstraint unit commitment (IRCUC) model is proposed in

2 Mathematical Problems in Engineering

this paper. In addition, the objective function of this modelis extended from the minimal production cost of traditionalUCmodel to the minimal sum of production cost and outageloss. That is, if the spinning reserve capacity is scheduled toomuch in the planning of power generation, then the outageloss would be reduced while the production cost would beincreased; on the contrary, if the reserve capacity is scheduledtoo little, then the production cost would be decreased whilethe outage loss would be increased.

Same as the traditional UC model, the IRCUC modelproposed in this paper is a combinatorial optimization prob-lem with time and nonlinearity, which contains continuousand discrete variables, so it is difficult to get the analyticalsolution. To solve the UC problem, extensive researcheshave been carried out by domestic and foreign scholars. Thesolutions are mainly attributed to the following categories:(1) priority list (PL) algorithm [7, 8]; (2) mathematicalalgorithms, including branch-bound algorithm [9], bender’sdecoupling algorithm [10], Lagrange relaxation algorithm[11, 12], dynamic programming algorithm and mixed integerlinear programming (MILP) algorithm [13]; (3) artificialintelligence algorithms, such as genetic algorithm (GA) [14–16], particle swarm optimization (PSO) [17], ant colonyoptimization (ACO) [18], and simulated annealing (SA) [19].

In the solving process of IRCUC model, the reliabilityof generation schedule is evaluated repeatedly to estimatethe power loss. Therefore, compared with the traditionalUC model, the solving of IRCUC model will become moredifficult. In this paper, a novel heuristic called artificial fishswarm algorithm (AFSA) is introduced into this topic for thefirst time [20]. This algorithm has an ability of optimizing,and it is found efficient and reliable in solving the UCproblem. AFSA has many advantages, including good globalconvergence, strong robustness, insensitivity to initial values,simplicity in implementing, and so forth [21]. The proposedAFSA has been widely used in many different applicationssuch as parameters analysis, neural network classifiers, signalprocessing, network combinatorial optimization, and com-plex function optimization [22–25]. For these reasons, thispaper uses the AFSA to solve the IRCUC model. Differentfrom the standard AFSA operations, an intelligent mutationoperator (MO) is employed to promote the optimizationperformance of the algorithm. To verify the advantages of theimproved AFSA method, the proposed method is tested andcompared to other methods on the systems with the numberof generating units from 10 to 100. The simulation resultsdemonstrate that the improved AFSA is superior to othermethods in terms of total production cost and computationaltime.

The rest of this paper is organized as follows. Section 2sets up themathematical model of IRCUC in detail. Section 3introduces a brief overview of reliability evaluation. Section 4proposes the improved AFSA and applies it to the solvingof IRCUC model. In addition, the model and the solutionalgorithmare examined and discussed through the utilizationof different testing systems in Section 5. Finally, Section 6draws the conclusion.

2. Mathematical Model of IRCUC

Under the satisfaction of the various conditions, the basic taskof UC model is to seek the planning of power generationand the arranging output with minimum cost. The totalproduction cost 𝐶𝑓 over the entire scheduling periods is thesum of the operating cost and start-up cost of the committedunits. Thus, the UC objective function of minimum cost canbe expressed as follows:

𝐶𝑓 =

𝑇

∑

𝑡=1

𝑀

∑

𝑖=1

[𝐹𝑖 (𝑃𝑖,𝑡) 𝑈𝑖,𝑡 + 𝑆𝑖,𝑡 (1 − 𝑈𝑖,𝑡−1) 𝑈𝑖,𝑡] , (1)

where 𝑇 is the dispatch period under consideration; 𝑀 isthe number of generating units; 𝑃𝑖,𝑡 is the power generationof unit 𝑖 at time 𝑡, which is the continuous variable foroptimization; 𝑈𝑖,𝑡 is the operating state of unit 𝑖 at time 𝑡,which is the discrete variable for optimization (1 if the unitis on and 0 if the unit is off); 𝐶𝑓 is the total cost of powergeneration, which is comprised of fuel cost and start-up costand given by part 1 and part 2 of (1), respectively, 𝐹𝑖(𝑃𝑖,𝑡) is thefuel cost of unit 𝑖 at time 𝑡, which is approximately expressedby the following quadratic function [9]:

𝐹𝑖 (𝑃𝑖,𝑡) = 𝑎𝑖 + 𝑏𝑖𝑃𝑖,𝑡 + 𝑐𝑖𝑃2𝑖,𝑡, (2)

where 𝑎𝑖, 𝑏𝑖, and 𝑐𝑖 are the fuel cost coefficients of unit 𝑖,respectively. The start-up cost of unit 𝑖 at time 𝑡 𝑆𝑖,𝑡 in (1) isgiven by the following exponential function:

