A New Fast Deterministic Economic Dispatch Method and Statistical Performance Evaluation for the Cascaded Short-Term Hydrothermal Scheduling Problem

Abstract

The Cascaded Short-Term Hydrothermal Scheduling (CSTHTS) problem is a highly non-linear, multi-modal, non-convex, and NP-hard optimization problem that has been solved by conventional and metaheuristic algorithms in the past. As the CSTHTS problem falls under the category of applied operational research, therefore, the work is still on-going to find new algorithms and variants of the existing algorithms that would better approximate the optimal global solution in a shorter computational time. This article proposes a novel deterministic thermal economic dispatch method embedded with the improved Accelerated Particle Swarm Optimization (APSO) algorithm to infinitesimally reduce the Big O time complexity for the standard benchmark test case of the CSTHTS optimization problem. Then, it discusses and presents the importance of performing standard statistical tests to establish the supremacy of one metaheuristic algorithm over the other one in solving the CSTHTS problem. The results obtained are better than the results of the many state-of-the-art algorithms applied to solve the considered test case of the CSTHTS problem in the literature, and the superiority of the improved APSO algorithm has been established statistically using the parametric independent samples t-test and the non-parametric Mann–Whitney U-test over the other metaheuristic algorithms such as particle swarm optimization in solving the chosen test case of the CSTHTS problem.

1. Introduction

The scheduling of the electrical power generated by different electrical power sources has been an important optimization problem in the domain of power system generation, operation, and control [1]. The hydel and thermal generation units are the major sources of electricity, and the scheduling of their generated power to meet the electrical load demand is called hydrothermal scheduling [2]. The hydrothermal scheduling for which the period of economic dispatch ranges from one day to one week is said to be a Short-Term Hydrothermal Scheduling (STHTS) problem [2]. In STHTS, if more than one hydel generation unit is present on the same water stream to generate electricity, i.e., one hydel generation unit is downstream another such that the outflow of the upstream reservoir becomes the inflow for the downstream reservoir, then the problem is called the Cascaded Short-Term Hydrothermal Scheduling (CSTHTS) problem [3]. CSTHTS is a highly multi-modal, non-linear, and non-convex NP-hard optimization problem. Its objective is to reduce the overall operational cost of the power system network for a given period. The CSTHTS problem has many equality, non-equality, non-linear, and time-coupled complex hydel and thermal constraints [4].

Many conventional and modern metaheuristic algorithms have been applied to optimize the CSTHTS problem in the past. As the CSTHTS problem falls under the category of applied operational research, therefore, the work is still on-going to solve the CSTHTS problem more effectively and efficiently in less computational time [5]. The motivation for research in this field is the “No Free Lunch theorems”, presented in [6], which state that, if an algorithm behaves well on one optimization problem, then it is not necessary that it will behave well with the same efficacy on other optimization problems. Thus, it is necessary to find for each optimization problem an efficient algorithm separately. These theorems demand that the algorithm provides a robust solution, not the best, because there is always a chance of having a new or existing algorithm give better results in a fast computational time, which has not been applied yet to the optimization problem [6].

Initially, classical methods were applied to solve CSTHTS problem. However, because they require initial guesses and approximations and still have a significant probability of being stuck at the local optimal point [7], they became unpopular in solving the CSTHTS problem. Then, the metaheuristic algorithms showed their effectiveness by eliminating the difficulties of the classical methods [6] and have become popular among researchers in solving CSTHTS problem. They offer many advantages, like not performing initial approximations or requiring initial guesses before starting the optimization process. They also have the characteristics of stochasticity and determinacy in their model [6]. The stochastic nature provides them with excellent exploration capability to avoid local optima, and the deterministic nature ensures that the solution would converge at the end of the iterations [6]. Therefore, the metaheuristic algorithms find their application in solving highly multi-modal, NP-hard, non-linear, and non-convex CSTHTS optimization problems in contrast to the conventional algorithms.

The most-promising and -used algorithm that appeared in the class of metaheuristic algorithms to solve different CSTHTS problem in the literature is PSO [5], proposed in [8]. It has gained wide fame because of the simplicity of its update process and the ease of coding for optimization problems [4]. The PSO algorithm is inspired by the food-searching technique of bird flocks and fish swarms. To date, many variants of it have been created and applied to the CSTHTS problem to obtain good results [4]. Figure 1 shows the distribution of the published articles on the CSTHTS problem using PSO and its variants.

Recently, the CSTHTS problem has been investigated using metaheuristic algorithms, including teaching–learning-based optimization [9], the chaotic hybrid differential evolution algorithm [10], the sine–cosine algorithm [11], grey wolf optimization [12], modified crisscross and binary particle swarm optimization [13], and a hybrid of the fish swarm and real coded genetic algorithm [14]. These algorithms provide a good approximation of the optimal global solution. However, they do not guarantee a robust solution more persistently.

The shortcomings in solving the CSTHTS problem with metaheuristic algorithms are that these algorithms have a high computational time and the final result obtained at the end of the iterations for every run applied to the same optimization problem is primarily different. If the result obtained is the same as the previous one, then the trajectory to reach the same answer would be different [7]. Therefore, a fast optimization method is needed to obtain a robust solution persistently with less Big O time complexity. Moreover, to demonstrate the superiority of one algorithm over another in solving the CSTHTS problem, rigorous statistical tests are needed because of the inclusion of stochasticity in their model [15,16,17]. The flowchart shown in Figure 2 presents the work performed in the literature to solve the CSTHTS problem and the convenience of the proposed technique over the previously applied techniques. The proposed fast economic dispatch method is deterministic and based on fuzzy logic to dispatch the thermal load demand of the CSTHTS problem among the available thermal units robustly. Being deterministic in nature, the formation of lookup charts for the thermal load demand is made possible, which is then used by the improved APSO to optimize the hydel powers of the reservoir-based generation units to meet the load demand. This proposed technique, in contrast to previously applied optimization algorithms to solve the CSTHTS problem, reduces the computational time of the improved APSO by avoiding the need to perform thermal dispatch, as it will be done implicitly by fetching the values from the lookup chart formed by the fast economic dispatch method.

To provide a robust solution more persistently and to make possible stochasticity and Big O time complexity reduction, this work proposes a novel deterministic economic dispatch of thermal units applied to the CSTHTS problem in combination with the improved APSO. Furthermore, it shows the importance of performing rigorous statistical tests to establish the supremacy of one metaheuristic algorithm over the other in solving the CSTHTS problem.

This article makes the following original contributions:

• A new fast deterministic economic dispatch method is proposed for the thermal units of the CSTHTS problem, which is applied in combination with the improved APSO to provide a more persistent solution by making possible stochasticity and Big O time complexity reduction.
• The lookup chart was made and used for the first time to mitigate the nested metaheuristic algorithm implementation to decrease the Big O time complexity and stochasticity occurring in solving the thermal dispatch part of the CSTHTS problem.
• The performance evaluation of the metaheuristic algorithms in solving the considered test case of the CSTHTS problem is presented for first time by performing the rigorous statistical parametric independent samples t-test and non-parametric Mann–Whitney U-test.

The paper is structured as follows: Section 2 presents the problem formulation of the CSTHTS problem. Section 3 and Section 4 describe the APSO and PSO algorithm with improved parameter control, respectively. Section 5 discusses the improved constraint handling technique. Section 6 describes the newly proposed economic load dispatch method for thermal units. Section 7 presents the methodology. Section 8 shows the results of the implementations. Section 9 gives the discussion on the results achieved, and finally, Section 10 concludes the paper.

