Day-ahead combined economic and emission dispatch with spinning reserve consideration using moth swarm algorithm for a data centre load

Abstract

Dynamic combined economic and emission dispatch is an important task in the power system that examines the optimal allocation of power generation resources that yield the least possible fuel and emission costs. In this study, the Moth Swarm Algorithm has been proposed for solving the combined economic and emission dispatch problem for a 24-hour period. The model has been implemented on a test system made up of a combination of thermal and solar photovoltaic plants, while also considering spinning reserve allocation. The results obtained are presented in comparison with commonly used state-of-the-art methods like Moth Flame Optimization, Whale Optimization Algorithm, Ant Lion Optimizer, and Tunicate Swarm Algorithm. Two test systems have been considered in the model implementation. The first system consists of a combination of six thermal plants and thirteen solar plants whose load demand is the hourly energy demand of an anonymous data centre in South Africa. The second test system is made up of three thermal plants and thirteen solar plants to service its unique hourly load demand. Results showed that the proposed MSA gave preference to solar photovoltaic generation over thermal generation given that environmental impact minimization is a major component of the overall objective function. The proposed MSA also scheduled thermal generators to provide the required spinning reserve capacity since solar energy is intermittent in nature. Overall, analyses show that the proposed MSA outperformed the other methods in finding the best fuel, emission, solar generation, and spinning reserve costs that best serve the energy demand of the data centre and the second test system for the 24-hour period.

Economic dispatch; Emission dispatch; Moth swarm algorithm; Solar photovoltaic

1. Introduction and literature review

Economic dispatch is a critical power system operation that involves determining the optimal operational levels of the electricity generation assets to meet a load demand in a power system while complying with all operational constraints [1]. A typical objective of this optimization task is to reduce the fuel costs associated with electricity generation. On the other hand, the emission dispatch task seeks to determine the optimal operational level of the generators in the system with the aim to minimize the environmental footprint [2]. A combination of the economic and emission dispatch tasks therefore forms a multi-objective optimization problem to be solved with the aim to minimize both fuel and emission costs.

There have been attempts to solve the economic dispatch, emission dispatch and Combined Economic and Emission Dispatch (DCEED) optimization problems in the literature using a variety of methods. These methods range from traditional methods like Newton-Raphson and Lambda Iteration to more intelligent methods like metaheuristic algorithms [3], [4]. A common demerit of the traditional methods is their inability to avoid getting trapped in local optima, especially when dealing with very complex and nonlinear systems. Metaheuristics have gained significant popularity over the years due to their improved ability to avoid local optima solutions by approaching every optimization problem as a black box [4]. Some of the metaheuristic methods that have been proposed for solving the energy dispatch problem in previous studies include Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), Moth Flame Optimization (MFO), Whale Optimization Algorithm (WOA), Genetic Algorithm (GA), Gravitational Search Algorithm (GSA), etc. Many studies also remain focused on the design of new metaheuristic methods and this creates an opportunity to solve complex optimization problems such as the Dynamic CEED problem using these new methods, especially since the No Free Lunch theorem suggests no single method is fit to solve all optimization problems [5]. In [6], the authors proposed the use of GA for solving an economic dispatch problem for the power system in Western Algeria. They also accounted for transmission losses in the system by using a Newton-Raphson approach. Results from their study showed that the GA had a superior performance than the conventional methods it was compared with in solving the economic dispatch problem. An improved GA was proposed for solving the dynamic economic dispatch problem for a system comprised of both thermal and wind generation plants in [7]. Results from their study showed the superiority of the improved GA in relation to the conventional GA and other methods like Evolutionary Programming (EP) and Dynamic Programming (DP). A modified PSO was proposed in [8] for solving a nonconvex economic dispatch problem using time-varying operators. Based on results from their study, the improved PSO outperformed the classical PSO in solving the economic dispatch problem.