𝑆𝑖,𝑡 = {𝑆ℎ𝑖, 𝑇𝑖,off ∈ [𝑇𝑖,down, 𝑇𝑖,down + 𝑇𝑖,cold] ,

𝑆𝑐𝑖, 𝑇𝑖,off ∈ [𝑇𝑖,down + 𝑇𝑖,cold, +∞] ,(3)

where 𝑆ℎ𝑖 and 𝑆𝑐𝑖 are the hot and the cold start-up cost of unit𝑖, respectively; 𝑇𝑖,off is the continuous off time of unit 𝑖 up totime 𝑡; 𝑇𝑖,down is the minimum down time of unit 𝑖; 𝑇𝑖,cold isthe cold start-up time of unit 𝑖.

The minimization of the objective function is subject to anumber of system and generating unit constraints.

(1) Generation Limit Constraints. The power produced byeach unit must be within certain limits and that is indicatedby

𝑃𝑖,min ≤ 𝑃𝑖,𝑡 ≤ 𝑃𝑖,max, (4)

where 𝑃𝑖,max and 𝑃𝑖,min are the maximum and minimumpower output of unit i, respectively.

(2) Load Balance Constraints. The total generated power ateach hour must be equal to the load of the correspondingtime. This constraint is given by

𝑃𝑙,𝑡 =

𝑀

∑

𝑖=1

𝑃𝑖,𝑡𝑈𝑖,𝑡, (5)

where 𝑃𝑙,𝑡 is the system demand at time 𝑡.

(3) Unit Minimum Uptime/Downtime Constraints. A unitmust be on for a certain number of hours before it can be

Mathematical Problems in Engineering 3

shut down. A unit must be off for a certain number of hoursbefore it can be brought online. Consider

𝑇𝑖,off ≥ 𝑇𝑖,down,

𝑇𝑖,on ≥ 𝑇𝑖,up,(6)

where 𝑇𝑖,on is continuously on time of unit 𝑖 up to time 𝑡 and𝑇𝑖,up is minimum uptime of unit 𝑖 up to time 𝑡.

(4) Generation Ramping Constraints. The ramping rate con-straints are activated, if a unit remains in operation for twosuccessive hours. In that case,

Δ 𝑖,min ≤ 𝑃𝑖,𝑡 − 𝑃𝑖,𝑡−1 ≤ Δ 𝑖,max, (7)

where Δ 𝑖,min and Δ 𝑖,max represent the lower and upperramping rate limits of the 𝑖th unit, respectively.

(5) Spinning Reserve Constraints. As mentioned before, spin-ning reserve constraints can be taken into consideration byusing either deterministic criteria or probabilistic techniques.These requirements can be specified in terms of excessmegawatt capacity, which is expressed by

𝑃𝑙,𝑡 + 𝑅𝑡 ≤

𝑀

∑

𝑖=1

𝑃𝑖,max𝑈𝑖,𝑡, (8)

where𝑅𝑡 is the system spinning reserve requirement at time 𝑡,which depends on the operating experience by the dispatcher.

In order to ensure the reliability of power system, thedemanded reserves of all time are predetermined by thedispatcher according to the operating experience and alsoare introduced into the UC model as constraint conditions.Actually, the demanded reserves are related to various factors.Therefore, it is not easy to predetermine the spinning reservecapacity of the power system. In general, the increased valueof 𝑅𝑡 would result in the number of start-up units growing,thereby increasing the production cost. On the contrary,the decreased value of 𝑅𝑡 will reduce the reliability of thesystem and increase the outage cost. Therefore, this paperwould remove the spinning reserve in (8) from the UCmodeland propose the IRCUC model which includes the implicitspinning reserve constraint. At the same time, the optimiza-tion model is extended from the minimal production cost oftraditional UC model to the minimal sum of the productioncost and the outage loss, namely,

𝑂𝑓 = 𝐶𝑓 +

𝑇

∑

𝑡=1

𝐸𝑡 (𝑈𝑖,𝑡) × 𝑉oll, (9)

where 𝑂𝑓 is the minimal sum of the production cost and theoutage loss; 𝐸𝑡 is the expected energy not served (EENS) attime 𝑡;𝑉oll is the value of lost load, which can be obtained viathe statistical survey of the users. It is needed to point out thatdue to ignoring the climbing constraints of the generatingunits, the value of 𝐸𝑡 is only associated with the start-upplanning of time and has nothing to do with output powerof each unit. That is to say, the reliability index EENS is

the function of discrete variables 𝑈𝑖,𝑡 and irrelevant to thecontinuous variables 𝑃𝑖,𝑡.