2. Problem Formulation

CSTHTS is an economic dispatch problem in which the power of hydel and thermal generation units connected at a single bus in the power system network to meet the load demand is scheduled for a specific period to minimize the overall operational cost while fulfilling all the power system network constraints.

The CSTHTS problem is a single-objective optimization problem, which can be modeled by Equations (1)–(10), as mentioned in [3]. The main objective of CSTHTS is to minimize the thermal fuel cost of the power system, as given in Equation (1).

$$min\left(f\right)=\sum _{m=1}^N\sum _{i=1}^T{n}_m{F}_{(m,i)}$$

(1)

where f is the total thermal generation units’ cost, $n_m$ is the number of hours in the scheduling interval m, $F_{(m,i)}$ is the fuel cost at the mth scheduling interval of the ith thermal generation unit, N is the total number of scheduling intervals, and T is the total number of thermal generation units.

The equality constraint of the CSTHTS problem is shown in Equation (2), which ensures that the total thermal $P_{t{h}_{i,m}}$ and hydel $P_{hy{d}_{j,m}}$ power generated must be equal to the sum of transmission line losses $P_l$ and load demand $P_d$.

$$\sum _{m=1}^N\left(\sum _{i=1}^{N_s}{P}_{t{h}_{i,m}}+\sum _{j=1}^{N_h}{P}_{hy{d}_{j,m}}\right)=P_d+P_l$$

(2)

where $N_s$ and $N_h$ are the total number of thermal and hydel generation units, respectively, and N is the total time of the scheduling intervals.

The hydel power $P_{hy{d}_{j,m}}$ is calculated from the volume $V_{hy{d}_{j,m}}$ and discharge $Q_{hy{d}_{j,m}}$ rate of the jth reservoir at the mth scheduling interval, as shown in Equation (3).

$$\begin{array}{c}\hfill P_{hy{d}_{j,m}}=\left\{\begin{array}{cc}& C_1^j·V_{hy{d}_{j,m}}^2+C_2^j·Q_{hy{d}_{j,m}}^2+C_3^j(V_{hy{d}_{j,m}}·Q_{hy{d}_{j,m}})+C_4^j·V_{hy{d}_{j,m}}+\hfill \\ & C_5^j·Q_{hy{d}_{j,m}}+C_6^j\hfill \end{array}\right.\end{array}$$

(3)

where $C_1^j$, $C_2^j$, $C_3^j$, $C_4^j$, $C_5^j$, and $C_6^j$ are the hydel generation coefficients.

The limits of the thermal power $P_{t{h}_{i,m}}$ and hydel power $P_{hy{d}_{j,m}}$ generated by the ith thermal and jth hydel generation units at the mth scheduling interval are presented by Equations (4) and (5).

$$P_{t{h}_i}^{min}\le P_{t{h}_{i,m}}\le P_{t{h}_i}^{max}$$

(4)

$$P_{hy{d}_j}^{min}\le P_{hy{d}_{j,m}}\le P_{hy{d}_j}^{max}$$

(5)

where $P_{t{h}_i}^{min}$ and $P_{t{h}_i}^{max}$ are the minimum and maximum thermal power, respectively, that can be generated by the ith thermal generation unit. $P_{hy{d}_j}^{min}$ and $P_{hy{d}_j}^{max}$ are the minimum and maximum hydel power, respectively, that can be drawn from the jth hydel generation unit. The operation of the hydel power units is presented by Equations (6) and (7).

$$V_{hy{d}_j}^{min}\le V_{hy{d}_{j,m}}\le V_{hy{d}_j}^{max}$$

(6)

$$Q_{hy{d}_j}^{min}\le Q_{hy{d}_{j,m}}\le Q_{hy{d}_j}^{max}$$

(7)

where $V_{hy{d}_j}^{min}$ and $V_{hy{d}_j}^{max}$ are the minimum and maximum value of the allowable volume $V_{hy{d}_{j,m}}$ of the jth hydel generation unit at the mth scheduling interval, respectively. $Q_{hy{d}_j}^{min}$ and $Q_{hy{d}_j}^{max}$ are the minimum and maximum value of the allowable discharge rate $Q_{hy{d}_{j,m}}$ of the jth hydel generation unit at the mth scheduling interval, respectively.

Equation (8) presents the overall value of the permissible discharge rate of the $j\mathrm{th}$ hydel generation unit during a total period of N scheduling intervals.

$$\sum _{m=1}^N{Q}_{hy{d}_{j,m}}=Q_{hy{d}_{j,total}}$$

(8)

The equation that balances the discharge rates and volumes of the hydel generation unit is the continuity equation and is presented by Equation (9).

$$\begin{array}{c}\hfill V_{hy{d}_{j,m+1}}=\left\{\begin{array}{cc}\hfill & \sum _{a=1}^{R_{u,j}}\left(Q_{{hyd}_{a,(m-t)}}+S_{{hyd}_{a,(m-z)}}\right)+V_{hy{d}_{j,m}}+I_{hy{d}_{j,m}}-\hfill \\ & Q_{hy{d}_{j,m}}-S_{hy{d}_{j,m}}\hfill \end{array}\right.\end{array}$$

(9)

where m is the scheduling period, j is the available hydel unit for power generation, $I_{hy{d}_{j,m}}$ is the inflow for the jth hydel unit at the mth scheduling period, $S_{hy{d}_{j,m}}$ is the spillage water of the jth hydel unit at the mth scheduling period, $R_{u,j}$ is the number of upstream hydel units present for the jth hydel power generating station, and z is the transport delay of water from hydel unit a to hydel unit j.

The $F_{(m,i)}$ is the thermal cost of the ith unit at the mth scheduling period. It is determined by the thermal power demand $P_{t{h}_{(i,m)}}$ of the ith unit at the mth scheduling period and is shown by Equation (10).

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{(m,i)}=& a_i+b_i{P}_{t{h}_{(i,m)}}+c_i{P}_{t{h}_{(i,m)}}^2+\left|d_{i}sin\left\{e_i(P_{t{h}_i}^{min}-P_{t{h}_{i,m}})\right\}\right|\hfill \end{array}\end{array}$$

(10)

where $a_i$, $b_i$, $c_i$, $d_i$, and $e_i$ are the thermal generation coefficients. Depending on the type of thermal generation unit, Equation (10) may be of a higher order, increasing the non-linearity of the CSTHTS problem.

The test case considered in this article was taken from [18]. It has four reservoir-based cascaded hydel and three thermal generation units for the production of electricity to meet the load demand. In this test case, the CSTHTS problem is a one-day-long scheduling problem in which each hour is considered as one interval, making a total of 96 decision variables and 24 scheduling intervals having varying load demands. The influence of the thermal generation units’ valve point loading is considered, and transmission line losses are ignored. The pictorial representation of the operation of the cascaded hydel generation units for the selected test case is shown in Figure 3.

Where I₁, I₂, I₃, and I₄ represent the inflow, V₁, V₂, V₃, and V₄ represent the volume, and D₁, D₂, D₃, and D₄ represent the discharge rate of Reservoir 1, Reservoir 2, Reservoir 3, and Reservoir 4, respectively.

