A solution to unit commitment problem using invasive weed optimization algorithm

Abstract

Unit commitment (UC) is one of the most important aspect of power generation in the world today. Though, there is no method to find the exact optimized solution, there exists several meta-heuristic algorithms to determine the close to exact solution. This paper proposes a novel solution to effectively determine UC and generation cost using the technique of invasive weed optimization (IWO). The existing technique distributes the load demand among all the generating units. The method proposed here utilizes the output of UC obtained by using the Lagrangian relaxation (LR) method and calculates the required generation from only the plants that are ON discarding the OFF generator units and thereby giving a faster and more accurate response. Moreover, the results show the comparison between the LR-particle swarm optimization (PSO) and LR-IWO, and prove that the cost of generation for a 4 unit, 8 hour schedule is much less in the case of IWO when compared to PSO.

Keywords

Lagrangian relaxation (LR) / invasive weed optimization (IWO) / economic dispatch / optimization / fuel cost / seed / fitness

Introduction

The problem of unit commitment (UC) is of high priority and is imperative in the world today. The continuous and increasing demand for power and the ever reducing fossil fuels make it impossible for continuous power supply to the people without any interruptions. Even though enough stress has been laid on the use of renewable sources of energy and a wide-scale implementation of different sources of energy like solar, wind, tidal, biomass etc., it can be seen that it has not helped in putting a check on the power deficit the world faces. Another way of tackling this problem is by producing electricity more efficiently in addition to going for new sources. The most widely used sources of power like thermal, hydro, gas etc., need to be utilized in a more optimal fashion so as to reduce any wastage of power and to generate only the required amount of power in the most economic fashion. Economic dispatch is the need of the hour and different methods are being analyzed to find the methods that bring out the most optimum generation at the least cost [1].

The different methods to effectively bring down the cost of generation are called evolutionary algorithm and generic population based meta-heuristic algorithms, some of which are the bacterial foraging technique, the particle swam method, the cuckoo search method, the firefly algorithms and invasive weed optimization (IWO). All these techniques are mapped from real life events such as growth of bacteria, foraging behavior of honeybees, nesting of birds and growth of weed plants. All these methods are known to give out the optimum cost of power generation over a scheduled period of time depending on the load requirements, assuming that all the generator units are ON. In this paper, the results derived from The Lagrangian derivation method for UC are mapped into IWO for economic dispatch. The output of the Lagrangian method serves as the input for IWO.

UC methodology

The LR method is used to find out the UC results for a particular duration. The objective function of LR is to discover the units that are the most economical for operation [2]. The economics of operation depends upon the fuel cost, uptime, downtime, cold-time, maximum and minimum generation limits. The fuel cost of a generation unit is given in the form of a second order polynomial function which depends on the power output of that particular unit.F(i)=a_i{P}_i^{\mathrm{2}}+b_i{P}_i+c_i,where F (i) is the fuel cost of the unit i; P_i, the power output of the unit i; and a_i , b_i and c_i are coefficients of the fuel cost polynomial.

The LR method finds out the derivative of the polynomial function F (i) and obtains the incremental cost of the unit λ, lambda. This λ gives the idea of the units with the least operational cost. The units are arranged in the order of increasing incremental cost starting with the one that has the least λ.

The different conditions considered for determining the UC are uptime; downtime; cold-time; initial status; maximum generation limit; and minimum generation limit.

IWO

Terms used

The terms that are used in IWO work are defined as follows:

Seeds — all units in the optimization problem that are assigned a value pertaining to the limiting conditions; Plants — seeds that grow into plants before being evaluated; Fitness value—a value that determines how good the plant is i.e., how much optimized the solution is; and Field — the probable solution area/search area.