The authors of [9] proposed the use of GSA for solving an economic dispatch problem in their study. The robustness of the GSA in comparison with other evolutionary methods was displayed based on the results obtained. A hybrid GSA and Differential Evolution (DE) which performed competitively with other methods was also proposed for solving an economic dispatch problem in [10]. In [11], the authors compared the performance of methods like PSO, GA, DE, EP, and Simulated Annealing (SA) in solving a dynamic economic dispatch problem. Their analyses showed that the DE outperformed the other four methods in terms of finding the least fuel cost and having the least execution time. An economic dispatch problem was solved using Tabu Search (TS) for a system where line flows were considered in [12] and the results presented showed that the TS method performed better than the Newton-Raphson and GA methods. The authors of [13] proposed the use of Exchange Market Algorithm (EMA) for solving a combined power and heat dispatch problem, and it was found that the EMA performed superiorly to the GA, PSO, Harmony Search, etc. The authors in [14] also carried out a comparative study on the use of ALO and EMA for solving the economic dispatch problem for a power system. Analyses of results showed a performance tradeoff between the two methods.

The ALO was used to solve a CEED problem in [15]. Although the results presented highlighted the ability of the ALO to solve the CEED problem, the results were not compared with existing methods to have a sense of the relative performance of the proposed method. ALO was also used to solve a DCEED problem for a 24-hour time horizon in [16] where only conventional generators were considered. Likewise, the results presented have not been compared with pre-existing methods. An improved HS method for solving a DCEED problem was proposed in [17]. The model was implemented on a variety of test systems all comprised of conventional generators. Results and analyses showed that the improved HS method outperformed other methods like PSO, DE, GA, Quantum-PSO, etc. for the various test systems considered. In [18], Cuckoo Search Algorithm (CSA) was proposed for solving a DCEED problem. The model was implemented on three different test systems and results show that the proposed CSA outperformed the GA in finding the best fuel and emission costs. In [19], MFO was proposed for solving a CEED problem using two test systems; one comprising of only thermal generators and the other comprising of diesel generators, fuel cells, and wind turbines. The effectiveness of the MFO in solving the CEED problem was estimated by comparing its results with those from other recorded in modern literature. A comparison of the performances of the WOA and PSO in solving a static CEED problem was conducted in [20]. The models were implemented on the standard IEEE 30 bus system and the results obtained showed the superior performance of the WOA to the PSO in solving the CEED problem. Although MSA was used in [21] to solve a CEED problem and showed promising results, the model only considered conventional generators and a static load demand for one hour of the day. The authors in [22] also used MSA to solve a CEED problem with consideration given to valve point effects. Their study also only considered a static load demand and strictly conventional generation options.

Subject to the review of the literature conducted, the contribution of the work presented in this article is such that it considers the use of MSA for solving the day-ahead economic and emission dispatch problem while providing for spinning reserve allocation for a data centre load using a combination of thermal generators and solar photovoltaic plants. Further, for integrating solar photovoltaics in solving the DCEED problem, real-life solar irradiation and environmental data specific to Cape Town, South Africa have been used in implementing the model. Hence, the work presented in this study is focused on the use of MSA for solving a DCEED problem to service a data centre load demand while considering a combination of thermal and solar photovoltaic generation with provision for spinning reserve. The rest of the paper is organized as follows; section 2 presents the DCEED formulation, section 3 presents the MSA methodology, results are presented in section 4 and discussed in section 5. The paper is then concluded in section 6.

2. Problem formulation

2.1. DCEED model

The DCEED mathematical model is described as follows:

Objective function: the objective function of the DCEED problem is represented as a total cost $C_T$ minimization task and is described as follows [23], [24], [25]:

$$Minimize{C}_T=\sum _{t=1}^N(\sum _{i=1}^n{F}_i\left(P_i\right)+E_i\left(P_i\right))$$ (1)

Where $P_i$ is the power output of the ith thermal generator, $F_i\left(P_i\right)$ and $E_i\left(P_i\right)$ are the fuel cost and emission cost of the ith thermal generator, n represents the number of generators in the system, N is the total number of hours in the day, t represents the current hour of the day.

