The solution of the economic dispatch problem via an efficient Teaching-Learning-Based Optimization method

: This paper is concerned with the economic generation dispatch problem. It is a well-known fact that practical aspects of power plant equipment, as well as the objectives to be met, may result in a nonconvex, nondifferentiable model that poses difficulties to conventional mathematical programming methods. This paper proposes the use of metaheuristic Teaching-Learning-Based Optimization to overcome such difficulties. This metaheuristic is well known for requiring a few parameters and, most importantly, it does not require the tuning of problem-dependent parameters. The algorithm proposed in this work is parameter-free; that is, the few parameters required by the Teaching-Learning-Based Optimization method are set automatically based on the power system’s data. In addition, the handling of constraints, such as generators’ prohibited zones and the generator-load-loss power balance, is performed in a very efficient way. Simulation results are shown for power systems containing 3 to 40 generation units, and the results provided by the proposed method are shown and discussed based on comparisons with other metaheuristics and a mathematical programming technique.


Introduction
The economic dispatch (ED) problem can be concisely defined as the setting of output powers from generating units to meet the demand at the minimum cost [1,2]. The ED problem can be regarded as a subproblem of the unit commitment problem [2]. The latter involves deciding which generation units must be turned on to meet a certain demand. Hence, the ED problem involves determining the optimal operating points of the committed generation units. The fuel rates are important factors that allow the generation agents to achieve their highest possible efficiency levels [3]. The minimization of fuel costs eventually results in lower energy costs for the consumers.
algorithm was proposed in [15], by considering that students also learn during tutorial hours by discussing with their classmates or even by a discussion with the teacher. In addition, students were sometimes self-motivated and tried to learn by themselves. Reference [16] shows a comparison of metaheuristics GA, PSO, Simulated Annealing (SA), Exchange Market Algorithm (EMA), and TLBO for solving the ED problem, and TLBO showed superior performance. An extensive comparison of metaheuristics for solving the optimal power flow problem was presented in [5]. Once again, TLBO showed to be very efficient.
TLBO presents many advantages, such as (a) fewer parameters, (b) simple algorithm, (c) easy to understand, (d) fast solution speed, (e) high accuracy, and (f) good convergence ability [13]. Being very flexible, TLBO allows different variations and improvements, which is very interesting and even desired in the optimization field. Because of its interesting features, TLBO has also been used for solving several power-system-related problems besides the ED. Some examples are the optimal capacitor placement in distribution systems [17], and distribution systems reconfiguration [18], among others.
The main goal of this paper is to apply an efficient version of the Teaching-Learning-Based Optimization (TLBO) metaheuristic for solving the ED problem. The main contributions of this paper are twofold: • TLBO was implemented such that no system parameters must be specifically defined. They are automatically set according to the system's characteristics. • The inequality constraints are dealt with such as to avoid the use of penalty factors and therefore the distortion of the objective function, making the convergence smoother.
The simulation results show that the proposed method leads to excellent results compared with those in the literature. In addition, TLBO is conceptually simple and computationally efficient. This paper constitutes a step forward in the development of optimization methods applied to power systems. The ability to solve real-life problems by overcoming problems resulting from non-convexities and non-differentiabilities is crucial.

Economic dispatch model
One of the requirements for an efficient power system operation is to make sure that the power is generated at the minimum cost. In practice, the overall minimum cost cannot be obtained due to operational constraints, which must be considered. In this case, the idea is to generate power at the least cost possible. This process is known as the economic dispatch problem, and its main features will be described in this section.

Objective function
As mentioned earlier, the ED problem consists of specifying the output powers from generation units to meet the load at the minimum cost. Each thermal generation unit i is usually associated with a quadratic cost function C i which depends on the output power P i [1]. For a power system with N generation units, the total generation cost function C t is given by, where coefficients a i , b i , and c i are previously known.

Equality constraint
The power generated by the several generation units must supply the load and the transmission power losses. This requirement is known as the power balance constraint, and it is given by, where P D is the total power demand and P L corresponds to the total transmission power losses. A simple, widespread way of representing the transmission power losses is by considering them a quadratic function of the generation output powers [1,2,16], as, where B 00 , B 0j , and B ij are known as the B coefficients.

Inequality constraint -generation limits
The output power of generation units must lie within the range, where P l i and P u i are respectively the lower and upper bounds associated to generation unit i.