It follows from (9) that the production cost and the outageloss are related to the system spinning reserve requirements.For this reason, the demand and reserve constraints arenot explicitly given in the IRCUC model. Rather, the trade-off between the production cost and the outage loss of theobjective function is implicit.

3. Reliability Evaluation of IRCUC

The foundation of solving the IRCUC model is to evaluatethe reliability of power generation. The reliability index cal-culated only depends on considering the load uncertaintiesand the random failures of units.

3.1. Reliability Model of Units. Two state models are adoptedby the units, which include the normal state and the faultstate. The IRCUC model in this paper is mainly used forshort-term scheduling of the power system (generally lessthan 24 hours). The possibility of repairing or replacing thefaulty units is ignored in such a short time [15]. Therefore,the probability of faulty unit 𝑓𝑖,𝑡 is expressed by the outagereplacement rate (ORR) of unit 𝑖 at time 𝑡. The index variesalong with time and can be expressed by

𝑓𝑖,𝑡 = 1 − exp (𝜆𝑖𝑡) ≈ 𝜆𝑖𝑡, (10)

where 𝜆𝑖 is the failure rate of unit 𝑖.

3.2. Load Fluctuation Model. In the IRCUC model, the load𝑃𝑙,𝑡 is the result of load forecasting at time 𝑡. The statisticsshow that the load forecasting result at a future momentcan be regarded as a random variable and its uncertainty isapproximately reflected by the normal distribution. 𝜇𝑃 and𝜎𝑃 are the mean and the standard deviation of the activeload forecasting, respectively, [26, 27]. Then, the probabilitydensity function of the active load 𝑃𝑙 is computed by

𝑓 (𝑃𝑙) =1

√2𝜋𝜎𝑃

exp[−(𝑃𝑙 − 𝜇𝑃)

2

2𝜎2𝑃

] . (11)

Assuming the load power factor as a constant, then thereactive power load 𝑄𝑙 varies with the 𝑃𝑙 by the power factor.

3.3. Calculation of the Reliability Index. For calculating the𝐸𝑡, the following assumptions are needed.

(1) The random fluctuation of power load and the ran-dom failures of units are assumed to be the indepen-dent random events.

(2) Random failure of each unit is independent of otherunits. Considering that the probability of faulty unit issmall, the multiple failures of units are ignored in thispaper. In other words, there is one unit out of order atmost in every moment.

It is supposed that there are 𝑚 units (𝑚 ≤ 𝑀) in start-up state at time 𝑡. Taking the random failures of units into


consideration, the available generating capacity of this period𝐺𝑡 is a discrete random variable, and the probability densityfunction is expressed as follows:

𝑃 {𝐺𝑡 = 𝐺𝑗} = 𝑝𝑗, 𝑗 = 0, 2, . . . , 𝑚, (12)

where𝐺0 is the available generating capacity of𝑚 units underthe normal condition and 𝑝0 is the probability correspondingto the event, which can be calculated by

𝐺0 =

𝑚

∑

𝑗=1

𝑃𝑗,max,

𝑝0 =

𝑚

∏

𝑗=1

(1 − 𝑓𝑗,𝑡) ,

(13)

where𝐺𝑗 (𝑗 = 1, 2, . . . , 𝑚) is the available generating capacitythat the unit 𝑗 of failure in the start-up state of𝑚 units, whilethe other (𝑚 − 1) units are in normal state. The value of 𝑝𝑗is the probability of the event 𝑗 occurrence, which can bedescribed by

𝐺𝑗 = 𝐺0 − 𝑃𝑗,max, 𝑗 = 1, 2, . . . , 𝑚,

𝑝𝑗 = 𝑝0

𝑓𝑗,𝑡

1 − 𝑓𝑗,𝑡

, 𝑗 = 1, 2, . . . , 𝑚.

(14)

Once the actual load of power system is greater than theavailable generating capacity, it would lead to part poweroutage. Therefore, the value of reliability index EENS at time𝑡𝐸𝑡 can be defined as

𝐸𝑡 =

𝑚

∑

𝑗=0

𝑝𝑗 ∫

+∞

𝐺𝑗

𝑓 (𝑥) (𝑥 − 𝐺𝑗) 𝑑𝑥. (15)

4. Solution to IRCUC Model Based onan Improved AFSA

According to the above analysis, the IRCUC problem belongsto a class of NP-hard problems, as much, is very difficultto solve. Artificial fish swarm algorithm (AFSA), a novelintelligent algorithm, was first proposed in 2002 [20]. It hasthe potential to be one of the excellent techniques to obtainoptimal or near-optimal solutions to realistic size IRCUCproblems.