3. Improved Accelerated Particle Swarm Optimization Algorithm

The improved APSO algorithm was proposed in [19]. It is a very promising version of the canonical APSO presented in [6]. The strength of this algorithm is that it has only one update equation, which makes it simple to apply to optimization problems and also results in a short computational time compared to other complex metaheuristic algorithms [4]. The update equation of the improved APSO algorithm was used in this work to update 96 decision variables, which are the discharge rates of the hydel generation units, as shown in Equation (11).

$$x_i^{t+1}=((1-\beta \left(t\right))\times p_i^t)+(\beta \left(t\right)\times g^t)+(\alpha \left(t\right)\times R_i^t)$$

(11)

where $\alpha$ and $\beta$ are the tuning parameters, $p_i^t$ is the local best value of the ith particle at iteration t, $g^t$ is the global best value among all the generated particles, and $R_i^t$ is a matrix of random numbers with $(p$ × $d)$ dimensions generated normally, as given in [19], where p is the population size and d is the total scheduling periods of the CSTHTS problem.

The values of $\alpha$ and $\beta$ should be taken in between 0 and 1 because increasing them beyond 1 could cause the APSO algorithm to give a divergent solution, and a value less than 0 makes the value negative and may lead the APSO algorithm to an undefined state [6]. The value of $\alpha$ decides the degree of randomization involved in finding the optimal global solution. The higher the value of $\alpha$, the greater would be the randomization involved to avoid premature convergence, and the lower the value of $\alpha$, the lesser would be the randomization involved, increasing the chance of being stuck at the optimal local solution [4]. The $\beta$ value decides the degree of influence taken from the particles’ local best and global best in finding the optimal global solution. A higher value of $\beta$ means more significant influence comes from the global best, and a lower value means more significant influence comes from the particles’ local best [4].

The values of $\alpha$ and $\beta$ are taken as fixed in the canonical version, but it has been established that the varying values of $\alpha$ and $\beta$ in correspondence to the iterations result in giving a better approximation of the global optimal solution [4]. Therefore, in this article, the parameter control of the values of $\alpha$ and $\beta$ were utilized, as given in Equations (12) and (13).

$$\alpha ={\alpha }_{max}-\left(({\alpha }_{max}-{\alpha }_{min})\times \frac{t_{current}}{t_{total}}\right)$$

(12)

$$\beta ={\beta }_{min}+\left(({\beta }_{max}-{\beta }_{min})\times sin\left(\frac{\pi }{2}\times \frac{t_{current}}{t_{total}}\right)\right)$$

(13)

where the values of ${\alpha }_{max}$, ${\alpha }_{min}$, ${\beta }_{max}$, and ${\beta }_{min}$ are taken as 0.81, 0.62, 0.81, and 0.62, respectively. At the start of the iterations, the value of $\alpha$ is large to increase the exploration, and a linear step reduction is made towards the end of the iterations to make the APSO algorithm converge. In the case of $\beta$, the value is kept small at the start of the iterations to have more influence from the particles’ local best to keep the solution space diversity alive, and a sinusoidal step increment is made towards the end of the iterations to have greater influence from the global best to make the particles converge. The test case of the CSTHTS problem considered in this study was simulated for 50 trials having 75 particles and $\mathrm{10,000}$ iterations using the above-mentioned improved APSO algorithm in combination with the proposed fast economic dispatch method.

4. Particle Swarm Optimization Algorithm

To make the proper rigorous statistical comparison of the proposed technique in solving the CSTHTS problem possible, the authors solved the test case with another metaheuristic algorithm, PSO, in combination with the fast economic dispatch method to obtain the required dataset for comparison. The PSO is a brilliant algorithm in its canonical form and has been applied to solve several non-linear, non-convex, and multi-modal optimization problems in the past [5]. There are almost more than two dozen variants of the PSO algorithm reported in the literature [6]. In this work, a new variant of PSO was used and applied to the CSTHTS problem by multiplying a uniform random number ${ϵ}_0$ in between 0 and 1 with the inertial factor w of the particles instead of the velocity ‘v’ to provide a better approximation of the optimal global solution. The update equation of the variant of the PSO is given as Equation (14).

$$\begin{array}{cc}\hfill v_i^{t+1}=& (w\times {ϵ}_0)+(\alpha \left(t\right)\times {ϵ}_1\times (g^t-x_i^t))+(\beta \left(t\right)\times {ϵ}_2\times (p_i^t-x_i^t))\hfill \end{array}$$

(14)

It has been established that the varying values of w, $\alpha$, and $\beta$ in correspondence with the iterations result in giving a better approximation of the global optimal solution [4]. Therefore, the parameter control of the tuning parameters was utilized in this work, as given in Equations (15)–(17).

$$w=w_{max}-\left((w_{max}-w_{min})\times \frac{t_{current}}{t_{total}}\right)$$

(15)

$$\alpha ={\alpha }_{max}-\left(({\alpha }_{max}-{\alpha }_{min})\times \frac{t_{current}}{t_{total}}\right)$$

(16)

$$\beta ={\beta }_{min}+\left(({\beta }_{max}-{\beta }_{min})\times sin\left(\frac{\pi }{2}\times \frac{t_{current}}{t_{total}}\right)\right)$$

(17)

where the values of $\alpha$ and w are linearly decreased from ${\alpha }_{max}=2.05$ to ${\alpha }_{min}=1.95$ and $w_{max}=0.1$ to $w_{min}=0$, respectively, and the value of $\beta$ is increased sinusoidally from ${\beta }_{min}=1.95$ to ${\beta }_{max}=2.05$$\left(0\mathrm{to}\frac{1}{4}\mathrm{of}\mathrm{wavelength}\right)$. The test case of the CSTHTS problem considered in this study was simulated for 50 trials having 75 particles and 10,000 iterations using the above-mentioned PSO algorithm in combination with the proposed fast economic dispatch method.

5. Constraint-Handling Technique

The most-challenging part while solving the CSTHTS problem is making the values of the volumes and discharge rates for the hydel power generation units within the defined limits, as mentioned in Equations (6)–(8). The volumes and discharge rates of the hydel generation unit for each scheduling interval should be balanced by the continuity Equation (9). Moreover, the final volume of each hydel generation unit at the end of the scheduling period must be fixed to a defined value. To satisfy all the equality and inequality equations and the final volume of each hydel generation unit, many different constraint-handling techniques such as the penalty factor approach have been presented in the literature [20].

The proper selection of the constraint-handling technique to keep the discharge rates and volumes within the defined boundaries improves the results obtained by applying a certain algorithm to the CSTHTS problem. The constraint-handling technique used to make the values of the discharge rates and volumes of the hydel generation units within the defined boundaries during the iterative process of finding the optimal solution set for the CSTHTS problem is as follows:

• Initialize the discharge rate matrix randomly, having each element value within the defined limit for each hydel generation unit, having rows equal to the scheduling intervals and columns equal to the particles generated (population size).
• For each discharge rate matrix of the hydel generation units, randomly select any entry of the discharge rate out of all entries from each column and make it a dependent entry, then find out the value of that dependent entry using Equation (18), assuming that the dependent entry is 24, which is derived from continuity Equation (9).

  $$Q_{i,24}=V_{i,initial}-V_{i,24}-\sum _{j=1}^{23}{Q}_{i,j}+\sum _{j=1}^{24}{I}_{i,j}+\sum _{j=1}^{24}\sum _{k=1}^{u_i}{Q}_{k,j-z}$$

  (18)

  where $I_{i,j}$ is the inflow in the ith reservoir in scheduling interval j, $u_i$ is the number of upstream reservoirs available for the ith reservoir, z is the transport delay of water from the kth hydel generation unit to the ith hydel generation unit, and $Q_{k,j-z}$ is the discharge rate of upstream hydel unit k for the ith hydel unit.
• If the dependent entry is within the limit mentioned in Equation (6), proceed normally according to the iterative process. Otherwise, make it in the limit using the steps mentioned in Algorithm 1.