Given that [26], [27];

$$F_i\left(P_i\right)=a_i⁎{\left(P_i^t\right)}^2+b_i⁎\left(P_i^t\right)+c_i+|e_i⁎\sin (f_i⁎(P_{imin}-P_i^t))|\mathrm{\$}/h$$ (2)

$$E_i\left(P_i\right)=x_i⁎{\left(P_i^t\right)}^2+y_i⁎\left(P_i^t\right)+z_i+q_i⁎\exp (r_i⁎P_i^t)\mathrm{\$}/h$$ (3)

Where $a_i$, $b_i$, $c_i$, $e_i$, and $f_i$ represent the fuel cost coefficients of the ith thermal generator and $x_i$, $y_i$, $z_i$, $q_i$ and $r_i$ represent the emission cost coefficients of the ith thermal generator. $P_{imin}$ is the lower bound of the ith thermal generator.

Since the combined economic and emission dispatch problem takes on a multi-objective form, a price penalty factor is introduced to transform the objective function into a single objective function. This price penalty factor is represented as follows [23], [28]:

$$k_i=\frac{a_i⁎{\left(P_{imax}\right)}^2+b_i⁎\left(P_{imax}\right)+c_i+|e_i⁎\sin (f_i⁎(P_{imin}-P_{imax}))|}{x_i⁎{\left(P_{imax}\right)}^2+y_i⁎\left(P_{imax}\right)+z_i+q_i⁎\exp (r_i⁎P_{imax})}$$ (4)

Where $P_{imax}$ is the upper bound of the ith thermal generator.

The DCEED optimization task is to be executed in compliance with some equality and inequality system constraints. These constraints are described as follows:

Power balance constraint: this ensures that the total electricity generated is equal to the load demand while considering losses in the system at a particular hour of the day. This is represented as [28]:

$$\sum _{i=1}^n{P}_i^t-P_{loss}^t-P_d^t=0$$ (5)

$P_d^t$ and $P_{loss}^t$ represent the load demand and power loss respectively in the system at time t.

Where [23];

$$P_{loss}^t=\sum _{i=1}^n\sum _{j=1}^n{P}_i^t{B}_{ij}^t{P}_j^t$$ (6)

$B_{ij}^t$ represents the system's loss coefficient matrix at that $time=t$.

Thermal generator limit constraint: this constraint ensures that all thermal generators in the system are operated within their safe operating limits and is represented as follows [25], [28]:

$$P_{imin}\le P_i^t\le P_{imax}$$ (7)

Ramp rate constraint: the extent to which each thermal generator in the system can be ramped up or down is guided by the following [24], [27]:

$$P_i^t-P_i^{t-1}\le U{R}_i$$ (8)

$$P_i^{t-1}-P_i^t\le D{R}_i$$ (9)

$U{R}_i$ is the ramp-up rate of the ith thermal generator and $D{R}_i$ is the ramp-down rate of the ith thermal generator. The effect of the ramp-up and ramp-down constraints on the output limits of the thermal generators is therefore represented as follows [23]:

$$\max (P_{imin},U{R}_i-P_i^t)\le P_i^t\le \min (P_{imax},P_i^{t-1}-D{R}_i)$$ (10)

DCEED with Solar Photovoltaic Integration

By integrating solar photovoltaics into the system, the objective function is transformed into [24]:

$$Minimize{C}_T=\sum _{t=1}^N(\sum _{i=1}^n{F}_i\left(P_i^t\right)+E_i\left(P_i^t\right)+\sum _{g=1}^m{C}_{gs}(Pg{s}_g^t))$$ (11)

Where $P_{gs}$ is the amount of power produced by the gth solar PV plant and $C_{gs}$ is the cost of solar PV generation.

The amount of power produced by the solar PV plant is calculated using [24], [25]:

$$P_{gs}=P_{rated}\{1+(T_{ref}-T_{amb})⁎\beta \}⁎\frac{P{V}_i}{1000}$$ (12)

Where $P_{rated}$ is the maximum amount of power that can be produced by the solar PV plant, $T_{ref}$ and $T_{amb}$ represent the reference and ambient temperatures. β represents the solar plant's loss coefficient and $P{V}_i$ represents the solar irradiation of the plant's location.

In a system with multiple solar PV generators, the share of solar energy in the system is described as follows [23], [24]:

$$Solarshare=\sum _{g=1}^{m}Pg{s}_g⁎S_g$$ (13)

Where m is the number of solar PV plants in the system, $S_g$ is a binary variable for which 1 represents that the solar PV plant is ON and 0 represents that the plant is OFF.