4.1. Artificial Fish Swarm Algorithm. In a water area, fishare most likely distributed around the region where foodsare the most abundant. A fish swarm completes its foodforaging process by each functioning several simple socialbehaviors. It is found that there are three most common fishbehaviors: (1) searching behavior, that is, fish tend to headtowards food; (2) swarming behavior, gregarious fish tend toconcentrate towards each otherwhile avoiding overcrowding;(3) following behavior, the behavior of chasing the nearestbuddy. Inspired by swarm intelligence, AFSA is an artificialintelligent algorithm based on the simulation of collectivebehavior of real fish swarms. It simulates the behavior of

a single artificial fish (AF) and then constructs a swarm ofAF. Each AF will search its own local optimum, pass oninformation in its self-organized system, and finally achievethe global optimum.

Suppose the searching space is D-dimensional and thereare𝑁 fish in the colony.The current state of an AF is a vector𝑋 = (𝑥1, 𝑥2, . . . , 𝑥𝑛), where 𝑥𝑖 (𝑖 = 1, 2, . . . , 𝑛) is the variableto be optimized. The food consistence of AF in the currentposition is represented by 𝑌 = 𝑓(𝑋), where 𝑌 is the objectivefunction. The distance 𝐷𝑖𝑗 between 𝑋𝑖 and 𝑋𝑗 of individualAF can be expressed as ‖𝑋𝑗 − 𝑋𝑖‖. 𝛿 is defined as the factorof crowded degree, which represents the crowded degree ofa location nearby and avoids more AF to gather together.Visual is the perception scope of an AF, which determinesthe moving direction of each AF. When the perception rangeof Visual becomes larger, the observation of an AF can bemore comprehensive. However, the number of fish shouldgrow more so as to reach the calculation amounts. Under theactual situation, the appropriate value should be selected bythe specific case. Step is the largest moving step of an AF.For fear of missing the optimum solution, the length of Stepshould not be set too large. Of course, the length of Step is toosmall to converge. Try number is represented as the numberof the biggest trial in the searching behavior.

In the initial state of the algorithm, the variable of trialnumber should be defined as the trial times of AF searchingfor food. Then, the following steps describe the fish swarmbehaviors.

(1) Searching Behavior. The fish follows the direction of foodwith high concentration when the fish is in the searchingbehavior. Suppose the current state of an AF is 𝑋𝑖 and itsfitness is 𝑌𝑖. A new state is randomly selected in its Visualfield. If, in the maximum problem, 𝑌𝑖 < 𝑌𝑗 (as the maximumproblem and minimum problem can be converted with eachother, the maximum problem is discussed as an examplein the following analysis), move a step in that direction;otherwise, select a state randomly again and judge whetherit satisfies the a forementioned condition. If it cannot besatisfied after predetermined times of Try number, it movesa step randomly.The searching behavior is shown in Figure 1.

(2) Swarming Behavior. Swarming behavior is a behavior inwhich a fish swims towards to the central position of swarmand avoids overcrowding. An AF at current state𝑋𝑖 seeks thecompanion’s number 𝑛𝑓 and its central position in its currentneighborhood (𝐷𝑖𝑗 <Visual); if 𝑌𝑐/𝑛𝑓 > 𝛿𝑌𝑖, this means that,at the center of the fish colony, there is enough food and itis not too crowded. The swarming behavior is illustrated inFigure 2.

(3) Following Behavior. Following behavior is the processwhere the fish captures the most active individual of swarmnearby. Suppose𝑋𝑖 is the current state of AF searching com-panion 𝑋max in the neighborhood with 𝑌max; if 𝑌max/𝑛𝑓 >

𝛿𝑌𝑖, this means that the current position of companion𝑋max has a higher food consistence and it is not crowdedenough. The AF will move a step towards the companion


Xi | n = 0

Yes Yes

No

No

Yi < Yj

n = n + 1

Xj = Xi + rand() · Visual

n ≥ Try number

= Xi + rand() · Step= Xi

Xi+ rand() · Step ·

Xj −

‖Xj − Xi‖Xi|next Xi|next

Figure 1: Searching behavior of AFSA.

𝑋max; otherwise, it will continue the searching behavior. Thefollowing behavior can be seen in Figure 3.

(4) RandomBehavior.The realization of the random behavioris relatively simple. It means selecting a state randomly in theVisual field and swimming towards that direction. In fact, thatis a default behavior of the searching behavior; namely,𝑋𝑖|nextis the next position of𝑋𝑖:

𝑋𝑖|next = 𝑋𝑖 + 𝑟 ⋅ Visual, (16)

where 𝑟 is the random number in [−1, 1].

