Ant Colony Search Algorithm For Solving Multi Area Unit Commitment Problem With Import And Export Constraints

Ant Colony Search Algorithm For Solving Multi Area Unit Commitment Problem With Import And Export Constraints

K. Venkatesan, C. Christober Asir Rajan

Research Scholar, Dept. Of Elect. Engg., J.N.T.U. Anantapur, Anantapur, 515 002, A.P., India Associate Professor,Department of EEE, Pondicherry Engg. College, Puducherry, 605 014, India.

Abstract Nomenclature

This paper presents a new approach to solve the multi area unit commitment problem (MAUCP) using a Ant Colony Search Algorithm (ACSA). The objective of this paper is to determine the optimal or a near optimal commitment schedule for generating units located in multiple areas that are interconnected via tie lines. The Ant Colony Search Algorithm is used to solve multi area unit commitment problem, allocated generation for each area and find the operating cost of generation for each hour. Joint operation of generation resources can result in significant operational cost savings. Power transfer between the areas through the tie lines depends upon the operating cost of generation at each hour and tie line transfer limits. The tie line transfer limits were considered as a set of constraints during optimization process to ensure the system security and reliability. The overall algorithm can be implemented on an IBM PC, which can process a fairly large system in a reasonable period of time. Case study of four areas with different load pattern, each containing 26 units connected via tie lines has been taken for analysis. Numerical results showed comparing the operating cost using Ant Colony Search method with conventional evolutionary programming (EP) and dynamic programming (DP) method. Experimental results shows that the application of this Ant Colony Search method have the potential to solve multi area unit commitment problem with lesser computation time.

Keywords

Ant Colony Search Algorithm (ACSA), Dynamic Programming (DP), Evolutionary Programming (EP), Multi Area Unit Commitment Problem (MAUCP).

F (Pk )

g

i

P

k gi

ak ,bk , ck

i i i

X

off i, j

P

k i, j

I

k

i, j

Pg

k i, j

D

k j

R

k j

E

k j

Pj

k

max

Pj

k

min

T

on

T

i off

i

Lj

k

max

Wj

sys

Pg

k

i max

Pg

k

i min

Production cost of unit i in area K

Power generation of unit i in area K

Cost function parameters of unit i in area K

Time duration for which unit i have been off at jth hour.

Power generation of unit i in area K at jth hour

Commitment state (1 for ON, 0 for OFF)

Power generation of unit i in area K at jth hour

Total system demand of area K at jth

hour

Spinning reserve of area K at jth hour

Total export power to area K at jth

hour

Maximum power generation in area K at jth hour

Minimum power generation in area K at jth hour

Minimum up time of unit i Minimum down time of unit i

Maximum total import power in area K at jth hour

Net power exchange with outside system

Marginal cost of supplying the last incremental energy to meet entire system demand.

Maximum power generation at area K at ith hour

Minimum power generation at area k at ith hour

1. Introduction

   In multi area, several generation areas are interconnected by tie lines, the objective is to achieve the most economic generation to meet out the local demand without violating tie-line capacity limits constraints [1]. The main goal of this paper is to develop a multi area generation scheduling scheme that can provide proper unit commitment in each area and effectively preserve the tie line constraints.

   In an interconnected multi area system, joint operation of generation resources can result in significant operational cost savings [2]. It is possible by transmitting power from a utility, which had cheaper sources of generation to another utility having costlier generation sources. The total reduction in system cost shared by the participating utilities [3].

   The exchange of energy between two utilities having significant difference in their marginal operating costs. The utility with the higher operating cost receives power from the utility with low operating cost. This arrangement usually on an hour to hour basis and is conducted by the two system operators.

   In the competitive environment, customer request for high service reliability and lower electricity prices. Thus, it is an important to maximize own profit with high reliability and minimize overall operating cost [4]. Multi Area unit commitment was studied by dynamic programming and was optimised with local

   this work, the parents are obtained from a predefined set of solution (i.e., each and every solution is adjusted to meet the requirements). In addition, the selection process is done using evolutionary strategy [8]-[10].