Hence, cost of solar power generation is represented as [23], [24]:

$$Solarcost=\sum _{g=1}^{m}Cos{t}_g⁎Pg{s}_g⁎S_g$$ (14)

Where $Cos{t}_g$ is the unit cost of solar power produced by the gth PV plant.

Since the objective of the DCEED task is to reduce fuel and environmental costs, it is logical to maximize the utilization of the solar energy resource. To achieve this, the objective function is modified as follows [23]:

$$\begin{gathered} Minimize{C}_T \\ =\sum _{t=1}^N\sum _{i=1}^n(a_i⁎{\left(P_i^t\right)}^2+b_i⁎\left(P_i^t\right)+c_i+|e_i⁎\sin (f_i⁎(P_{imin}-P_i^t))|+k_i\left((x_i⁎{\left(P_i^t\right)}^2+y_i⁎\left(P_i^t\right)+z_i+q_i⁎\exp (r_i⁎P_i^t))\right))+(\sum _{g=1}^{m}Cos{t}_g⁎Pg{s}_g^t⁎S_g^t)+(\sum _{g=1}^{m}Pg{s}_g^t-\sum _{g=1}^{m}Pg{s}_g^t⁎S_g^t) \end{gathered}$$ (15)

The corresponding effect of the modified objective function on the power balance constraint is represented as follows [23]:

$$\sum _{i=1}^n{P}_i^t+\sum _{g=1}^m(Pg{s}_g^t⁎S_g^t)-P_{loss}^t-P_d^t=0$$ (16)

2.2. Introduction of a spinning reserve

The spinning reserve is primarily spare generation capacity that has been set aside and is dispatchable to maintain continuity and security of supply in the power system. Because of the intermittent nature of solar energy, the required spinning reserve capacity is to be provided by the thermal generators in the system.

Hence, part of the objective of the DCEED problem is the minimization of spinning reserve cost and this is described as follows [27]:

$$MinimizeSRCost=\sum _{t=1}^N{C}_{SRi}^t\left(P_{SRi}^t\right)$$ (17)

Where $P_{SRi}^t$ and $C_{SRi}^t$ represent the spinning reserve provided by the ith generator at time t and its associated cost respectively.

Therefore the objective function becomes [27], [29]:

$$\begin{gathered} Minimize{C}_T \\ =\sum _{t=1}^N\sum _{i=1}^n(a_i⁎{\left(P_i^t\right)}^2+b_i⁎\left(P_i^t\right)+c_i+|e_i⁎\sin (f_i⁎(P_{imin}-P_i^t))|+k_i \\ ⁎\left((x_i⁎{\left(P_i^t\right)}^2+y_i⁎\left(P_i^t\right)+z_i+q_i⁎\exp (r_i⁎P_i^t))\right))+(\sum _{g=1}^{m}Cos{t}_g⁎Pg{s}_g^t⁎S_g^t)+(\sum _{g=1}^{m}Pg{s}_g^t-\sum _{g=1}^{m}Pg{s}_g^t⁎S_g^t)+C_{SRi}\left(P_{SRi}^t\right) \end{gathered}$$ (18)

The introduction of spinning reserve results in the following additional constraints:

Allocated spinning reserve being equal to required capacity: this ensures that the total spinning reserve provided by the thermal generators is equal to the spinning reserve $T_{SRreq}^t$ required in the system at time t [27].

$$\sum _{i=1}^n{P}_{SRi}^t=T_{SRreq}^t$$ (19)

Thermal generators providing spinning reserve must operate within their permissible bounds. The bounds are represented as follows [27], [28]:

$$0\le P_{SRi}^t\le \min (U{R}_i,P_{SRimax}^t)$$ (20)

$P_{SRimax}^t$ is the maximum spinning reserve capacity that can be provided by the ith thermal generator at time t, and it is derived as follows [28]:

$$P_{SRimax}^t=P_{imax}-P_i^t$$ (21)

3. Moth swarm algorithm

This is an optimization method inspired by the movement of moths towards the moonlight. In the MSA framework, the position of the light source represents the possible solution to the optimization problem while the fitness of the solution represents the light source's luminous intensity [30]. The MSA also categorizes the population of moths into three types based on the role they play while searching for the optimal solution. These three moth groups are namely pathfinders, prospectors, and onlookers [31]. The pathfinders are moths that can discover new regions of the search space using the First In-Last Out principle. The pathfinders guide the movement of the swarm towards the best positions by describing it as a light source. The prospectors explore areas around the light sources which have been identified by the pathfinders and onlookers move in the direction of the best solution that has been found by the prospectors.