Inequality constraint -prohibited operating zones
Prohibited zones are due to the steam valve operating or vibration in a shaft bearing [6]. The prohibited zones associated with generation unit i are modelled as, where NP i is the number of prohibited zones of generation unit i. P l i,j and P u i,j are the lower and upper limits of prohibited zone j, respectively. Figure 1 shows an example cost curve for generation unit i as a function of its generated power. Note the presence of two prohibited zones within the allowed power range as described in Sec. 2.3.

Proposed ED model
The ED model proposed this paper is, the objective Eq. (2) and the power balance constraint Eq. (3) are quadratic. It is worth pointing out once more that the presence of prohibited zones implies in difficulties to conventional mathematical programming methods.

Other aspects of ED models
The basic model of the economic dispatch is theoretically simple; however, it may become increasingly complex due to the size of the problem, the coordination the different characteristics and operating costs due to different generation technologies and sources, the variations in load over daily and seasonal cycles, and the need to operate the system reliably, abiding by transmission line operating limits [19]. In addition, the security-constrained economic dispatch adds more difficulties to the problem since it must consider the possibility outages (contingencies), either in the generation or the transmission systems.
Still, according to [19], the economic dispatch has gotten more complex because of the incorporation of public policy changes, technological innovation, and the ever-increasing penetration of stochastic, intermittent generation, such as wind and solar generation, as well as energy storage.
Several other practical aspects can be included in the ED model. It is worth noting that the inclusion of such aspects does not affect the performance of the metaheuristic algorithm proposed in this paper. Some of those aspects are included here for the sake of example, without the intention of being comprehensive.
It is possible to include the valve-point effects, which appears in the objective function, by including an additional term to (OC) [7,16,20], resulting in, where e i and f i are coefficients related to the valve loading of generation i. Also, [20] modelled the fuel cost as a cubic function and discussed the inclusion of other aspects to the objective function, such as the cost of emission of pollutants and fuel limitation.
A constraint related to imposing limits to transmission line power flows are represented as, where L is the number of transmission lines. The ramp rate limit constraint is related to maximum generation changes between time periods, being given by, where DR i and UR i are respectively the down-ramp and up-ramp limits from generation unit i. P i,t and P i,(t−1) are the generation output powers of generation unit i at two consecutive time instants. Figure 2 illustrates the case where P i,t > P i,(t−1) . Note that P i,t must lie within the range delimited by UR i . It is worth noting that the aspects mentioned in this section can be included in the model and solved by metaheuristics despite their characteristics as far as linearity and convexity are concerned.

Teaching-Learning-Based Optimization
The algorithm of the TLBO implemented in this paper was based on [12] and [21]. Both the teacher and student phases were applied to all individuals. Also, the removal of duplicate individuals was not included. As a result, the number of evaluations of the objective function is deterministic, given by, where IT is the number of iterations and N p is the population size.

Basic algorithm
The following pseudo-code (Algorithm 1) shows the basic steps of the algorithm proposed and implemented in this paper.

Implementation details and comments
The following comments regarding the algorithm are important for its implementation.
• Both teacher and student phases consist of greedy selection processes, where a new individual (solution candidate) solution is accepted whenever it is better than the current one. • The population size is defined automatically in terms of the number of generation units N, as, therefore, the proposed algorithm is parameter-free. • The teaching factor T F can be either set as a constant [21] or as a random value [12]. In this paper, T F was set according to [12].
where r is a random number in the range [0, 1] and T F is the teaching factor randomly chosen as either 1 or 2; Choose an individual X j randomly; b. if X i is better than X j then Better = X i and Worse = X j ; else Stopping criterion was met? If so, stop. Else, go back to step 2; • The stopping criterion adopted in this paper is based on the evolution of the objective function. If the objective function remains the same for (10 · N) consecutive iterations, it is assumed that its optimal value has been reached and the process is interrupted. • Whenever new individuals are generated, such as in steps 1, 5(a), and 6(d), it is necessary to verify whether they meet the equality and inequality constraints. In other words: -The power balance constraint Eq. (3) must be met.
-The generated powers must be within the lower-upper range Eq. (5), and outside the prohibited zone regions Eq. (6).
The way to deal with these inequalities is described in detail in the next section.

Dealing with constraints
Consider the following mathematical programming problem, where f is the objective function, and g and h are respectively the sets of inequality and equality constraints. u is the array of decision variables, bound by lower and upper limits u l and u u . N, NE, and N I are respectively the numbers of decision variables, equality constraints, and inequality constraints. According to [22], the conventional strategies of handling constraints in an optimization problem can be broadly classified as Eq. (1) inclusion penalty functions, Eq. (2) decoders, Eq. (3) special operators, and Eq. (4) separation of objective function and constraints. The latter strategy is used in this paper. As mentioned earlier, whenever new individuals are generated, it is necessary to verify whether they meet the equality and inequality constraints. This verification is described in detail ahead.