   For the last few years, the algorithms inspired by the observation of natural phenomena to help solving complex combinatorial problems have been increasing interest. In this study, a new Ant Colony Search Algorithm (ACSA), which was derived by the observation of the behaviors of ant colonies, is proposed [11]. In analyzing the behaviors of real ants, it was found that the ants are capable of finding shortest path from food sources to the nest without using visual cues. In the application of this method to UC problem, the initial population of colony can be first randomly generated within the search space of problem. Then, the fitness of ants is individually assessed based on their corresponding objective function. With the selection of trial, the ant dispatch can be activated based on the level of pheromone and the distance of selected trial in order to find the best tour or the shortest path.

2. Problem Formulation

   The cost curve of each thermal unit is in quadratic form [1]

   demands with a simple priority list scheme on a personal computer with a reasonable execution time [5]. Even though the simplicity and execution speed are well suited for the iterative process, the commitment schedule may be far from the optimal, especially when

   F(Pgik ) ak (Pg k )2 bk (Pg k ) ck

   i i i i i

   k = 1 NA

   Rs/hr (1)

   massive unit on/off transitions are encountered. The tie- line constraint checking also ignores the network topology, resulting in failure to provide a feasible generation schedule solution [5]. The transportation model could not be used effectively in tie line constraints, as the quadratic fuel cost function and exponential form of start up cost were used in this study.

   An Evolutionary algorithm is used for obtaining the

   The incremental production cost is therefore

   i i i

   2ak Pg k bk

   (or)

   i i

   Pgik bk / 2ak

   (2)

   (3)

   initial solution which is fast and reliable [6]. Evolutionary Programming (EP) is capable of

   The startup cost of each thermal unit is an exponential function of the time that the unit has been off

   o ff

   determining the global or near global solution [7]. It is based on the basic genetic operation of human chromosomes. It operates with the stochastic

   S ( X off )

   A B (1 exi , j i )

   (4)

   i, j

   i

   i

   mechanics, which combine offspring creation based on the performance of current trial solutions and competition and selection base on the successive generations, from a considerably robust scheme for large scale real valued combinational optimization. In

   The objective function for the multi-area unit commitment is to minimize the entire power pool generation cost as follows [1].

   k k k

   X on

   T on I

   I 0

   N A t

   Nk Ii, j Fj (Pi, j

   i, j1 i

   i, j 1

   i, j

   (12)

   min

   I,P

   k 1 j1i1I

   (1I

   )S (X off

   ))

   X off

   T off I I

   0

   i, j

   i, j1 i

   i, j1

   (5)

   i, j 1 i

   i, j

   i, j1

   (13)

   To decompose the problem in above equation (5), it is rewritten as

   1. In each area, power generation limits caused by tie line constraints are as follows

      Upper limits

      t

      min

      F (P

      )

      Pk

      Dk E k

      (14)

      j 1

      gi , j

      (6)

      gi , j

      i

      j jmax

      N A

      k k

      g

      Lower limits

      F Pg

      i , j

      F

      k 1

      P

      i , j

      (7)

      P

      j

      k gi , j

      Dk

      j

      • Lk

        max

        (15)

        F P

        Subject to the constraints of equations (9), (11) and i

        (1418). Each

        k k

        gi , j

        for K=1 NA is

        Import/Export balance

        represented in the form of schedule table, which is the solution of mixed variable optimization problem

        Ek Lk W

        0 (16)

        min I k F k Pk I 1 I S X off (8)

        j j j

        i k

        I ,P i

        i, j i

        i, j

        i, j

        i, j1 i

        i, j

   2. Area generation limits

      Subject to following constraints are met for optimization.

      P

      1. System power balance constraint

         k gi , j

         P

         i

         k gimax

         P

         i

         Rk ; k=1. N

         j A

         j=1.t (17)

         j

         g j

         k Dk

         k k

         (9)

         k gi , j

         P

         i

         k

         P

         gi min

         i

         ; k=1. NA

         Sum of real power generated by each thermal unit must be sufficient enough to meet the sum of total demand of each area while neglecting transmission losses.

         j=1.t (18) The objective is to select sys at every hour to minimize the operation cost.