Algorithm 1 Pseudo-code for the constraint handling technique, influenced by [4]

If ($Q_{min}$ > $Q_{i,m}$)        Find the difference = $Q_{min}$ − $Q_{i,m}$        Make $Q_{i,m}$ = $Q_{min}$     For m = 23 to 1        Adjust the difference in the remaining 23 entries while ensuring they are in the limits.        Start from the 23rd entry to the 1st entry of the discharge rate matrix.     End For End If If ($Q_{max}$ < $Q_{i,m}$)        Find the difference = $Q_{i,m}$ − $Q_{max}$        Make $Q_{i,m}$ = $Q_{max}$     For m = 1 to 23        Adjust the difference in the remaining 23 entries while ensuring they are in the limits.        Start from the 1st entry to the 23rd entry of the discharge rate matrix.     End For End If

6. Proposed Fast Economic Dispatch Method

The economic dispatch of thermal generation units is one of the major tasks performed in power system operation and control to minimize the electricity generation cost. Many algorithms have been developed to efficiently perform the economic dispatch of thermal generation units, and researchers are still finding new algorithms to perform the economic dispatch more efficiently than the previous ones because the economic dispatch problems are NP-hard problems [21]. The economic dispatch of thermal generation units is a subproblem of CSTHTS having more than one thermal generation unit.

The test case considered has more than one thermal generation unit for electricity generation. Utilization of a lookup chart to dispatch different thermal load demands among thermal generation units to solve the CSTHTS problem has not been presented in the literature; instead, metaheuristic algorithms are again used at runtime for the thermal dispatch part of the CSTHTS problem, significantly resulting in the increase of stochasticity and computational time.

It was found that the lookup charts cannot be formed using metaheuristic algorithms as the final dispatch values for each load demand keep on changing for each run of the metaheuristic algorithm. Secondly, one has to find the optimized values of the tuning parameters for each thermal load demand that need to be dispatched among the thermal generation units, which in itself forms many optimization problems that need to be solved first before optimizing the given problem. Therefore, a new fuzzy-logic-based fast deterministic economic dispatch method to make a lookup chart is proposed to reduce the computational time and stochasticity. The time complexity in performing the thermal power dispatch by utilizing the proposed lookup chart technique is $\mathit{O}\left(1\right)$, which is extremely less than the time complexity $\mathit{O}(I\times P)$ of metaheuristic algorithms performing the thermal power dispatch, where I is the number of iterations and P is the number of particles generated. In combination with the improved APSO and PSO algorithms, the proposed technique provides a robust solution with a low standard deviation in less time. This algorithm was designed by looking at the minimum peaks in the fuel cost curves of the thermal generation units, as shown in Figure 4.

The steps of the proposed algorithm to perform the economic dispatch of the thermal generation units are as follows:

• Determine the minimum peaks’ power values of the fuel cost curve occurring for each of the three thermal generation units.
• Choose the peaks of the thermal unit X and compare them with the load demand given to dispatch among the thermal generation units.
• Select the peak value closer to the load demand as the power value for thermal unit X.
• Find the remaining load demand by subtracting the power value of thermal unit X from the total load demand.
• Choose the peaks of the thermal unit Y, and compare all of them with the remaining load demand.
• Select the peak value closer to the remaining load demand as the power value for thermal unit Y.
• Find the remaining load demand by subtracting the power value of thermal unit Y and thermal unit X from the total load demand.
• Choose the peaks of thermal unit Z, and compare all of them with the remaining load demand.
• Select the peak value closer to the remaining load demand as the power value for thermal unit Z.
• Repeat the steps from Step 2 to Step 9 for the six combinations of thermal units shown in

  Table 1. Combination of thermal generation units to select the minimum peak power values to perform economic dispatch.

  Combination	Thermal Unit X	Thermal Unit Y	Thermal Unit Z
  One	Thermal Unit 1	Thermal Unit 2	Thermal Unit 3
  Two	Thermal Unit 1	Thermal Unit 3	Thermal Unit 2
  Three	Thermal Unit 2	Thermal Unit 1	Thermal Unit 3
  Four	Thermal Unit 2	Thermal Unit 3	Thermal Unit 1
  Five	Thermal Unit 3	Thermal Unit 2	Thermal Unit 1
  Six	Thermal Unit 3	Thermal Unit 1	Thermal Unit 2

  .
• Find the mismatch value of power for each combination of Table 1 by subtracting the sum of the power values selected for the three thermal units from the total load demand.
• Compensate the mismatch value for each combination in Table 1 by adding it equally to the selected peak power values.
• Evaluate the cost of each combination in Table 1 for the values of the power obtained for the three thermal generation units in Step 12.
• Select the combination for which the cost obtained is minimum as compared to the cost of the other combinations in Table 1.
• The selected combination in Step 14 gives the optimized power values for the three thermal generation units for a given load demand.

The dispatched fuel cost curve for the thermal generation units obtained after utilizing the above-mentioned economic dispatch method is shown in Figure 5. It can be seen that the sinusoidal valve point loading effect of the thermal generation units approximately tends towards linearization (smooth curve) using the proposed economic dispatch method.

The computational time required to form the lookup chart for all the thermal load demands ranging from 20 MW to 975 MW with different sampling sizes by implementing the proposed economic dispatch method is shown in

Table 2. Computational time required to make a thermal load demand lookup chart for different sample sizes.

Sampling Size (MW)	Computational Time (s)
1	0.2
0.1	0.7
0.01	3.25
0.001	26

.

It can be seen that, as the sample size decreases, the computational time increases, but not too much, and remains acceptable. Making the lookup chart is a one-time task because of the deterministic nature of the proposed economic dispatch method; the time required for smaller sample sizes is still acceptable. However, a smaller sample size is not required to deal with the practical scheduling problems because of the physical limitation of maneuvering between the power values of thermal generation units distanced at smaller sample sizes. Therefore, the proposed economic dispatch method has the highly favorable advantages of being stochastic-independent and time-efficient in solving the CSTHTS problem to provide a robust solution.

7. Methodology

The standard benchmark test case of the CSTHTS problem was solved by implementing the suggested variants of the APSO and PSO algorithms along with the utilization of the lookup charts formed by the newly proposed fast economic dispatch method using the following steps:

• Randomly initialize the discharge rate vectors (particles) for 24 scheduling intervals within the defined limits, as mentioned in Equations (7)–(9). The volume vectors can also be the candidate particles for solving the CSTHTS problem. However, they were not selected as particles to solve the CSTHTS problem in this work.
• Calculate the volume vectors for the hydel generation units from the discharge rate vectors within the defined limits using the constraint-handling technique discussed in Section 5.
• Calculate the hydel generation for 24 scheduling intervals of each hydel generation unit from the discharge rate and volume values, as mentioned in Equation (3).
• Calculate the load demand for thermal generation by subtracting the sum of hydel generation obtained in Step 3 from the total load demand for each scheduling interval, as mentioned in Equation (2).
• Calculate the cost of generation using the lookup chart formed by the newly proposed economic dispatch method discussed in Section 6 for each particle.
• Select the minimum cost value and its corresponding vector (particle) as the global best.
• Select the minimum cost value obtained so far by each particle as the particles’ local best.
• Update all the particle values using the update equation of the suggested variants of the APSO and PSO algorithms, mentioned in Section 3 and Section 4.
• Iterate from Step 2 to Step 8 until the stopping criteria have been reached.
• Generate the result.