For every iteration, to find the luminous intensity of a light source, every moth is incorporated into the optimization problem. The best fitness values are used to represent the positions of pathfinders, and it is used to guide the movement of the swarm in the following iteration [30]. The optimization process takes place in stages and these are described in the following section.

3.1. Initialization phase

The first solution set is randomly created for a d-dimensional problem and n number of population as follows [30]:

$$x_{ij}=rand(0,1)⁎(x_j^{max}-x_j^{min})+x_j^{min}$$

$$\forall i\in \{1,2,\dots ,n\},j\in \{1,2,\dots ,d\}$$ (22)

Where $x_j^{max}$ and $x_j^{min}$ are the upper and lower bounds of each search agent.

After initialization, the fitness of each moth is calculated and the swarm of moths is divided into three distinct groups. The best performing moths are put in the pathfinder group, the next best performing moths are put in the prospector group, and the least performing moths make up the onlooker group.

3.2. Reconnaissance phase

To avoid being stuck in local optima, MSA delegates a portion of the moths to compulsorily discover less-crowded areas of the search space. The pathfinders usually bear and execute this responsibility through crossover operations and adaptive crossover with levy mutation. These principles are described as follows:

Diversity index for crossover points:

The strategy for selecting crossover points which ensure diversity of solutions is represented as follows [31]:

$${\sigma }_j^t=\frac{\sqrt{\frac{1}{n_p}{\sum }_{i=1}^{n_p}{(x_{ij}^t-\overline{x_j^t})}^2}}{\overline{x_j^t}}$$ (23)

Where ${\sigma }_j^t$ is the degree of dispersal of moths in the jth dimension at iteration t.$\overline{x_j^t}=\frac{1}{n_p}{\sum }_{i=1}^{n_p}{x}_{ij}^t$, and $n_p$ is the number of pathfinders in the population.

The variation coefficient ${\mu }^t$ which represents the relative dispersion is calculated as follows [31]:

$${\mu }^t=\frac{1}{d}\sum _{j=1}^d{\sigma }_j^t$$ (24)

Pathfinders with a low degree of dispersal are added to the crossover points group $c_p$ which changes dynamically as described in the following [30].

$$jϵc_p,if{\sigma }_j^t\le {\mu }^t$$ (25)

Levy flights:

This describes random processes which have the ability to travel over huge distances using variable step sizes. The methodology of the levy flight is adequately described in [30]. For crossover points, the MSA develops a sub-trial vector by perturbing selected elements of the host-vector using related elements of the donor vector. The mutation strategy is represented as follows [21]:

$$\overline{v_p^t}=\overline{v_{r^1}^t}+L_{p1}^t⁎(\overline{x_{r^2}^t},-,\overline{x_{r^3}^t})+L_{p2}^t⁎(\overline{x_{r^4}^t},-,\overline{x_{r^5}^t})\forall r\ne r^2\ne r^3\ne r^4\ne r^5\ne p\in \{1,2,\dots ,n_p\}{x}_{r^1}^t$$ (26)

Where $L_{p1}$ and $L_{p2}$ are the mutation scaling factors which are obtained using the following expression [30], [31]:

$$L_p\sim random\left(n_c\right)\oplus Levy(\alpha )$$ (27)

Adaptive crossover operation based on population diversity:

The host-vector which represents pathfinder solutions updates its position through crossover options by incorporating the mutated variables of the sub-trail vector to the corresponding host vector. The complete trail solution is therefore described as follows [30]:

$$V_k^p=\{\begin{array}{cc}{v}_{pj}^t,\hfill & ifj\in c_p\hfill \\ x_{ph}^t,\hfill & ifj\notin c_p\hfill \end{array}$$ (28)