      2. Spinning reserve constraint in each area

         k Dk

         P

         j

         g

         j

         • E k

         • Lk

           (19)

           Pk

           Dk Rk E k Lk

           (10)

           j

           j

           gi , jmax

           i

           1. j j j

              where

              Nk

              P

              P

           2. k

           g j gi , j

           (20)

      3. Generation limits of each unit

         i1

         Since the local demand D k is determined in

         P

         P

         k jmax

         k i, j

         Pk

         j

         min

         (11)

         j

         accordance with the economic dispatch within the pool,

         i=1..Nk, j=1.t, k=1NA

         changes of P k

         g

         j

         will cause the spinning reserve

      4. Thermal units generally have minimum up time Ton and down time Toff constraints, therefore

         constraints of equations (10) to change accordingly and redefine equation (8). Units may operate in one of the following modes when commitment schedule and unit generation limits are encountered [16].

         1. Coordinate mode : The output of unit i is determined by the system incremental cost

            An area may reach its lower generation limit according to equation (15) or (18) in this case because

            min,i sys max,i

            (21)

            of higher generation cost

         2. Minimum mode : Unit i generation is at its minimum level

            k

            min

            sys

            (27)

            min,i sys

            (22)

3. Tie Line Constraints

To illustrate the tie-line flow in a multi-area system, the four area system given in Fig.1 is studied.

1. Maximum mode : unit i generation is at its

   maximum level

   An economically efficient area may generate more power than the local demand, and the excessive power

   max,i sys

   (23)

   will be exported to other areas through the tie-lines [1]. For example assume area 1 has the excessive power the tie line flows would have directions from area1 to other

2. Shut down mode : unit i is not in operation,

Pi = 0

Besides limitations on individual unit generations, in a multi- area system, the tie-line constraints in equations (12), (13) and (15) are to be preserved. The operation of each area could be generalized into one of the modes as follows.

1. Area coordinate mode

   areas, and the maximum power generation for area1 would be the local demand in area1 plus the sum of all the tie-line capacities connected to area1.

   If we fix the area 1 generation to its maximum level than the maximum power generation in area 2 could be calculated in a similar way to area 1. Since tie line C12 imports power at its maximum capacity, this amount should be subtracted from the generation limit of area

   1. According to power balance equation (9) some areas must have a power generation deficiency and requires generation imports. The minimum generation limits in these areas is the local demand minus all the connected

      k

      sys

      tie-line capacities. If any of these tie-lines is connected to an area with higher deficiencies, then the power flow directions should be reserved.

      Dk Lk

      Pk

      Dk E k

      (24)

      j max

      gi , j

      i

      (or)

      j max

      1

      100 300

      • L

      k

      max

      k gi , j

      P

      i

      • Dk

      k

      E

      max

      (25)

      4

      300

      150

      2

      100

      j

2. Limited export mode

   When the generating cost in one area is lower than the cost in the remaining areas of the system, that area may generate its upper limits according to equations

   (14) or (17) therefore

   3

   Figure 1. Multi-area connection and tie-line limitations

   k

   sys

   (26)

   1. ACSA Paradigm

      For area k, area k is the optimal equal incremental cost which satisfies the generation requirement.

3. Limited import mode

4.1. Behavior of Ants

Ant colony search (ACS) studies are inspired from the behavior of ants colonies that are used to solve

function or combinatorial optimization problems. Currently, most work has been done in the direction of applying ACS to combinatorial optimization. The first ACS system was introduced by Marco Dorigo [12] and was called ant system. Ant colony search algorithm, to some extent; mimic the behavior of real ants. As is well known, real ants are capable of finding the shortest path from food sources to the nest without using visual cues. They are also capable of adapting to changes in the environment. Pheromone trails, which ants use to communicate information among individuals regarding path and to decide where to go. Ants deposit a certain amount of pheromone while walking, and each ant probabilistically prefers to follow a direction rich in pheromone rather than a poorer one. The behavior of ant is given in fig. 2.

(a)

(b)

(d)

Figure 2: Behavior of ants : (a) Real ants follows a path between nest and food source. (b) An obstacle appears on the path: ants choose whether to turn left or right with equal probability. (c) Pheromone is deposited more quickly on the shorter path. (d) All ants have chosen the shorter path.