(5) Bulletin. Bulletin is used to record the AF’s optimal stateand the optimal value of the problem. Each AF updates itsown state and compares it with the bulletin after makingmovements. If its current state of AF is better, then the valueon the bulletin will be replaced. At the end, the value on thebulletin is the optimal solution of the problem.

In the process of AFSA, searching behavior lays thefoundation for the AF, swarming behavior enhances theconvergence of stability, following behavior ensures the con-vergence of quickness, and behavior selection guarantees thehigh efficiency and stability of the algorithm. Through thebehavior selection, the AFSA forms an optimization strategywith high efficiency.

4.2. AFSA with Added Mutation Operator. The AFSA hasthe ability to grasp the searching direction and avoid fallinginto the local optimum. But when some fish move in aimlessrandom or gather around the local optima, the speed ofconvergence will be slowed down greatly and the searchingaccuracy is greatly reduced. To avoid premature convergence,an intelligent mutation operator (MO) similarly to geneticalgorithm is introduced to enhance the ability escaping fromthe local optima in this paper. If the state of AF is notimproved during the iterations and the AF has entered intoa state of partial mining, it needs to be mutated. The specificsteps are as follows.

(1) Randomly select one of the variables in the positionto plus one, and choose the nonnull to minus one.

Xi

and it’s central position Xc

Seek the companion’s number nf

𝛿Yi Xi in searching behavior

Yes

NoYc

nf

>

= Xi + rand() · Step ·Xc − Xi

‖Xc − Xi‖Xi|next

Figure 2: Swarming behavior of AFSA.

Xi

Xi in searching behavior

Yes

NoYmax

nf

> 𝛿Yi

Seek the companion’s number nf and

companion Xmax in the neighborhood with Ymax

= Xi + rand() · Step ·Xmax − Xi

‖Xmax − Xi‖Xi|next

Figure 3: Following behavior of AFSA.

(2) If the value of state is better than that of the currentstate, update the state of position; otherwise, go backstep (1) until the initial number of the mutatingoperation is satisfied.

By adding the mutation mechanism into the AFSA, itachieves the aim of altering the AF. Through adjusting theswarms, the rate of convergence and the global searchingability of AFSA are both improved.The selection of mutatingprobability will have a great influence on the performance ofthe proposed algorithm, which has a positive correlation tothe elapsed time. According to the experimental experience,the probability of mutation operator (PMO) selected as1/(30D)∼1/(10D) (D is the dimension) can obtain a goodperformance. Usually, the PMO of an AF is assumed as 0.03∼0.1.

4.3. IRCUC Model with Improved AFSA. In this paper, thebinary coded matrix is used to denote the AF swarm. Eachindividual of AF swarm corresponds to a start-up planningof IRCUC model.

In this matrix, the element 0 refers to the outage stateof the units; on the contrary, the element 1 corresponds tothe power-on state. The AF in swarm is divided into two


categories: the valid AF and the invalid AF. If the availablegenerating capacity of AF is greater than the load forecastingat any time of the start-up planning, the AF is valid, or isotherwise invalid.

The food concentration of AF in current state is one ofthe important factors of AFSA, and its value determines thebehavior selection in the optimization process. In the solvingprocess of IRCUCmodel, economic dispatch (ED) should becomputed for calculating the food concentration.The essenceof ED is following the principle of the lowest fuel cost andassigns the power load to the state variables marked as 1of units in the start-up planning. Considering that the EDcalculation is time consuming, this algorithm only takes EDcalculation by the valid AF. The invalid AF in the swarm isdirectly given poor food concentration and made to be siftedout gradually in the evolution process. After ED calculation,the total fuel cost of power generation is counted by the fuelcost function of each unit, which pluses the start-up costof units to obtain the corresponding power generation costof AF. It is needed to point out that the AF in accordancewith the start-up planning may violate the constraints of theunit minimum uptime and downtime as mentioned in (6).Therefore, the penalty function algorithm is used to dealwith the inequality constraints and to acquire the positionconcentration 𝑉fit ultimately; namely,

𝑉fit = 𝑂𝑓 (𝑃𝑖,𝑡, 𝑈𝑖,𝑡) + 𝜂𝑝 × 𝛽. (17)

In (17), the first term is objective function of IRCUCmodel, 𝜂𝑝 is the predetermined large penalty coefficient, and𝛽 is the number of times of violating the constraints of theunit minimum uptime and downtime.

When using the PL algorithm to solve the UC problem,the start-up order of each unit is determined by the averagefull-load cost of units. According to the start-up order, theMO operation should be adopted in this paper. The detaileddescriptions are as follows.

