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EP Based PSO Method for Solving Multi Area Unit Commitment Problem with Import and Export Constraints

  • Venkatesan, K. (Dept. of Electrical Engg, J.N.T.U. Anantapur) ;
  • Selvakumar, G. (Excel Engineering College) ;
  • Rajan, C. Christober Asir (Dept. of Electrical and Electronic Engineering, Pondicherry Engg. College)
  • 투고 : 2013.07.11
  • 심사 : 2013.10.29
  • 발행 : 2014.03.01

초록

This paper presents a new approach to solve the multi area unit commitment problem (MAUCP) using an evolutionary programming based particle swarm optimization (EPPSO) method. The objective of this paper is to determine the optimal or near optimal commitment schedule for generating units located in multiple areas that are interconnected via tie lines. The evolutionary programming based particle swarm optimization method 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. Case study of four areas with different load pattern each containing 7 units (NTPS) and 26 units connected via tie lines have been taken for analysis. Numerical results showed comparing the operating cost using evolutionary programming-based particle swarm optimization method with conventional dynamic programming (DP), evolutionary programming (EP), and particle swarm optimization (PSO) method. Experimental results show that the application of this evolutionary programming based particle swarm optimization method has the potential to solve multi area unit commitment problem with lesser computation time.

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 has cheaper sources of generation to another utility having costlier generation sources. The total reduction in system operating cost will be shared by the participating utilities [3].

In the competitive environment, customers 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 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 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 initial solution which is fast and reliable [6]. Evolutionary Programming (EP) is capable of determining the global or near global solution [7]. It is based on the basic genetic operation of human chromosomes. It operates with the stochastic mechanics, which combine offspring creation based on the performance of current trial solutions and competition and selection based on the successive generations, from a considerably robust scheme for largescale real-valued combinational optimization. In this proposed work, the parents are obtained from a predefined set of solutions (i.e., each and every solution is adjusted to meet the requirements). In addition, the selection process is done using evolutionary strategy [8-10]. The application on this 26 unit shows that we can find the optimal solution effectively and these results are compared with the conventional methods.

PSO [11] is an exciting new methodology in evolutionary computation that is similar to GA and EP in that the system is initialized with a population of random solutions. In addition, it searches for the optimum by updating generations and population evolution is based on previous generations. In PSO, the potential solutions, called particles, are flown through the problem space by following the currently optimal particles. Each particle adjusts its flying according to its own flying experience and the flying experience of other particles in the swarm. PSO was traditionally considered for homogeneous swarms of potential solution vectors. The homogeneity of the swarm is not practically feasible because the loads vary continuously between the maximum and the minimum. Hence in this paper, we propose that PSO be solved as a non-homogeneous recurrence relation[12]. The fitness of the particular particle to the swarm as a whole in checked through statistical fitness tests on each one of them. This increases the convergence rate and the accuracy of the solution.

From the literature review, it has been observed that there exists a need for evolving simple and effective methods, for obtaining an optimal solution for the MAUCP. Hence, in this paper, an attempt has been made to couple EP with PSO for meeting these requirements of the MAUCP, which eliminates the above mentioned drawbacks. In case of PSO, the demand is taken as control parameter. Hence, the quality of solution is improved. Classical optimization methods are direct means for solving this problem. EP seems to be promising and is still evolving. EP has the great advantage of good convergent property and hence the computation time is considerably reduced. The EP combines good solution quality for PSO with rapid convergence for EP. The EP-based PSO (EPPSO) is used to find the multi area unit commitment.

In this paper, section 2 describes the problem formulation of multi area unit commitment. Section 3 discusses the tie line constraints taken into account for power transfer between the areas. Section 4 describes general algorithm of evolutionary programming method. Section 5 describes the overview of particle swarm optimization method and its general algorithm. Section 6 describes EP based PSO method for multi area unit commitment problem. Section 7 describes about the numerical results. Finally, section 8 describes about the conclusion of this work.

 

2. Problem Formulation

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

The incremental production cost is therefore

(or)

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

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

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

Subject to the constraints of Eqs. (9, 11) and (14-18). Each Fk(Pkgij) for K=1 ……NA is represented in the form of schedule table, which is the solution of mixed variable optimization problem

Subject to following constraints are met for optimization.

1) System power balance constraint

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.

2) Spinning reserve constraint in each area

3) Generation limits of each unit

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

5) In each area, power generation limits caused by tie line constraints are as follows

Upper limits

Lower limits

Import/Export balance

6) Area generation limits

The objective is to select λsys at every hour to minimize the operation cost.