The simulation study was performed on an Intel(R) Core (TM) i3-3220 CPU at $3.30$ GHz with 8 GB RAM and in MATLAB. The considered CSTHTS problem was simulated for 50 trials, each for the improved APSO and PSO algorithms having $\mathrm{10,000}$ iterations and a population size of 75. The population size and the number of iterations were chosen based on empirical testing, which produced the best results for the test system.

8. Results

A highly non-convex and multi-modal test case of the CSTHTS problem was solved using the improved APSO and PSO algorithms in combination with the newly proposed fast economic dispatch method. The solution obtained satisfies all the complex and time coupling hydel and thermal network constraints. The convergence behavior of the improved APSO and PSO algorithms applied to the chosen test case of the CSTHTS problem is shown in Figure 6 and Figure 7.

The detailed comparison of the results obtained by implementing the variants of the APSO and PSO algorithms on the selected test case of the CSTHTS problem with the results reported in the literature is shown in

Table 3. Detailed cost comparison of different algorithms applied to the CSTHTS problem.

Algorithm	Minimum Cost (USD)	Average Cost (USD)	Maximum Cost (USD)
Fuzzy EP [22]	45,063.000	NA	NA
DE [23]	44,526.100	NA	NA
MDE [23]	42,611.140	NA	NA
PSO [24]	42,474.000	NA	NA
IPSO [25]	44,321.236	NA	NA
CSA [25]	42,440.574	NA	NA
RCGA-AFSA [14]	$\mathrm{40,913.000}$	$\mathrm{41,236.000}$	$\mathrm{41,363.000}$
RCGA [14]	$\mathrm{42,886.000}$	$\mathrm{43,032.000}$	$\mathrm{43,261.000}$
TVAC-PSO [26]	41,263.000	41,324.000	41,374.000
DNLPSO [27]	41,231.000	41,783.000	42,367.000
TLBO [18]	423,85.880	42,407.230	42,441.360
GWO [12]	41,700.000	41,901.400	42,035.000
GWO-MS [12]	41,501.000	41,600.800	41,750.000
SCA [11]	41,471.900	41,473.520	41,476.600
MCPSO-BPSO [13]	41,181.000	41,308.000	41,616.000
PSO	41,563.506	41,809.772	42,142.861
Improved APSO	41,178.296	41,342.470	41,576.270

NA → Not Available.

. It can be observed that the costs obtained by implementing the variants of the APSO and PSO algorithms are better than the cost obtained by many other algorithms applied to the CSTHTS problem reported in the literature [11,12,13,22,23,24,25,26,27]. The metaheuristic algorithms are stochastic in nature, due to which the final answers obtained by them for each run on the same optimization problem mostly appear different. Therefore, to make a performance comparison between the improved APSO and PSO algorithms, the statistical parametric independent samples t-test and non-parametric Mann–Whitney U-test were performed on the dataset obtained by running the improved APSO and PSO algorithms each 50 times on the selected test case of the CSTHTS problem.

Table 4. Statistical non-parametric Mann–Whitney U-test results.

Test Statistics
Mann–Whitney U	$2.000$
Wilcoxon W	$1277.000$
Z	$-6.603$
Asymp. Sig. (2-tailed)	$0.000$

and

Table 5. Rank statistics non-parametric Mann–Whitney U-test results.

Ranks
Group	N	Mean Rank	Sum of Ranks
Improved APSO	50	25.54	1277.00
PSO	50	75.46	3773.00
Total	100

show the result of the non-parametric Mann–Whitney U-test. The result of the parametric independent samples t-test is shown in

Table 6. Group statistics results of independent samples t-test.

Group Statistics
Group	N	Mean	Std. Deviation	Std. Error Mean
Improved APSO	50	41,342.46946	88.87130686	12.56830075
PSO	50	41,809.77222	134.8096419	19.06496239

and

Table 7. Independent samples t-test for equality of means results.

Independent Samples Test	Levene’s Test for Equality of Variances				t-Test for Equality of Means
	F	Sig.	t	df	Sig. (2-Tailed)	Means Difference	Std. Error Difference	95% Confidence Interval of the Difference
								Lower	Upper
Equal variances assumed	7.808	0.006	−20.46	98	0.00	−467.302 764	22.8349 5073	−512.617 981	−421.987 548
Equal variances not assumed			−20.46	84.824	0.00	−467.302 764	22.8349 5073	−512.706 125	−421.899 404

.

Table 8. Power flow and cost optimization for the CSTHTS problem by implementing the improved APSO variant (a).

Interval	Q1 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	Q2 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	Q3 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	Q4 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	V1 $\times {10}^4{\mathbf{m}}^3$	V2 $\times {10}^4{\mathbf{m}}^3$	V3 $\times {10}^4{\mathbf{m}}^3$	V4 $\times {10}^4{\mathbf{m}}^3$
1	10.35834621253340	7.86511864696029	29.88575564405790	7.13813014401904	99.6416537874666	80.1348813530397	148.2142443559420	115.6618698559810
2	6.06124270768322	6.45856110668036	29.99925473585590	6.01345535575891	102.5804110797830	81.6763202463594	126.4149896200860	112.0484145002220
3	8.28380269456061	6.03516551537061	29.96095666274160	6.00106909922879	102.2966083852230	84.6411547309887	110.8123791698780	107.6473454009930
4	8.24411848998457	6.85511820422048	17.10735374428990	6.10607867016303	101.0524898952380	86.7860365267683	109.6313867802320	101.5412667308300
5	8.24691397515824	6.05381443664776	16.81707911362790	6.25625445562105	98.8055759200799	88.7322220901205	110.5566714678450	125.1707679192670
6	10.93314544831380	8.48842011006336	16.89920319870660	12.43305868122130	94.8724304717661	87.2438019800571	111.9367522744930	142.7369639739020
7	7.98734074135218	6.17958734213721	17.21567799238310	12.84999352649430	94.8850897304140	87.0642146379199	112.8231064614890	159.8479271101490
8	8.97305272613104	8.20713835282111	16.09801967884600	16.96040016090000	94.9120370042829	85.8570762850988	115.7120466676050	159.9948806935390
9	9.35121722915190	6.73182711589459	16.73345545078250	16.81209001520290	95.5608197751310	87.1252491692042	116.4543520682380	159.9998697919640
10	7.39225726223708	7.56566133426718	18.39449921568210	16.90890782618960	99.1685625128939	88.5595878349370	114.2124929208240	159.9901651644810
11	8.23041495489539	8.28533814212756	16.79919936795280	17.23189056845270	102.9381475579990	89.2742496928095	115.9716491348440	159.9739525884110
12	6.94773255972133	8.14891695412473	18.38216208612010	16.07265571625170	105.9904149982770	89.1253327386848	113.7135714268560	159.9993165510060
13	9.34338530394867	9.08841287862872	17.13083439476090	16.73356006326110	107.6470296943290	88.0369198600560	116.3788133212570	159.9992119385270
14	9.00908572632989	8.61297251213748	16.26635234379020	18.39420688714820	110.6379439679990	88.4239473479186	118.3455316793160	159.9995042670610
15	7.93614047477703	8.18226312185546	16.82985035964180	16.80036662530620	113.7018034932220	89.2416842260631	122.0079835777480	159.9983370097070
16	10.42481868258380	11.41542616047650	14.57023172133070	18.38080599059890	113.2769848106380	85.8262580655866	127.5352504613750	159.9996931052290
17	9.85315613912648	10.58919453112490	13.74352312323890	17.13072648353220	112.4238286715110	82.2370635344617	132.3408403250510	159.9998010164570
18	9.33289993783836	10.85328022217320	15.68122256603440	16.66108822148590	111.0909287336730	77.3837833122885	137.2666995634560	159.6050651387620
19	5.70570017674210	6.07215736496447	16.22509640902250	16.72315774880200	112.3852285569310	78.3116259473240	143.3101854540360	159.7117577496010
20	9.20442565573267	11.51071349567860	13.60341932519700	18.19420932269590	109.1808029011980	74.8009124516454	150.6288605978030	156.0877801482360
21	5.50171411481344	10.13849407531630	10.01622988312390	16.91415157892080	110.6790887863850	73.6624183763291	159.1716111135940	152.9171516925540
22	6.61201516640393	12.88185297629290	13.75570569056100	19.93224275444090	112.0670736199810	69.7805654000362	162.6924884437300	148.6661315041480
23	5.91419448870659	9.10550796939795	11.06441614685360	19.97561268696800	115.1528791312740	68.6750574306383	169.6404999073690	144.9156152262020
24	5.15287913127433	6.67505743063828	16.39100914908900	18.51903455139900	120.0000000000000	70.0000000000000	170.0000000000000	140.0000000000000