Step 1. Compute the percentage of the value of reliabilityindex 𝐸𝑡 in load forecasting 𝛿𝑡 at time 𝑡. If the value 𝛿𝑡 isgreater than the predetermined upper threshold, this meansthe number of combination units is too small (that is, under-saturation); otherwise, if the value 𝛿𝑡 is less than the presetlower threshold, this means the number of combination unitsis too big (that is, oversaturation). Even though the outageloss is smaller at this time, it corresponds to the uneconomicalstart-up scheme.

Step 2. If it is undersaturation at that time, the outage unitsshould take MO operation (start-up) according to the orderof start-up, until the state of that time is not under-saturation.

Step 3. If it is over-saturation at that time, the operationalunits should take MO operation (closedown) according tothe order of start-up, until the state of that time is not over-saturation. It is noticeable that the MO operation should betried to avoid violating the constraints of the unit minimumuptime and downtime in (6) as far as possible.

Start

Initialize the state offish swarm

Evaluate all individuals and assign thebest individual to the bulletin board

Update itself by searching, swarming, andfollowing behavior and generate a new

fish swarm

Evaluate all individuals; if oneindividual is superior to the bulletin board,update the bulletin board for the individual

Add the mutationoperator to AFSA

Evaluate reliability assessmentand economic dispatch

Refresh the bulletin board

No

Yes

Does it mee the terminationconditions of the AFSA?

End

No

Yes

Figure 4: Flow chart of solving the IRCUC model with improvedAFSA.

Note that the MO operation is executed in accor-dance with a certain probability which was mentioned inSection 4.2. In conclusion, the flow chart of solving theIRCUC model is shown in Figure 4.

5. Numerical Analysis

The efficiency of the proposed method is verified by solvingthe IRCUC model. The improved AFSA method is initiallytested by systems with different sizes based on a basic systemof 10 units from the literature [17–19]. The scheduling timehorizon 𝑇 is chosen as one day with 24 intervals of one houreach.

5.1. Solution Quality and Convergence Characteristics. Forassessing the reliability of the generation schedule, the relia-bility parameters of units are given in Table 1 and the standarddeviation of random fluctuation of the load is accounted for3% of the load forecasting. For estimating the outage cost


Table 1: Reliability parameters of units.

Number of unit Failure rate (times/hour)1-2 0.000333–5 0.000506–7 0.001518–10 0.00075

Table 2: Parameters of the proposed algorithm.

Parameter ValueSizepop 500Step 0.3Visual 1.8PMO 0.03∼0.1Try number 10MAXGEN 100Penalty coefficient 𝜂𝑃 106

Upper limit of 𝛿𝑡 0.5%Lower limit of 𝛿𝑡 0.1%𝛿 0.618

of the power system, the value of lost load is supposed as1,000$/MWh; namely, 𝑉oll = 1,000$/MWh.

This paper uses the improved AFSA to schedule theplanning of power generation and the output arrangement.The parameters of the proposed algorithm are shown inTable 2.

The solid line in Figure 5 shows the fitness function of theoptimal location in the process of evolution. It can be seenfromFigure 5 that this algorithmhas a good convergence.Thefinal result converges to the 37th generation and the corre-sponding total cost is $554,105. In addition, the optimizationresult of AFSA without MO is also given in the dotted line ofFigure 5. The results show that the proposed MO can largelyimprove the optimization ability of the algorithm.

For comparison, this paper also uses the traditional UCmodel to optimize the planning of power generation and thescheduling of output power. In the process of optimization,the spinning reserve is assumed to be 5% and 10% of the loaddemand in each system. The results are illustrated in Figures6 and 7 and Table 3.

The assessments of reliability under the optimal scheduleof power generation are provided with each of the schedulingtimes in Figure 6. The reliability indexes of different casesare described in round, plus sign, and asterisk, respectively.Figure 7 points out the production cost and outage losscorresponding to the various optimization results. Table 3shows the optimal scheduling of power generation underdifferent situations. When the traditional UC model is usedto optimize the generation schedule and to arrange the outputpower, the power generation schedule in Table 3 shows thatthe production cost and the outage loss are related to thepredetermined spinning reserve requirements. In response tothe increasing of spinning reserve, there will be more gen-erating units in start-up state when the spinning reserves ofsystem are gained from the 5%percentage of the load demand

0 10 20 30 40 50 60 70 80 90 1005.5

5.6

5.7

5.8

5.9

6

6.1

6.2

6.3

6.4

6.5

Iteration number

Tota

l cos

t ($)

AFSA without MOAFSA with MO

×105

Figure 5: Total cost in the process of iterations.