Hence, the fitness of the trail solutions is computed and compared with the fitness of the host solution. The better-performing solutions are chosen to survive into the next iteration. This process is described as follows for a minimization optimization problem [30], [31]:

$$\overline{x_p^{t+1}}=\{\begin{array}{cc}{\bar{x}}_p^t,\hfill & iff(\overline{V_p^t})\ge f(\overline{x_p^t})\hfill \\ {\bar{v}}_p^t,\hfill & iff(\overline{V_p^t})<f(\overline{x_p^t})\hfill \end{array}$$ (29)

The probability value $P_p$ is derived as follows [31]:

$$P_p=\frac{fi{t}_p}{{\sum }_{p=1}^{n_p}fi{t}_p}$$ (30)

For a minimization problem, the luminescence intensity is derived as follows [30]:

$$fi{t}_p=\{\begin{array}{cc}\frac{1}{1+f_p},\hfill & for{f}_p\ge 0\hfill \\ 1+\left|f_p\right|,\hfill & for{f}_p<0\hfill \end{array}$$ (31)

3.3. Transverse orientation phase

Here, the size of the prospector group over the iterations is decreased as follows [30], [31]:

$$n_f=round((n-n_p)⁎(1-\frac{t}{T}))$$ (32)

Every prospector updates its position as guided by the following [30]:

$$\begin{gathered} x_i^{t+1}=|x_i^t-x_p^t|⁎e^{\theta }⁎\cos \left(2\pi \theta \right)+x_p^t \\ \forall p\in \{1,2,3,\dots ,n_p\};i\in \{n_p+1,n_p+2,n_p+3,\dots n_f\} \end{gathered}$$ (33)

Where $\theta \in [r,1]$ represents a random number used to establish the spiral shape, and $r=-1-(t/T)$.

The methodology of the MSA is based on two major modifications to the Moth Flame Optimization. The first modification is that the MSA tries to reduce the computational cost by dealing with each moth as an integrated unit. For the second modification, the light source is chosen using a probability function with hopes of improving the algorithm's exploitation ability. The type of each moth is dynamically changed in the MSA. If a prospector can find a light source that outperforms the current sources, the prospector moth is then promoted to become a pathfinder. Thereby making sure that there are new lighting sources and moonlight at the end of the phase [21], [30], [31].

3.4. Celestial navigation phase

As the number of prospectors reduces, the number of onlookers $n_o=n-n_f-n_p$ in the group increases proportionally [30]. This has the potential to increase the convergence speed of the MSA towards global optima. The MSA forces the onlookers which are the moths with the least fitness in the group to search the objective space more effectively by critically exploring hot spots of the prospectors. The onlookers are therefore divided as follows:

Gaussian walks:

Here, the focus is on promising areas of the objective space. Due to the ability of Gaussian stochastic distribution to limit distributions of random samples and consequently limiting the variations in the following generation, it is used in this phase. The size of this group of onlookers is calculated as $to{n}_G=round(n_o/2)$. The movement of this onlooker group is described as follows [30], [31]:

$$x_i^{t+1}=x_i^t+{\epsilon }_1[{\epsilon }_2⁎bes{t}_g^t-{\epsilon }_3⁎x_i^t]\forall i\in \{1,2,3,\dots ,n_G\}$$ (34)

Where ${\epsilon }_1$ is a random sample based on Gaussian stochastic distribution, $bes{t}_g$ is the global best solution obtained by both the pathfinders and prospectors. ${\epsilon }_2$ and ${\epsilon }_3$ are uniformly distributed random numbers within the range of [0,1].

Associative learning mechanism with immediate memory:

This helps the moth population to communicate useful information from one generation to the next. The second proportion of onlooker moths are used to move towards the moonlight using associative learning operators. The size of this group is determined using $to{n}_A=n_o-n_G$. The positions of the onlooker moths here are updated as follows [30]:

$$x_i^{t+1}=x_i^t+0.001⁎G[x_i^t-x_i^{min},x_i^{max}-x_i^t]+(1-\frac{g}{G})⁎r_1⁎(bes{t}_p^t-x_i^t)+\frac{2g}{G}⁎r_2⁎(bes{t}_g^t-x_i^t)\forall i\in \{1,2,3,\dots ,n_A\}$$ (35)

Where $\frac{2g}{G}$ represents the social factor, $1-\frac{g}{G}$ represents the cognitive factor, $r_1$ and $r_2$ are random numbers within the range of [0,1]. $bes{t}_p$ is a randomly selected light source from the new pathfinder group based on the probability factor of its corresponding solution.