and

Table 9. Power flow and cost optimization for the CSTHTS problem by implementing the improved APSO variant (b).

Demand (MW)	Phydel-1 (MW)	Phydel-2 (MW)	Phydel-3 (MW)	Phydel-4 (MW)	Pthermal-1 (MW)	Pthermal-2 (MW)	Pthermal-3 (MW)	Individual Cost $\left(\mathbf{\$}\right)$
750	87.4610632070913	61.2820228955224	0.0000000000000	142.8499665404980	20.000000000000	299.203478174859	139.203478174859	1558.14115245862
780	61.9619904824369	53.1820195469510	0.0000000000000	124.8559899706130	20.000000000000	292.500004704191	227.500004704191	1767.93920840051
700	77.5548359573697	51.9038723263958	0.0000000000000	120.5412917162360	20.000000000000	292.500004704191	137.500004704190	1531.34095855611
650	76.9468371544748	58.7585744413596	33.6158179470411	115.9550499406590	20.000000000000	294.723728669443	50.000000000000	1298.85112675393
670	76.2718470801085	54.2352667189548	34.9287793569813	139.8403863274910	20.000000000000	294.723728669442	50.000000000000	1298.85112675393
800	87.8268553365352	69.1444492257355	35.2711592121463	228.4781733979480	20.000000000000	40.000000000000	319.279370307697	1337.27565210228
950	73.3980203064325	54.2526984206744	34.6511493212851	247.6981319516080	20.000000000000	292.500004704191	227.500004704191	1767.93920840051
1010	79.0494603149234	66.7216086958158	39.0923054969572	285.1366254923040	20.000000000000	292.500004704191	227.500004704191	1767.93920840051
1090	81.2443437188795	58.1143383097645	37.6643254081952	283.9769925631620	20.000000000000	292.500004704191	316.500004704190	2032.76283973782
1080	70.9110724338915	64.3389187318644	31.0204892716276	284.7295195626170	20.000000000000	292.500004704191	316.500004704191	2032.76283973782
1100	77.4102043501448	69.1047521196849	37.2722282495994	287.2128152805730	20.000000000000	292.500004704190	316.500004704189	2032.76283973783
1150	69.5040100713974	68.2169604105381	30.8481669667138	277.9680469244430	174.487605208969	299.487605208969	229.487605208969	2275.40888509516
1110	85.1551848597465	72.9219775016898	36.4291717684704	283.3543559954380	20.000000000000	296.569657088928	315.569657088928	2045.72717677310
1030	84.0675162150863	70.5202677539673	39.7183525857604	295.6938634451870	20.000000000000	292.500004704191	227.500004704190	1767.93920840051
1010	78.0130351735683	68.4788620493664	39.6251576048097	283.8829451722560	20.000000000000	292.500004704191	227.500004704191	1767.93920840051
1060	92.0869230563228	82.4264597712109	46.5838364467433	295.6013171838700	20.000000000000	296.650733871673	226.650733871673	1781.60491483892
1050	89.0850327832692	76.7189137060104	49.1525657643732	286.4615667756960	20.000000000000	299.290964487874	229.290964487874	1795.65618140108
1120	85.9985652929790	74.6305862075596	47.9608869636010	282.4099615358610	20.000000000000	292.500004704191	316.500004704191	2032.76283973782
1070	60.7198476182527	48.5013847374656	48.7751134523418	283.0036541919400	20.000000000000	292.500004704191	316.500004704191	2032.76283973782
1050	84.8064865416902	75.4103329222418	54.5332343288842	290.4532113119500	20.000000000000	297.398372343048	227.398372343047	1785.61051126601
910	58.7337268438706	68.9519143899836	54.3194215619207	277.9949372042250	20.000000000000	292.500004704191	137.500004704191	1531.34095855610
860	68.1003991624652	76.1510425310689	57.3525648022735	293.6722729877280	20.000000000000	294.723728669442	50.000000000000	1298.85112675393
850	62.8272198862834	60.4335737545745	57.6631568301017	289.7966867014040	20.000000000000	40.000000000000	319.279370307697	1337.27565210229
800	56.4472475818915	47.2549383825380	55.8215978171655	275.7524957019420	20.000000000000	294.723728669440	50.000000000000	1298.85112675393
Total cost ($)								41,178.296790857

show the power flow and cost optimization for the considered test case of the CSTHTS problem by implementing the improved APSO algorithm mentioned in Section 3.

Table 10. Power flow and cost optimization for the CSTHTS problem by implementing the PSO variant (a).