The IWO technique is a meta-heuristic technique that mimics the colonizing behavior of weeds. It uses the technique of growth of weed plants spread out over an area [3]. First, the seeds are randomly spread out into the open boundary field and allowed to grow. As the plants grow, their fitness functions are evaluated and are sorted in the order of decreasing fitness, the best fit plant being at the top. Depending upon this fitness values, the new seeds of these plants get planted, and the fitness of the parent weed and the seed are evaluated now together and assessed. The new seeds are distributed from the location of the plant at a distance determined by a normalized standard deviation value. The least fit weeds are removed from the list. This technique of converging to the optimum result, in other words, finding out the best fit parent-seed combination proves to be quite accurate as compared to other evolutionary algorithms. The objective function of this algorithm depends upon the type of application the algorithm is used for. The objective function is utilized as the fitness function to achieve the optimized results using convergence technique.

Steps involved in conventional IWO

The steps involved in the conventional IWO are as follows:

Step 1 The seeds are initialized depending upon the number of selected variables involved in the process over the probable search boundary. The initialization of seeds is random which means that the seeds are dispersed in a random manner in the solution space [4].

Step 2 The fitness of the seeds initialized is evaluated depending upon the fitness function (or) the objective function chosen for the optimization problem. These seeds then evolve into weed plants capable of producing new units.

Step 3 The plants evolved are arranged in a definite order (increasing or decreasing) and new seeds are produced by these plants depending upon its position in the sorted list of plants, starting with the maximum number of seeds produced by the best fit plant [5].

Step 4 The number of seeds to be produced by the plants varies linearly from N_max to N_min which is decided by

(1)\mathrm{Number}\mathrm{of}\mathrm{seeds}={\displaystyle \frac{F_i-F_{\mathrm{worst}}}{F_{\mathrm{best}}-F_{\mathrm{worst}}}}(N_{\max }-N_{\min })+N_{\min },

where F_i is the fitness of the ith plant; F_worst, the fitness value of the worst plant; and F_best, the fitness value of the best plant.

Step 5 The generated seeds are distributed normally over the search space with zero mean and a standard deviation that is varying σ_iter which is given by

(2){\sigma }_{\mathrm{iter}}={\left({\displaystyle \frac{ite{r}_{\max }-iter}{ite{r}_{\max }}}\right)}^n({\sigma }_{\mathrm{0}}-{\sigma }_f)+{\sigma }_f,

where, iter_max is the maximum value of the iteration assigned by the user; iter, current iteration value; σ₀ and σ_f = initial and final values of standard deviations pre-assigned by the user; and n is the non-linear modulation index.

The non-linear modulation index is used to traverse around the search space more efficiently and is generally assumed to be between 2 and 3.

Step 6 The fitness of each seed produced in the above steps is calculated along with the parent weeds and by means of competitive exclusion. The seed-parent combinations that are least in fitness are eliminated and the number of weed plants is limited to the maximum of number of weeds allowed.

Step 7 The above steps are carried out until maximum iterations and the plant with the best fitness value at the end of it is the optimized solution.

Problem formulation

The objective function is to minimize the fuel cost of all units. This also serves as the fitness function required to evaluate the fitness value of the individual units [6]. The objective function is subjected to certain constraints like,

1) The power generated from all the units at a particular hour should be equal to the load demand of that particular hour.

2) The power generated by the individual unit should adhere to the minimum and maximum generation limits and

3) The constraints of minimum up time and down time have also been taken into consideration in the UC result.

Objective function

(3)\min F{C}_{\mathrm{total}}={\displaystyle \sum F{C}_i}

Subjected to the constraints,

(4)\sum P_i=P_d

and

(5)P_{i\min }\le P_i\le P_{i\max },

(6)(T_{i,\mathrm{ON}}(t-1)-T_{i,\mathrm{UP}})(I_i(t-1)-I_i(t))\ge \mathrm{0},

(7)(T_{i,\mathrm{OFF}}(t-1)-T_{i,\mathrm{DOWN}})(I_i(t-1)-I_i(t))\ge \mathrm{0},

where T_i,ON is the number of hours the unit has been ON for at time t; T_i,OFF, the number of hours the unit has been OFF for at time t; T_i,UP, the minimum up time of unit i; and T_i,DOWN, the minimum down time of unit i.