Equality constraints
A conventional way of dealing with equality constraints is by squaring and adding them to the objective function using a penalty factor, resulting in, where the penalty factor µ is usually a large number. According to [23], the penalty factor has several drawbacks, such as (a) it is system-dependent, therefore, the user must search for the best factor through a trial-and-error procedure whenever any parameter undergoes any change; (b) its value influences significantly the solution of the problem; and (c) the penalty factor µ causes a distortion of F and, depending on its value, this distortion may lead to artificial, local optimal solutions. To cope with this problem, in this paper the equality constraints in Eq. (13) are replaced by, where ε is a small, positive threshold value.
It is a common practice to add the constraints to the objective function with the inclusion of penalty factors. In this paper, the constraints are treated separately, following the principles presented in [23]. Whenever a candidate solution is evaluated, the following rules apply.
• Any feasible solution is preferred to any infeasible solution.
• Among two feasible solutions, the one having a better objective function value is preferred. • Among two infeasible solutions, the one having smaller constraint violation is preferred.
It will be shown that by handling the constraints described above, the drawbacks associated with the use of penalty factors mentioned earlier can be overcome. Setting ε in Eq. (15) is significantly easier than that of µ in Eq. (14), and its value does not significantly affect the results. This constraint-handling technique has been used in several studies [14]. Considering that the model in Eq. (7) adopted in this study, there is only one equality constraint represented by Eq. (3). Parameter ε was set to 0.05 MW, which is the same for any power system.

Inequality constraints
In this paper, the inequality constraints included in Eq. (7) are: • The lower and upper power bounds of the generation units, and • The prohibited power generation regions.
In this paper the following, simple rule is used to consider inequality constraints: "if a generation unit violates a certain limit, its output value is set to the closest violated limit." As an example, assume a generation unit with lower and upper limits corresponding to 100 MW and 300 MW, respectively. In addition, we consider a prohibited zone [140 − 170] MW. If, in a certain iteration, the generation is set to 320 MW (violation of the upper limit), it is reset to 300 MW. If the generation is set to 160 MW (within the prohibited zone), it is reset to 170 MW.

Simulation results
This section presents the simulation results for power systems with 3, 6, 15, and 40 generation units. These systems are well known in the literature and their respective data are widely available. Therefore, the results of the proposed method can be compared with those available in the literature. The results provided by the proposed method were compared with those of other metaheuristics and mathematical programming methods. The latter is successive quadratic programming, implemented using the SQP function from Octave [24]. SQP is assumed to provide the exact solution, and it is taken as a reference for comparison with other methods. Each simulation was run 100 times, and the averages and standard deviations are shown.

3-unit system
The data from the 3-unit system was taken from [1]. The load for this system was set to 850 MW. In this case the generators do not present prohibited zones, therefore, model Eq. (7) does not include Eq. (6). Table 1 presents a comparison of the results obtained with TLBO with those provided by running function SQP from Matlab. Function SQP (Sequential quadratic programming) is an iterative method for constrained nonlinear optimization problems.
The results provided by TLBO are very close to those obtained by SQP, since the global optimum operating point provided by SQP lies within the range of values provided by TLBO (average ± std deviation). Note also that the 3-unit system is considered small, since it has only three decision variables. Also, since this system does not present any prohibited zones, there are not non-convexities in the model.   Figure 3 shows the evolution of the value of the objective function along the iterative process. The presence of the equality constraint Eq. (3) leads the objective function to increase as the iterative process progresses. In the first iterations, the individuals may have lower generation costs, however, the constraints are violated. Later on, the situation is reversed, and TLBO moves toward the optimal solution through feasible individuals.

6-unit system
The complete data of the 6-unit system, including the cost coefficients, loss coefficients, generation limits, and prohibited zones was taken from [2]. The system's load was set to 1, 263 MW. Table 2 shows the results obtained by the proposed method.
A comparison of the best results obtained by the proposed TLBO, as well as by PSO and GA [2] are shown in Table 3. Column SQP is also included for comparison purposes. Table 4 shows the average and best total generation costs provided by the proposed TLBO, as well as by PSO and GA [2], by modified algorithms of PSO [25,26]. According to [2], PSO and GA were run 50 times. The proposed TLBO was run 100 times. Row SQP is also included for comparison purposes. Table 2 to Table 4 clearly show that TLBO performed very well in comparison with the other metaheuristics, outperforming the other methods as far as the average and best generation costs, as compared with SQP. Figure 4 shows the evolution of the values of the objective function for the 6-unit system. The general behavior of the objective function, in this case, is like the one from Figure 3.  Table 4. 6-unit system -comparison of average and best results.