Interval	Q1 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	Q2 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	Q3 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	Q4 $\times {10}^4{\mathbf{m}}^3/\mathbf{h}$	V1 $\times {10}^4{\mathbf{m}}^3$	V2 $\times {10}^4{\mathbf{m}}^3$	V3 $\times {10}^4{\mathbf{m}}^3$	V4 $\times {10}^4{\mathbf{m}}^3$
1	7.42623568722762	7.69765330616148	29.80608755354510	9.13270539179275	102.5737643127720	80.3023466938385	148.2939124464550	113.6672946082070
2	10.64667135140800	7.72784345789273	30.00000000000000	11.08971617461390	100.9270929613640	80.5745032359458	126.4939124464550	104.9775784335930
3	8.39040846030579	6.10622659607950	17.94844863976680	9.99608100295718	100.5366845010590	83.4682766398663	119.9716994939160	96.5814974306361
4	9.60054869160366	9.07872552469969	29.78846135065110	8.64558354416515	97.9361358094549	83.3895511151666	110.5275628008340	87.9359138864710
5	7.72120681096571	6.86682665610047	17.42812177581010	7.47361017205651	96.2149289984892	84.5227244590661	112.2176929432230	110.2683912679600
6	7.61185550774672	7.31728143765288	28.06398647595050	12.72137622101180	95.6030734907425	84.2054430214132	103.8604817549550	127.5470150469480
7	10.31450513002390	8.01689863191699	18.26352436434690	14.24298726844710	93.2885683607185	82.1885443894963	105.3968897262740	131.2524764182670
8	8.72736647657246	6.80359224830012	16.29743793076000	10.02006743848860	93.5612018841461	82.3849521411961	105.5781339593610	151.0208703304300
9	8.80024252451287	8.10926872469328	16.73349247226930	17.42402610703840	94.7609593596332	82.2756834165029	107.4764280547680	151.0249659992020
10	6.32431873883526	8.08917951746842	19.61383920939590	19.29126938192550	99.4366406207979	83.1865038990344	105.6068539538620	159.7976830932270
11	6.34247805527876	9.99163396036437	17.41626460138510	18.26410738089040	105.0941625655190	82.1948699386701	104.7944241252900	159.7971000766830
12	7.54599944269790	7.21020844915692	16.25394290189570	16.49483225913040	107.5481631228210	82.9846614895131	104.9740686869230	159.5997057483130
13	10.42138048452660	8.63819532680409	15.63809174076130	17.15098879214200	108.1267826382950	82.3464661627090	107.7676345189090	159.1822094284400
14	8.79202166461899	8.85863779174274	16.58124215121600	19.55683792887700	111.3347609736760	82.4878283709663	111.7240257707550	159.2392107089590
15	8.72774248166076	6.27053652924975	17.00355231492560	17.78940880065970	113.6070184920150	85.2172918417165	115.3520623895130	158.8660665096850
16	6.75395676315217	8.80463527915517	20.58766743968060	15.27304416316850	116.8530617288630	84.4126565625614	114.1946119412550	159.8469652484120
17	10.60800800392650	11.55929870737120	14.22551731610400	17.08191270695580	115.2450537249360	79.8533578551902	119.5554748985550	158.4031442822170
18	9.26077510137644	10.28818428469150	15.32633997927020	18.28723864774060	113.9842786235600	75.5651735704987	119.2536282116860	156.6971477856930
19	6.96053801412530	8.13945120525827	13.72996698518900	14.56006385135140	114.0237406094350	74.4257223652404	125.9363045095790	159.1406362492670
20	8.22368789168311	11.19121211236810	11.41824946021730	19.90530924536820	111.8000527177510	71.2345102528723	136.3381288581090	159.8229944435790
21	5.39375468936804	12.17550313462620	10.51288816259240	15.46599824034630	113.4062980283830	68.0590071182461	145.0739629943340	158.5825135193370
22	9.04014549229315	8.85350897556900	10.75282653265320	19.60832958147610	112.3661525360900	68.2054981426771	152.6842755586220	154.3005239171310
23	6.34948467353705	8.19015343963477	11.40572461796640	19.98105460322510	115.0166678625530	68.0153447030423	158.8635177423920	148.0494362990950
24	5.01666786255325	6.01534470304234	10.07916636931130	19.46768575931220	120.0000000000000	70.0000000000000	170.0000000000000	140.0000000000000

and

Table 11. Power flow and cost optimization for the CSTHTS problem by implementing the PSO variant (b).

Demand (MW)	Phydel-1 (MW)	Phydel-2 (MW)	Phydel-3 (MW)	Phydel-4 (MW)	Pthermal-1 (MW)	Pthermal-2 (MW)	Pthermal-3 (MW)	Individual Cost $\left(\mathbf{\$}\right)$
750	72.0784991783815	60.3744485426079	0.0000000000000	164.9504904077030	20.000000000000	296.298283257467	136.298283257467	1543.42883573304
780	89.1471802991791	60.7245821147456	0.0000000000000	176.6711358064670	20.000000000000	296.728552941741	136.728552941740	1545.62884909990
700	77.6738306573954	51.7545173996943	35.1740081932226	156.1296111601990	20.000000000000	40.000000000000	319.268040068781	1337.32031792881
650	83.3593594555837	70.1257973623686	0.0000000000000	131.8233619987770	20.000000000000	294.691489386065	50.000000000000	1298.97204286862
670	72.1724978606793	57.5744462793036	33.6844859729983	141.8754485121690	20.000000000000	294.693129575109	50.000000000000	1298.96589115775
800	71.2700998949537	60.3256937505866	0.0000000000000	216.6033738386290	20.000000000000	295.900418829279	135.900418829279	1541.38809486387
950	84.7365592357725	63.4379096255817	27.5260309970654	234.3108558063400	20.000000000000	292.497165789782	227.491487957402	1767.98557447568
1010	77.2193909158885	55.9248945247135	34.2762927628650	209.0620058762150	20.000000000000	297.258709679567	316.258709679567	2049.61908071529
1090	78.0637930707204	64.0350422729856	33.8398047891089	279.9212225411800	20.000000000000	297.570073235347	316.570073235347	2051.37089173240
1080	63.2753799894693	64.4630530093719	21.7035685434958	301.4459874631410	20.000000000000	295.056008598260	314.056008598260	2037.10549878323
1100	64.7228091232745	73.9677788249381	30.4092965229094	294.5918138175180	20.000000000000	298.654153388133	317.654153388133	2057.43674692256
1150	74.1046563771047	58.9325970421629	34.1313328867463	281.0697875139030	173.920542060028	298.920542060028	228.920542060028	2273.93435403529
1110	90.6147628680204	67.0984124087596	36.9284708662465	285.8443799664670	20.000000000000	295.256989920192	314.256989920192	2038.25594536573
1030	82.9605777537205	68.3945238959253	36.1274796831358	302.5469604783200	20.000000000000	292.492619256048	227.477848350497	1768.05982746462
1010	83.0693306649412	53.8893460239460	36.3967049570217	290.3056292662630	20.000000000000	298.169497988333	228.169497988333	1789.71997291599
1060	69.8757678129572	69.2643363714842	20.9329469863054	270.9531126704230	20.000000000000	292.493463748001	316.480381827417	2032.86412002343
1050	93.4317376468780	79.1004468935980	44.2269737320564	284.5745718172740	20.000000000000	299.333138719570	229.333138719570	1795.87848180659
1120	86.2729173088558	70.9490724532172	42.2491748039920	291.7021550242180	20.000000000000	292.456674833798	316.370015038656	2033.43359335351
1070	71.0821901783471	59.2249256052618	47.0576816489922	263.8692326788350	20.000000000000	292.441497213030	316.324482157312	2033.66845064022
1050	79.5381331825137	71.6114740981092	50.7268788009569	305.2564640483580	20.000000000000	296.433527172015	226.433527172015	1780.43729493199
910	58.1187486039353	72.6833609372922	52.1330405788681	271.5076481896120	20.000000000000	297.778605025101	137.778605025101	1550.96731466570
860	84.6511670161003	58.7971267386904	54.1149585606971	297.7831101996970	20.000000000000	294.653645746086	50.000000000000	1299.11398082886
850	66.4249412232629	55.0718598087890	56.0664158006416	293.2758221107800	20.000000000000	40.000000000000	319.160968528557	1337.74242973984
800	55.1765612245909	42.8065750481708	56.2069525800963	281.4483082653400	20.000000000000	294.361611594329	50.000000000000	1300.20932484399
Total cost ($)								41,563.5069148969

show the power flow and cost optimization for the considered test case of the CSTHTS problem by implementing the PSO algorithm mentioned in Section 4. It can be seen that no hydel, thermal, or network constraint was violated.