The MSA is an improvement on the Moth Flame Optimizer (MFO) in two key areas. Firstly, the reduction in computational cost is enhanced by dealing with each variable as an integrated unit. Secondly, the light source is selected based on a probability function $P_p$ as represented in equation 30 [30]. These modifications allow for better exploitation and exploration of the objective space. Thus, the overall implementation flow for MSA in solving the DCEED problem is presented in Fig. 1.

Figure 1.

MSA Flowchart for Solving the DCEED problem.

4. Results

The DCEED problem was solved for two test systems respectively. While implementing for both systems, the stopping criterion used was the maximum number of iterations which was set to 10000. This was determined by starting with a maximum number of iterations of 1000 and increasing it in steps of 1000 till no significant improvement was noticed in the results. The number of search agents used was also set to 30.

For the two test systems considered, the MSA has been used to solve the DCEED problem thirty different times to have an understanding of the model's performance across multiple simulations. The results obtained from the thirty simulations in comparison with methods like MFO, ALO, WOA, and Tunicate Swarm Algorithm (TSA) are presented in this section in detail for both test systems. The total costs presented are therefore a sum of the fuel cost, emission cost, solar PV cost, and the spinning reserve cost.

4.1. Test System 1

This system consists of six thermal generators and thirteen solar PV units for the day ahead. The load demand in the system is the hourly energy demand of an anonymous data centre in South Africa, which is presented in Table 4. The data centre operator has been kept anonymous subject to a Non-Disclosure Agreement. Data for $P_{rated}$, $T_{ref}$ and β variables were obtained from [23], [24]. The solar irradiation for Cape Town, South Africa was obtained from [32] and used to compute the amount of power generation from the solar plants. The required spinning reserve capacity used for the simulations is 14.9131MW.

Table 4.

Per hour total generations in MW and associated costs in USD ($) for test system 1.

Hour	Total Thermal Generation	Total Solar Generation	Total Generation	Total Spinning Reserve	$P_d$	Fuel Cost	Emission Cost	Solar Cost	Spinning Reserve Cost	Total Cost
1	213.939	0	213.939	14.913	213.939	3250.939	242.380	0	365.626	3858.945
2	209.085	0	209.085	14.913	209.085	3192.906	242.610	0	365.626	3801.143
3	209.981	0	209.981	14.913	209.982	3202.113	240.143	0	365.626	3807.882
4	211.166	0	211.166	14.913	211.166	3214.576	244.928	0	365.626	3825.131
5	211.664	0.775	212.438	14.913	212.438	3245.329	231.239	0.202	365.626	3842.397
6	192.243	22.818	215.061	14.913	215.061	3016.015	236.699	5.930	365.626	3624.271
7	152.023	64.228	216.251	14.913	216.251	2665.912	203.469	16.692	365.626	3251.699
8	113.002	105.096	218.097	14.913	218.097	2300.496	202.515	27.403	365.626	2896.040
9	81.541	141.079	222.620	14.913	222.620	2072.945	181.397	36.810	365.626	2656.779
10	72.141	161.172	233.313	14.913	233.313	1902.846	204.487	41.435	365.626	2514.394
11	52.002	197.447	249.449	14.913	249.449	1704.125	208.001	51.601	365.626	2329.354
12	58.637	194.073	252.710	14.913	252.710	1818.382	190.909	50.046	365.626	2424.963
13	91.115	152.933	244.048	14.913	244.048	2077.993	204.525	40.055	365.626	2688.199
14	78.108	168.549	246.658	14.913	246.658	1971.370	199.184	43.804	365.626	2579.984
15	131.009	110.577	241.587	14.913	241.587	2537.173	184.498	28.721	365.626	3116.018
16	164.438	90.288	254.726	14.913	254.726	2785.921	223.969	23.465	365.626	3398.980
17	186.055	54.798	240.853	14.913	240.851	2988.744	223.670	14.241	365.626	3592.281
18	215.003	27.240	242.243	14.913	242.243	3279.370	238.431	7.079	365.626	3890.507
19	244.874	5.938	250.812	14.913	250.812	3580.146	241.575	1.554	365.626	4188.901
20	242.575	0.028	242.603	14.913	242.603	3521.764	250.975	0.007	365.626	4138.373
21	226.741	0	226.741	14.913	226.741	3415.378	227.714	0	365.626	4008.718
22	218.634	0	218.634	14.913	218.634	3285.314	244.574	0	365.626	3895.515
23	217.541	0	217.541	14.913	217.541	3270.103	247.600	0	365.626	3883.330
24	211.797	0	211.797	14.913	211.797	3252.399	227.877	0	365.626	3845.903