15-unit system
The data from this system, including the cost coefficients, loss coefficients, generation limits, and prohibited zones was taken from [2]. The system's load is 2, 630 MW. Table 5 contains the simulation results obtained by TLBO.
A comparison of the best results provided by TLBO with those provided by PSO and GA [2] are shown in Table 6. Once more, column SQP was included for comparison purposes.   Table 5. 15-unit system -simulation results provided by TLBO. Table 7 shows the average and best generation costs provided by TLBO, as well as by PSO and GA [2], by MIPSO [25], and by QPGPSO-W [26]. Again, TLBO was run 100 times, while PSO and GA were run 50 times [2]. Row SQP is also included for comparison purposes. Table 5 to Table 7 show that TLBO performed very well as compared to the other methods and to SQP. The results provided by TLBO for this system stand among the best shown in Table 7.  Table 7. 15-unit system -comparison of average and best results. Figure 5 shows the evolution of the values of the objective function along the iterative process. Again, the behaviour of the objective function in this case is similar to the ones shown in Figure 3 and Figure 4.

40-unit system
The complete data, including the cost coefficients and generation limits, can be found in [27]. The demand of the 40-unit system is 10, 550 MW.  Table 8. Simulation results for the 40-unit system.   Table 9. Output powers for the 40-unit system. Figure 6 shows a comparison of the results obtained by the proposed TLBO with those obtained by CSO, PSO, and GA [27]. The figure allows a visual comparison among the output generations provided by each method. By taking one generation unit at a time, it is possible to see that all four methods provide compatible, close results.
The average total generation and costs provided by CSO, PSO, and GA [27], as well as by the proposed TLBO are shown in Table 10. Note that the method adopted in this paper for handling the equality constraints allows a very good precision without creating numerical problems. Row SQP has been included for comparison purposes.
It is not clear in [27] whether the results are the best ones, or average values, the reason why both the average and best results provided by TLBO were presented. Also, the total costs provided by TLBO are compatible with those provided by GA and SQP; however, they are larger than those provided by CSO and PSO. It seems odd that CSO and PSO show similar costs; however, the sum of all generations, as shown in [27], is quite different. In particular, the total generation reported for PSO is more than 1, 000 MW shorter than the specified value. The evolution of the objective function value is shown in Figure 7. Again, its general behavior is similar to that of previous ones.

Discussion
The simulation results presented in Section 5 clearly indicate the excellent characteristics of TLBO. In all simulations, TLBO provided the best or close-to-best results. All metaheuristics are based on a random initial population and random exploration and exploitation search. However, they may provide different results depending on the manner in which the searches are defined. In this work no changes in the TLBO algorithm were performed to adapt to the particular problem (ED). The idea of automatically defining the number of individuals in the population and handling constraints may be applied to any other optimization problem. Therefore, TLBO proved to have excellent potential for solving the economic dispatch problem and any other problem in the power system area.

Conclusions
This paper tackled the problem of economic generation dispatch. Even though this problem is well known, its formulation may result in a nonlinear, nonconvex model, which may pose numerical difficulties to mathematical programming methods. This work showed that metaheuristic TLBO is a simple and efficient method for solving ED. The automatic setting of its parameters, regardless of the system, is an important feature of the proposed method, and has been shown to be very effective. In addition, dealing with these constraints did not result in numerical problems, which are commonly found when penalty methods are used. In contrast, the iterative process was smooth, and high-quality results were obtained successfully. The simulation results obtained using the proposed method were compared with those reported in the literature. Also, simulation results from a mathematical programming method based on successive quadratic programming were also shown for comparison. The proposed method exhibited excellent performance. The use of TLBO to solve the ED problem has now expanded. The next steps consist of including actual aspects into the model, such as the effect of valve points and the minimization of the greenhouse effects, as described in Sec. 3.6. Also, the active power losses and power balance are more appropriately represented by the power flow equations. These aspects would result in a more realistic ED model and more precise representation of the electric system. The inclusion of nonlinear terms regarding the valve point effect would not significantly change the performance of TLBO because one of the strong characteristics of metaheuristics is their capability to handle nonlinearities and nonconvexities. The use of power flow equations does not at all affect the performance of TLBO, because its idea is to represent power losses in a more precise way and to automatically meet the generation-load balance (constraint).   3-unit Table 14. Data from the 15-unit generation system (cont.).