1. Ant colony search Algorithm

   1. ACS State Transition Rule

      In ACS the state transition rule is as follows: An ant positioned on node choose the city S to move by applying the given rule, if q qo

      Where

      q is a random number uniformly distributed in (0..1) q0 is a parameter (0qo1)

      S is random variable selected according to the probability distribution given in (28)

      The state transition rule used by ant system, called a random proportional rule, is given by Eq(8), which gives the probability with which ant k in city r chooses to move to the city s.

      r, s.r, s

      r,u.r,u

      ifsJ ( r) (28)

      pk r, s

      uJk r

      r k

      0 otherwise

      (c)

      Where

      is the pheromone

      Jk(r) is the set of cities that remain to be visited by ant k positioned on city r (to make the solution feasible)

      is a parameter, which determines the relative importance of pheromone versus distance (>o)

      1/ is the inverse of the distance (r,s)

   2. ACS Global Updating Rule

      Global updating is performed after all ants have completed their tours. The pheromone level is updated by applying the global updating rule of equation (29).

      (r,s) (1-)(r,s)+ (r,s) (29)

      Step 2 : Assessment of Fitness

      In this step, the fitness of all ants is assessed based on the corresponding objective function, which can be expressed as following:

      where

      L

      1

      f () tcs

      i

      , s i1

      (31)

      (r, s) =

      gb

      0 otherwise

      if(r,s) global-best-tour

      Where

      tc(si,sj) is the transition cost between state si and sj

      is the pheromone decay parameter

      Lgb is the length of the globally best tour from the beginning of trial

   3. ACS Local Updating Rule

      While building a solution of the MAUCP, ants visit edges and change their pheromone level by applying the local updating rule of equation (30)

      (r,s) (1-)(r,s)+ (r,s) (30)

      where

      is a heuristically defined coefficient o – is the initial pheromone level

   4. ACS Parameter Setting

In this program of the following sections the numeric parameters, except when indicated differently, are set to the following values: =2, qo= 0.9, ==0.1.

4.3. Ant Colony Search General Algorithm

To solve MAUCP by ACSA, the search space of generation scheduling problem is established using multi-process decision making concept. The main computations are discussed below.

Step 1 : Ant Generation

In the first step, the colonies of ants are first generated. Ants are positioned on initial state while the initial pheromone value of 0 is also given at this step. Based on the concept multi state process, the search space of thermal generation scheduling problem can be established. All the possible permutations constitute this search space. Each stage contains several states, while the order of state selected at each stage can be combined as an achievable tour that is deemed a feasible solution to the problem.

µ(i) for i=1.n defines a permutation

With the evaluated fitness of the corresponding ants, the pheromone can be added to the particular direction in which the ants have selected.

Step 3: Ant Dispatch

In this step, the ants are dispatched based on the level of pheromone and distance. Each ant chooses the next state to move taking into account of ij and ij values. When the value of ij gets larger, there has been a lot of traffic on this edge; hence it is more desirable to reach the optimal solution. When the value of ij increases, it represents that the closer state should be chosen with a higher probability.

Step 4: Convergence

The computation process continues until the number of iterations reaches the predefined maximum threshold, of the iteration counter without improving the best objective function reaches the maximum allowable value. All the tour visited by ants in each iteration should be evaluated. If a better path found in the process, it will be saved for later reference. The best path selected among all iterations implies the optimal scheduling solution to the problem.

Algorithm for ACSA for SCUC Step 1 : Ant generation

Step 2 : Assessment of fitness

Step 3 : Ant dispatch

Step 4 : Update pheromone by applying local updating rule

Step 5 : Check for all ants are completed their tours.

If No go to step 3

Step 6 : Apply Global pheromone update rule

Step 7 : Check for convergence. If No go to step 2

1. Ant colony search algorithm for MAUCP

ACSA is conducted to solve MAUCP by following sequence of operations.