2 4 6 8 10 12 14 16 18 20 22 240

1

2

3

4

5

6

7

8

Time (hour)

EEN

S (M

Wh)

5% 10% 0%

Figure 6: Reliability index of optimal generation schedule.

to 10% percentage. Therefore, the production cost increasesfrom $509,165 to $516,517; on the contrary, the outage lossdecreases from $65,425 to $51,731. Unlike the traditional UCmodel, the IRCUC model proposed in this paper containsan implicit reserve constraint. That is to say, the systemspinning reserve would not be predetermined. The demandand reserve constraints imply in a tradeoff between theproduction cost and the outage loss. The calculation resultsshow that more generating units of the power system wouldbe scheduled to achieve the balance of the reliability andeconomy. Accordingly, the production cost further increasesfrom $516,517 to $526,921, while the outage loss substantiallyfurther reduces from $51,731 to $27,184. Taking the effect of


Table 3: Optimal scheduling of power generation (5%, 10%, and 0% of the load demand, resp.).

Unit no. Hour1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 04 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 05 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 06 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 07 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 08 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 09 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 04 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 05 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 06 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 07 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 18 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 09 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 05 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 08 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 09 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0

the reliability into consideration, the better scheduling ofpower generation can be obtained by the IRCUC model.

Moreover, it should be emphasized that the IRCUCmodelis not needed to specify the capacity of spinning reserve inadvance. Therefore, the IRCUC model is more practical andflexible than the normal UC model.

5.2. Effect of Optimization Results Caused by the Value ofLost Load. In the model of IRCUC, the value of 𝑉oll has adirect impact on the outage loss and further affects the finalscheduling of power generation. To testify the influence of𝑉ollon the optimization results, the value of𝑉oll in 10 units systemis assumed to be from 1,000$/MWh down to 500$/MWh.At this time, Figure 8 and Table 4 show the productioncost, and the EENS cost and the optimal scheduling ofpower generation which are optimized by the IRCUCmodel,respectively.

It can be seen from Figure 8 and Table 4 that the valueof 𝑉oll has a significant effect on the optimization results.The decreasing of 𝑉oll signifies that the percentage of outage

loss accounting for objective function is diminished. And themain goal of the IRCUC model in optimization schedulingof power generation focuses on reducing the productioncost. Consequently, in order to meet the balance betweenthe reliability and economy, the number of committed unitsin generation schedule is significantly dropped in Table 4.The production cost falls from $526,921 to $516,467 and thefalling range is $10,454. However, the EENS correspond-ing to the generation schedule increases from 27.18MWhto 40.74MWh, the outage loss increases from $13,592 to$20,370, and the additional amount is only $6,778.

5.3. Influence of Unit Number on Solution Quality and Com-putational Time. To verify the feasibility and effectiveness ofthe proposed method for solving large-scale IRCUC model,the simulations are tested on systems with 20, 40, 60, 80, and100 units, respectively. For the systemswith 20, 40, 60, 80, and100 units, the basic 10 units system is duplicated and the totalproduction costs are adjusted proportionally to the systemsizes. To avoid any hazardous interpretation of optimization


Table 4: Scheduling of power generation with different 𝑉oll (1,000$/MWh and 500$/MWh, resp.).

Unit no. Hour1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 04 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 07 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 08 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 09 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 05 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 08 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 09 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0

Table 5: Comparison of the total production costs ($) for up to 100 units using IRCUC model with other methods (𝑉oll = 1,000$/MWh).

Units GA PSO ACO IAFSABest Worst Best Worst Best Worst Best Worst

10 554,943 556,121 555,404 556,231 556,686 558,006 554,105 554,57920 1,110,068 1,125,790 1,119,244 1,133,793 1,112,292 1,117,054 1,108,979 1,110,64340 2,219,103 2,229,024 2,219,523 2,240,085 2,220,116 2,239,146 2,218,163 2,223,11760 3,340,066 3,352,886 3,342,108 3,359,012 3,340,407 3,353,921 3,338,979 3,343,12580 4,460,202 4,472,478 4,465,349 4,480,939 4,465,177 4,477,753 4,457,032 4,461,943100 5,583,977 5,658,925 5,617,388 5,680,841 5,590,607 5,651,827 5,581,284 5,608,291

results, related to the choice of particular initial states, 10 trialsfor each units system are performed.The best and worst totalproduction costs of the improvedAFSA (IAFSA) are obtainedwith different 𝑉oll in Tables 5 and 6, respectively.