9. Discussion

The CSTHTS problem is a highly complex, non-linear, non-convex, and multi-modal optimization problem. The selected test case of the CSTHTS problem in this work was solved by forming the deterministic lookup chart for the thermal load demands obtained by the newly proposed fast economic load dispatch method in combination with the improved APSO and PSO algorithms. The comparison of the minimum cost, maximum cost, and average cost obtained presented in Table 3 shows that the minimum cost achieved by the improved APSO is better than the minimum cost achieved by PSO and other state-of-the-art algorithms, which manifests the effectiveness of the proposed technique.

The time complexity $\mathit{O}(I\times P\times D)$ for the considered test case of the CSTHTS problem is limited to the time complexity of a single metaheuristic algorithm, which is a significant improvement as compared to the time complexity $\mathit{O}(I\times P\times D\times (I_1\times P_1))$ of the nested metaheuristic algorithm implementation on the considered test case of the CSTHTS problem, where I, P, and D are the iterations, particles, and dimensions of the problem, respectively. $I_1$ and $P_1$ are the iterations and particles for the nested-metaheuristic-algorithm-based economic dispatch part of the CSTHTS problem, which the proposed technique avoids in this article. The CPU runtime mainly depends on the computer system specifications and the software used for simulation purposes. The comparison of the CPU runtime cannot be made with other algorithms reported in the literature as they used different software and computer system for simulation. Hence, it is convenient to the present time complexity in Big O notation. However, the CPU runtime is approximately 45 s in simulating one trial for the considered CSTHTS problem, which is infinitesimally small compared to the day-ahead scheduling problem having an interval of 1 hour.

The power flow and cost optimization for the CSTHTS problem obtained by implementing the improved APSO algorithm is shown in Table 8 and Table 9, and the power flow and cost optimization for the CSTHTS problem obtained by implementing the PSO algorithm is shown in Table 10 and Table 11. It can be observed that no constraint was violated, and all parameters are within the allowed limits. It was further observed that the parameter control of the tuning parameters gives significantly better results than the constant values of the tuning parameters while solving the CSTHTS problem. The convergence behavior of the APSO and PSO algorithms to optimize the CSTHTS problem is shown in Figure 6 and Figure 7, respectively. The oscillatory trend seen in the convergence behavior of the improved APSO and PSO algorithms is because they did not incorporate greedy search; rather, they took the updated solution of each iteration as it is irrespective of being better or worse than the previous iteration’s solution to increase their searchability in finding the optimal global solution.

It can be noticed in the solution of the CSTHTS problem that, for some intervals, the combination of discharge rates and volumes gave zero values of the hydel power, which in fact became negative because there is a flaw in the mathematical modeling of the considered benchmark test case of the CSTHTS problem [28]. Since this is a non-pumped storage CSTHTS problem, these negative hydel power values mean nothing and were fixed to a zero value. During such intervals, the power plant is simply shut down and the water is spilled out [29]. To date, this issue of the CSTHTS problem has been adopted as it is considered by all the research articles present in the literature, and therefore, the optimal scheduling is taken as it is. Furthermore, the metaheuristic algorithms are stochastic and mostly give different final approximations of the optimal global solution for each run on the same optimization problem. Therefore, it could not be claimed by just comparing the final approximation of the optimal global solution of the two different algorithms that one metaheuristic algorithm is better than the other. For example, two different metaheuristic algorithms were applied and simulated for 100 times on the same optimization problem of minimization; the first one gave the best minimum value to the optimization problem one time out of hundred trials, and most of the time, it gave a bad approximation of the optimal global solution. The second one did not give the best minimum value to the optimization problem, but gave persistently good approximations of the optimal global solution close to the best minimum value achieved by the first algorithm. If this is the case, which would one call the best out of the two metaheuristic algorithms?

This question could be answered by performing the proper statistical tests on the datasets of the final approximate optimal global solutions obtained by the two algorithms applied to the same optimization problem.

Therefore, proper rigorous statistical tests were performed to demonstrate the supremacy of one algorithm over the other algorithm on the selected benchmark test case of the CSTHTS problem in this work, which was previously not present in the literature. The non-parametric Mann–Whitney U-test and parametric independent samples t-test were implemented on the dataset obtained by running 50 times the improved APSO and PSO algorithms. The results obtained are reported in Table 4, Table 5, Table 6, and Table 7, respectively. The results show that the APSO algorithm outperformed the PSO algorithm on the selected test case of the CSTHTS problem statistically. It can be seen that the mean rank of the PSO is greater than that of the improved APSO. Therefore, it was concluded that the PSO algorithm performance is not better than the APSO algorithm on the solved CSTHTS problem. Furthermore, it also establishes that we could not say that an algorithm is best fit for an optimization problem by just looking at the final result; rather, we have to perform a rigorous and proper statistical test to establish its supremacy and reliability.

It can be seen that only the so-far best and robust solution provided in the literature by [14] for the considered benchmark test case of the CSTHTS problem is $0.64\%$ less than the obtained robust solution. The difference is minimal, and it has no significance because of the lack of statistical tests and the stochastic nature of the applied algorithm by [14], as it could not be claimed which algorithm is better than the other.

The technique proposed in this article to solve the CSTHTS problem is better to adopt as it has the highly favorable advantages of being time-effective (the Big $\mathit{O}$ time complexity is reduced) and least-stochastic compared to other applied algorithms in the literature to solve the CSTHTS problem, because the thermal load dispatch part of the CSTHTS is solved deterministically in constant time $\mathit{O}\left(1\right)$ to provide a robust solution, and the improved APSO algorithm solves the hydel dispatch part of the CSTHTS problem. The APSO is very simple and easy to program for any optimization problem with a single update equation and two tuning parameters, resulting in further complexity and computational time reduction to optimize the CSTHTS problem.

10. Conclusions

A highly non-convex, non-linear, and NP-hard CSTHTS optimization problem test case was solved by implementing the newly proposed fast economic dispatch method in combination with the improved APSO algorithm. The result obtained is better with a low standard deviation than the results of the many other state-of-the-art algorithms reported in the literature; the minimum cost achieved was USD 293.604, USD 322.704, and USD 2.704 better than the minimum cost achieved by the SCA, GWO-MS, and MCPSO-BPSO algorithms, respectively. It was established that the proposed technique offers the highly favorable advantages of being time-effective and least-stochastic. The proposed technique reduced the Big O time complexity by solving the thermal dispatch part of the CSTHTS problem implicitly using a lookup chart formed by the fast economic dispatch method. The proposed technique can be applied to real-world power systems to lower the operational cost, as most of the generated power comes from hydel and thermal units. Furthermore, the importance of performing rigorous standardized statistical tests was shown in establishing the superiority of one metaheuristic algorithm over the other. The statistical results obtained established the proposed technique’s supremacy over the other PSO algorithm. In future works, fetching the data entries from the fuzzy-logic-based fast economic dispatch method lookup chart could be further improved by exploring modern data structure techniques to provide a robust solution in a shorter computational time.