Methodology proposed

The algorithm proposed utilizes IWO for economic dispatch using UC outputs obtained from the LR method [7]. The algorithm has been modified to suit UC outputs adhering to the up-down time constraints instead of assuming that all units are ON and distributing the load demand among all units. The load is shared only among the units that are ON during that particular interval.

Step 1 Read the input from the UC matrix and assign a matrix of size depending upon the units that are ON. This is analogous to the search area, and the number of variables is equal to the number of units that are ON. The UC output satisfies the up-down time constraints [8].

Step 2 Randomize the values obtained for generation for each unit subjected to the constraints P_{i\min }\le P_i\le P_{i\max } and \sum P_i=P_d. These values of generation, otherwise called as seeds, assume random values in the search space. The search space is reduced in this method since it is reduced to only the units that are ON.

Step 3 Determine the fuel cost for the obtained combination of generation values and repeat this procedure for a total of 100 iterations. The fuel cost, otherwise called fitness values are assigned to the respective seeds.

Step 4 Arrange the values of fuel cost obtained from the objective function in an increasing order. These values of fuel cost represent the fitness value of each seed. The minimum amount serves as the best fitness value. The next generation of seeds is generated based on a value of normalized standard deviation as given in Eq. (2) and the number of seeds depends upon the fitness value according to Eq. (1).

Step 5 The procedure is repeated until the maximum number of iterations is not met. The values with the best fitness values are taken and are put forward as the generation values provided are within the individual generation limits and meet the load requirements [9].

The primary advantage of the methodology proposed is that it takes the up-down time into consideration and reduces the execution time of computation for optimal solution since the search space is considerably reduced as the units that are ON are only considered as variables for the process of IWO. Moreover, the convergence is much faster in the case of IWO and since this method carries a large number of iterations which begin with random dispersing of values in the search area, and the extent of search is wide when compared to other algorithms. Hence, the chances of obtaining a solution that is much closer to the exact solution is very much possible when compared to other algorithms. In this paper, the fuel cost obtained using this methodology is compared with the fuel cost obtained using another evolutionary algorithm known as particle swarm optimization (PSO).The above mentioned step by step procedure of the methodology proposed was given in Fig. 1

Results and discussion

The algorithm developed was tested with a 4 unit, 8 hour system and a 10 unit, 24 hour system. The detailed analyses are listed in Tables 1 to 8.

Case study 1: the 4 unit, 8 hour system

The input data of the 4 unit, 8 hour system in case study 1 are listed in Table 1. In Table 2 the load data for 8 hour system was given. The UC scheduling of all the 4 generators for 8 hours is given in Table 3. The generation dispatch using IWO is listed for each hour and is compared with PSO in Table 4. The total cost of IWO is reasonably less when compared to PSO, which is represented in the form of a graph in Fig. 2 where the generation cost of every hour using IWO and PSO is demonstrated [10].

Case study 2: the 10 unit, 24 hour system

Similar to the first case study, a second analysis was performed on a 10 unit, 24 hour system. The input data of the 10 unit, 24 hour system are given in Table 5 [11]. The UC scheduling of all the 10 generators for 24 hours is presented in Table 6. In Table 7 the unit commitment output using IWO is listed for 10 unit 24 hour system, and in Table 8 , the economic dispatch output for each hour is given. The total cost of IWO is reasonably less when compared to PSO.

Conclusions

This paper evidently proves that the IWO technique is much more effective than PSO in the case of cost optimization for generating power [12]. The method proposed effectively reduces the time of execution along with more optimized results which could be very useful in the case of short term load scheduling. This technique could be extended for any number of generating units and for any duration of load scheduling [13]. Future works could rely on the possibilities of obtaining UC output using the IWO technique itself instead of obtaining the outputs of UC from some other techniques which will effectively reduce the speed of execution and improve the convergence of results. The area of applications of IWO is vast. This algorithm could also be employed in distribution of power in transmission systems [14]. Obtaining data for optimized transmission of power flow using IWO could lead to a completely optimized power grid, right from generation of power to distribution to consumers.

Notation

References

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