As shown in Tables 5 and 6, the average production costsof 10 trials using the improved AFSA generated variationin a small range and the standard deviations are small andtolerable. It is demonstrated that the improved AFSAmethodhas a better quality of solution and robustness for the IRCUCmodel. To validate the results obtained with the proposedmethod, we compare the performance of the improvedAFSA to those of other approaches with respect to the totalproduction costs. Tables 5 and 6 provide the comparison ofthe total production costs from the improved AFSA methodto those of other methods with different value of 𝑉oll. It isclearly shown that the total production costs by the improvedAFSA for large-scale systems constantly outperform thoseobtained by GA, PSO, and ACO. Meanwhile, it is obviousthat the value of 𝑉oll still plays an important role in the

optimization of the power generation. Considering moreinformation of AF to control the mutation operation, theproposed method can find the best solution. Therefore, theimproved AFSA has the potential applications to the realsystem.

In addition, the computational times of the improvedAFSA and other methods are another important evaluationindex. The computational times of the above methods tofind the optimal solutions with various numbers of units tobe committed are shown in Table 7. Analysis of the resultspresented in Table 7 shows that the computational time ofthe improved AFSA is in direct proportion to the numberof units and the system sizes. From Table 7, it is clear thatthe computational efficiency of the improved AFSA has beenimproved significantly compared with that of other methods.All these situations have shown that the improved AFSAmethod provides excellent performances: fast computationspeed, stable convergence, and cost saving system. Moreover,the computational times of the improved AFSA are not


Table 6: Comparison of the total production costs ($) for up to 100 units using IRCUC model with other methods (𝑉oll = 500$/MWh).

Units GA PSO ACO IAFSABest Worst Best Worst Best Worst Best Worst

10 537,582 540,312 537,515 549,530 538,797 546,660 536,837 537,27920 1,076,156 1,085,332 1,088,494 1,102,101 1,078,919 1,090,458 1,075,279 1,079,03440 2,152,751 2,168,248 2,159,903 2,180,850 2,160,012 2,174,762 2,148,968 2,155,00860 3,238,565 3,259,686 3,240,161 3,278,542 3,245,471 3,269,129 3,232,139 3,240,65480 4,388,004 4,426,215 4,395,974 4,438,793 4,395,848 4,434,752 4,381,521 4,388,022100 5,428,795 5,479,510 5,439,588 5,492,534 5,430,955 5,485,750 5,407,844 5,429,125

5 10 0 0

1

2

3

4

5

6

Spinning reserve (%)

Production costOutage loss

×105

Prod

uctio

n co

st an

d ou

tage

loss

($) Total cost

= $574,590Total cost= $568,248

Total cost= $554,105

Figure 7: Production cost and outage loss.

0

1

2

3

4

5

6

Voll = Voll =

Different Voll

27.18 MWh 40.74 MWh

1000$/MWh 500$/MWh

×105

Prod

uctio

n co

st an

d EE

NS

cost

($)

Production costEENS cost

Figure 8: Production cost and EENS cost with different 𝑉oll.

Table 7: Comparison of the computational times (s) for up to 100units using IRCUC model with other methods.

Units Computational timesGA PSO ACO IAFSA

10 90 55 25 1920 255 158 60 4540 728 440 180 9560 995 885 355 18280 1,780 1,624 595 242100 3,310 2,825 860 448

increasing exponentially with respect to the system size ofIRCUC model, which is favorable for the large-scale system.By the way, it is noticed that the increasing rate of thecomputational times of the improved AFSA is polynomial tothe problem size.

It follows fromTable 5 to Table 7 that the total productioncosts of the improved AFSA are less expensive than thoseobtained by using other methods on all of the generatingsystems. Obviously, the improved AFSA improves perfor-mance vastly more than other methods in terms of bothsolution quality and computational time especially on thelarge-scale implementation.The proposedmethod convergesto the solution at a faster rate than the above-mentionedmethods.

6. Conclusions

To overcome the deficiency of UC model, this paper triedto remove the constraint of spinning reserve from thetraditional UC model and the IRCUC model was proposed.The objective function was extended from the minimalproduction cost of traditional UCmodel to the minimal sumof the production cost and the outage loss. The demand andreserve constraints were not explicitly given in the IRCUCmodel. Rather, the tradeoff between the production cost andthe outage loss of the objective function was implicit.

The IRCUC model was solved by using the AFSA.In addition to the standard operation of AFSA, the MOoperation was designed with the idea of PL algorithm anddramatically improved the optimizing ability of AFSA.

The efficiency of the proposed method was demonstratedby using the testing systems with the number of gener-ating units from 10 to 100. The numerical results showed


the improvements in effectiveness and computational timecompared to the results obtained from other methods. Fur-thermore, the results proved that the method is capable ofsolving realistic UC problems.

Acknowledgments

Thiswork is supported by theGraduate Education InnovationProject in Jiangsu Province (no. CXZZ12 0228). The authorswould like to thank the editor and anonymous reviewers fortheir suggestions in improving the quality of the paper.

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