Optimal unit commitment of a power plant using particle swarm optimization approach

Received Dec 4, 2018 Revised Oct 4, 2019 Accepted Oct 12, 2019 Economic load dispatch among generating units is very important for any power plant. In this work, the economic load dispatch was made at Egbin Thermal Power plant supplying a total load of 600MW using six generating units. In carrying out this study, transmission losses were assumed to be included into the load supplied. Also, three different combinations in the form of 6, 5and 4-units commitment were considered. In each case, the total load was optimally dispatched between committed generating units using Particle Swarm Optimization (PSO). Similarly, the generation cost for each generating unit was determined. For case 1, the six generators were committed and the generation cost is 2,100,685.069$/h. For case 2, five generators were committed and the generation cost is 2,520,861.947$/h. For case 3, four generators were committed and the generation cost is 3,150,621.685$/h. From all considered cases, it was found that, the minimum generation cost was achieved when all six generating units were committed and a total of 420,178.878$/h was saved.


INTRODUCTION
For efficient and reliable operation of power system, proper analysis of operating the system economically is of great importance [1]. For economic operation of generators many variables are considered such as fuel, labour and maintenance cost. For thermal and nuclear plants, the most important variable considered is the fuel cost [2]. Economic load dispatch problem is defined as allocating loads to generating units at minimum cost while satisfying various operational constraints [3][4][5][6][7][8]. The generators are to be scheduled in such a way that generators with minimum cost are used as much as possible [6]. Several factors contribute in generation cost of a thermal power plant such as the location of load centres and the fuel cost. The cost of power generation particularly in fossil fuel plants is high and economic dispatch helps in saving a significant amount of revenue for a utility company [4]. Most generating stations are faced with the problem of allocation of generators and this lapse leads to non-economical operation of the plants. Non-economical operation of the plants directly leads to higher incremental fuel cost, thus; leading to high tariff of electricity on consumers.
In this study, Particle Swarm Optimization (PSO) technique was used to economically dispatch the load between the generating units and determine the minimum generation cost at Egbin Thermal Power Plant station in Nigeria. These were done after determining the generation cost function of each generating unit using Least Square Estimation Technique. PSO technique was used in this study due to its mathematical simplicity, fast convergence and robustness to solve hard optimization problems. The study will benefit utility companies and consumers of electricity. This will help in reduction of production cost of the utility companies and minimize tariff on consumers.
The solution of economic dispatch problem has been proposed and different algorithms have been developed. Traditional algorithms such as Lambda iteration, gradient, Newton-Raphson methods, etc. were widely employed in solving the Economic Load Dispatch (ELD) problem if their cost functions are piecewise linear [4,9]. These methods were challenged by the introduction of transmission losses and prohibited zones due to environmental impacts; thus, the use of advanced techniques such as genetic algorithms, evolutionary programming, particle swarm optimization, artificial immune systems, harmony search, Tabu search, artificial neural network, among others are preferred [10][11][12][13][14][15][16][17][18][19].

RESEARCH METHOD
The Egbin electric power generation station used for this study is a steam thermal plant that makes use of steam to drive its turbines in order to generate electricity. The power station uses natural gas as fuel to fire the boiler. The station was established in 1985 and is located in Egbin village near Ijede town in Ikorodu Local Government Area of Lagos state, Nigeria. At present, the installed capacity of the generating station is 1320MW which consists of six (6) steam-turbine generators each having maximum plant capacity of 220MW [20].
The major concern of an economic dispatch problem is to minimize the fuel cost for a given thermal power plant considering a given total load demand subject to operating constraints of a power system. Therefore, it can be formulated mathematically with the objective function and constraints. In any practical case, the fuel cost function of any generating unit is represented by a quadratic function of the real power generation.
The incremental fuel-cost curve is a measure of how costly it will be to produce the next increment of power.
Thus, the objective function is formulated as where CT is the total fuel cost, Pi are the cost functions and real power output of generator i respectively n is the number of committed generators.

System constraints
In this study, the system constraint was classified into equality constraint and inequality constraint

Equality Constraints
As stated in [21], the total power generation must cover the total demand PD and the real power loss in transmission lines PL . It is also called power balance equation and is expressed as

Inequality constraints
As stated in [22] the output power of each generator should lie between minimum and maximum limits, so that

Overview of PSO
The PSO algorithm which was first proposed by Kennedy and Eberhart has been inspired by the Social behavior of a simple system (flock of birds). This algorithm can be effectively useful in solving many non-linear hard optimization problems [10]. Unlike the mathematical methods for solving optimization problems, this algorithm does not need any gradient information about objective or error function and it can obtain the best solution independently [23]. According to the PSO algorithm, a swarm of particles that have predefined restrictions starts to fly on the search space. The performance of each particle is evaluated by the value of the objective function and considering the minimization problem, in this case, the particle with lower value has more performance [24]. The best experience for each particle in iterations is stored in its memory and called personal best (Pbest).
The best value of Pbests (less values) in iterations determines the global best (Gbest). By using the concept of Pbest and Gbest the velocity of each particle is updated in (6) where X k i 1  : Current particle position at iteration k+1 X k i : Particle position at iteration k With numerical analysis method, the marginal cost of each unit was determined using least square estimation technique. The incremental cost (marginal cost) for each generating unit was obtained by solving the following equations By solving (8) and (9), the marginal cost function was given as The generation cost function of a unit is determined by the area under the marginal cost function; hence the generation function is given by the integration of the marginal cost function of a considered unit. The load dispatch between generating units and minimum generation cost was done using Particle Swarm Optimization (PSO) considering system constraints. For the reason of comparing results, 3 cases were considered namely: test with six generating units, test with 5 generating units and test with 4 generating units. Sample data used in this study are given given in Tables 1-3. Full details of the data used can be found in [25] The data contains power output and energy generated from each of the six generating units at Egbin power plant. A generation cost of 0.07$/KWh was considered [26].

RESULTS AND ANALYSIS
Using the values given in Tables 1 to 6 the marginal cost function for each unit was obtained as follows: Unit-1: 35 0   (22) The solved problem was given by (3) as follow: Test Case 1:

P IC
In this case, six generating units were committed. The power outputs and the generation costs are presented in Table 4. In this case, only five generating units were committed. The power outputs and the generation costs are presented in Table 5. In this case, only four generating units were committed. The power outputs and the generation costs are presented in Table 6. The reason for considering only 3 cases was because of no optimization was found using PSO for the other 3 combination of generating units. From the results obtained, a difference of 420178.878$/h was recorded when six generating units' combination was used ahead of the five and four generating units combinations.

CONCLUSION
Using the plant generation data for 5 years from 2002 to 2006 the generation cost functions of 6 generating units at Egbin thermal plant were determined. The economic load dispatch and optimum generation cost for the considered period was determined using Particle Swarm Optimization. Considering a combination of six generating units all operational, the minimum generation cost was obtained as 2,100,685.069$/h. The other possible unit combinations were analysed where the 5-units and 4-unit combinations were considered. From the results obtained from each combination, it was seen that the minimum generation cost was achieved if all six generating units were committed and the power plant was saving a total of 420,178.878$/h.