1. Initialize area A=1.

2. Read unit data, tie-line data, load demand profile and number of iterations to be carried out.

3. Find initial feasible solution.

4. Colonies of ants are generated and ants are positioned with initial phermone value of 0 .

5. Calculate the fuel cost and start up cost of each power plant at each ant position.

6. Calculate total production cost.

7. Calculate the fitness function of all ants.

8. Update ant position based on ij and ij values.

9. Update phermone by applying local updating rule.

10. Check for all ants completed their tour. IF No go to step 8. Otherwise go to next step.

11. Apply global phermone update rule.

12. Check for N number of areas completed. If yes go to step 2, else go to step 14.

13. Export power from lower operating cost areas to higher operating cost area by following tie-line constraints.

14. Print the commitment schedule of N areas and tie- line flows

1. Repair mechanism

   A repair mechanism to restore the feasibility of the constraints is applied and described as follows

   • Pick at random one of the OFF units at one of the violated hours.

   • Apply the rules in section 4.2 to switch the selected units from OFF to ON keeping the feasibility of the down time constraints.

   • Check for the reserve constraints at this hour. Otherwise repeat the process at the same hour for another unit.

2. Making Offspring Feasible

   While solving the constrained optimization problem, there are various techniques to repair an infeasible solution [8] [11]. In this paper, we have chosen the technique, which evolves only the feasible solutions. That is, the schedule which satisfies the set of constraints as mentioned earlier. Here, in this paper,

   the selection routine is involved as curling force to estimate the feasible schedules. Before the best solution is selected by evolutionary strategy, the trial is made to correct the unwanted mutations.

3. Implementation

Software program were developed using MATLAB software package, and the test problem was simulated for ten independent trials using Ant Colony Search Algorithm (ACSA). The training and identification part as implemented in the ACSA technique is employed here and considered as a process involving random recommitment, constraint verification, and offspring creation.

6. Numerical Results

The test system consists of four areas, and each area has 26 thermal generating units [1]. Units have quadratic cost functions, and exponential start up cost functions. Table 1 lists generating unit characteristics like the minimum up/down times, initial conditions and generation limits of units in every area. Table 2 to Table 5 lists the cost functions of units given in the four area [1], where variables ai, bi and ci are defined in equation 1. Ai, Bi and Ci are defined in equation 4. Load demand profile for each area is different and is given in Fig. 3. The hourly operating cost of four areas by Dynamic Programming (DP), Evolutionary Programming (EP) and Ant Colony Search Algorithm (ACSA) method is given in Table 6 to Table 8 respectively. The total operating cost in pu coparison between DP, EP and ACSA method is shown in Table

9. Comparison of total operating cost in each area by DP, EP and ACSA method is shown in Fig. 4. The proposed algorithm quickly reaches smallest total operating cost compared to DP and EP method, which indicates that the proposed algorithm could determine the appropriate schedule within a reasonable computation time. It is noted that cost in one iteration may be lower than that of the previous iteration, indicating that our optimization rules always comply with the equal incremental cost criterion for dispatching power generation among thermal units. The tie-line flow pattern at 11 am and 4 pm are shown in Fig. 5 and Fig. 6 respectively.

Figure 3. Load demand profile in each area

Figure 4. Comparison of Total Operating cost by DP, EP and ACSA method

598 Area Generation

1

81 0

846 4

89 2

1200

20 20

3

857

Figure 5. Tie line flow pattern at 11 am

Table 1. Generating unit characteristics

Table 2. Cost functions for generating units in area 1

td>

175.057

Table 3. Cost functions for generating units in area 2

Table 4. Cost functions for generating units in area 3

td>

150

Table 5. Cost functions for generating units in area 4

Table 6. Hourly cost of each area by DP method

Table 7. Hourly cost of each area by EP method

Table 8. Hourly cost of each area by ACSA method

Table 9. Comparison of total operating cost for 26 unit

466

1

19 0

784 4

177

2 1177

0 0

3

787

Figure 6. Tie line flow pattern at 4 pm

7. Conclusion

This paper presents ACSA method for solving multi area unit commitment problem with import and export constraints. In comparison with the results produced by the technique DP and EP method obviously proposed method displays satisfactory performance. Test results have demonstrated that the proposed method of solving multi area unit commitment problem with import and export constraints reduces the total operating cost of the plant. An effective tie-line constraint checking procedure is implemented in this paper. This method provides more accurate solution for multi area unit commitment problem with import and export constraints.

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