Since the local demand Djk is determined in accordance with the economic dispatch within the pool, changes of Pkgj will cause the spinning reserve constraints of Eqs. (10) to change accordingly and redefine Eq. (8). Units may operate in one of the following modes when commitment schedule and unit generation limits are encountered [16].

a) Coordinate mode : The output of unit i is determined by the system incremental cost

b) Minimum mode : Unit i generation is at its minimum level

c) Maximum mode : unit i generation is at its maximum level

(d) Shut down mode : unit i is not in operation,

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

(i) Area coordinate mode

(or)

(ii) 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 Eqs. (14) or (17) therefore

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

(iii) Limited import mode

An area may reach its lower generation limit according to Eqs. (15) or (18) in this case because of higher generation cost

 

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.

An economically efficient area may generate more power than the local demand, and the excessive power will be exported to other areas through the tie lines [1]. For example assume area 1 has the excessive power. The tie line flows from area1 to other 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 2. According to power balance Eq. (9) some areas must have power generation deficiency and requires generation imports. The minimum generation limits in these areas is the local demand minus all the connected 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.

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

 

4. Evolutionary Programming Based Particle Swarm Optimization

4.1. EP-Based PSO

In the PSO technique for solving MAUCP, initial operating schedule status in terms of maximum real power generation of each unit is given as input. As we that PSO is used to improve any given status by avoiding entrapment in local minima, the offspring obtained from the EP algorithm is given as input to PSO, and the refined status is obtained. In addition, evolutionary strategy selects the final status. EP based PSO method for solving multi area unit commitment problem is given in Fig. 2.

Fig. 2.Flowchart for EPPSO algorithm for MAUCP

4.2. Repair mechanism

A repair mechanism to restore the feasibility of the constraints is applied and described as follows [13] Pick at random one of the OFF units at one of the violated hours. Apply the rules in section 6.1 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.

4.3. Making offspring feasible

While solving the constrained optimization problem, various techniques are there to repair an infeasible solution [7, 13]. 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.

4.4. Implementation

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

 

5. Numerical Results

5.1 Case study-7 unit (NTPS) system

A NTPS (Neyveli Thermal Power Station) in India with seven generating units, each with a capacity of 210MW, has been considered with different generation at each area for time period of 24h is considered as a case study [7]. The test system consists of four areas, and each area has 7 thermal generating units. The hourly operating cost of four areas is given in Table 1. Figs. 3-6 show the sum of the total number of units switched ON during every hour for EPPSO method of area 1 to area 4 respectively. Tie-line flow pattern of 11 am and 4 pm is given in Figs. 7 and Fig. 8 respectively. . Load demand profile for each area is different and is given in Fig. 10. The proposed algorithm quickly converged after two iterations and reaches smallest total operating cost, which indicates that the proposed algorithm could determine the appropriate schedule within a reasonable computation time.

Table 1.Hourly operating cost of each area of EPPSO method for 7 Unit(NTPS)

Fig. 3.Number of units ON during every hour of area 1 of EPPSO method for 7 unit system

Fig. 4.Number of units ON during every hour of area 2 of EPPSO method for 7 unit system

Fig. 5.Number of units ON during every hour of area 3 of EPPSO method for 7 unit system

Fig. 6.Number of units ON during every hour of area 4 of EPPSO method for 7 unit system

5.2 Case study- 26 unit system

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. Generating unit characteristics like the minimum up/down times, initial conditions and generation limits of units in every area, the cost functions of units given in the four area [1] are taken for analysis. Ai, Bi and Ci are defined in Eq. (4). Table 2 indicates the generation levels and committed capacities of all areas at 11 am and 4 pm. Comparison of total operating cost in each area by DP, EP, PSO and EPPSO method is shown in Fig. 9. Load demand profile for each area is different and is given in Fig. 10. The tie line flow pattern at 11 am and 4 pm are shown in Figs. 11 and Fig. 12 respectively. The convergence characteristic of EPPSO method for MAUCP is shown in Fig. 13. Comparison of cost for 7 Unit and 26 Unit system are shown in Table 3. The proposed algorithm quickly reaches smallest total operating cost compared to DP, EP and PSO 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.

Fig. 7.Tie line flow pattern at 11 am of EPPSO method for 7 Unit system

Fig. 8.Tie line flow pattern at 4 pm of EPPSO method for 7 Unit system

Fig. 9.Total operating cost in each area by DP, EP, PSO and EPPSO method

Fig. 10.Load demand profile in each area

Fig. 11.Tie line flow pattern at 11 am of EPPSO method for 26 Unit system

Table 2.Generations and committed capacity in each iteration of EPPSO method of 26 unit system

Fig. 12.Tie line flow pattern at 4 pm of EPPSO method for 26 Unit system

Fig. 13.Convergence characteristics of EPPSO method for 26 Unit systems

Table 3.Comparison of cost for 7 Unit (NTPS) and 26 Unit systems

 

6. Conclusion

This paper presents EPPSO method for solving multi area unit commitment problem for 7 unit (NTPS) and 26 unit with import and export constraints. In comparison with the results produced by the technique DP, EP and PSO 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 reduced 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 for both 7 unit (NTPS) practical system and 26 unit system with import and export constraints

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피인용 문헌

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