4.2. Test System 2

In this case, the test system is made up of a combination of three thermal generators and thirteen solar PV units for the day ahead. The generation system parameters, cost coefficients, and load demand for the system are adapted from [33] and presented in Table 8. For the solar photovoltaic generation plants, similarly, as in the first test system, data for $P_{rated}$, $T_{ref}$, and β variables were also obtained from [23], [24]. The solar irradiation for Cape Town, South Africa was obtained from [32] and used to compute the amount of power generation from the solar plants. The required spinning reserve capacity used for the simulations is 14.9131MW.

Table 8.

Per hour total generations in MW and associated costs in USD ($) for test system 2.

Hour	Total Thermal Generation	Total Solar Generation	Total Generation	Total Spinning Reserve	Pd	Fuel Cost	Emission Cost	Solar Cost	Spinning Reserve Cost	Total Cost
1	140	0	140	14.91313	140	6062.34	154.69	0	1885.84	8102.87
2	150	0	150	14.91313	150	6289.72	162.99	0	1885.84	8338.54
3	155	0	155	14.91313	155	6369.60	159.00	0	1885.84	8414.44
4	160	0	160	14.91313	160	6461.92	160.32	0	1885.84	8508.08
5	164.11	0.89	165	14.91313	165	6601.17	172.88	0.23	1885.84	8660.12
6	152.63	17.37	170	14.91313	170	6324.27	154.84	4.66	1885.84	8369.61
7	129.02	45.98	175	14.91313	175	5827.00	150.40	12.04	1885.84	7875.28
8	128.10	51.90	180	14.91313	180	5808.28	150.34	14.27	1885.84	7858.73
9	128.32	81.68	210	14.91313	210	5813.84	149.54	21.48	1885.84	7870.70
10	127.03	102.97	230	14.91313	230	5786.44	150.27	26.53	1885.84	7849.07
11	127	113.17	240.17	14.91313	240	5785.83	150.29	28.92	1885.84	7850.87
12	128.70	121.30	250	14.91313	250	5820.59	150.38	32.14	1885.84	7888.96
13	127.55	112.45	240	14.91313	240	5797.43	150.00	29.80	1885.84	7863.07
14	129.98	90.02	220	14.91313	220	5846.61	150.48	23.42	1885.84	7906.35
15	127.76	72.24	200	14.91313	200	5802.61	150.56	18.81	1885.84	7857.83
16	128.70	51.30	180	14.91313	180	5823.98	151.28	13.75	1885.84	7874.85
17	127.03	42.97	170	14.91313	170	5786.59	150.31	11.16	1885.84	7833.90
18	160.24	24.76	185	14.91313	185	6522.44	172.38	6.39	1885.84	8587.05
19	194.85	5.15	200	14.91313	200	7234.86	189.36	1.33	1885.84	9311.39
20	239.98	0.02	240	14.91313	240	8288.26	244.76	0.01	1885.84	10418.87
21	225	0	225	14.91313	225	7984.94	243.42	0	1885.84	10114.20
22	190	0	190	14.91313	190	7208.54	206.82	0	1885.84	9301.19
23	160	0	160	14.91313	160	6517.65	172.34	0	1885.84	8575.83
24	145	0	145	14.91313	145	6169.63	156.96	0	1885.84	8212.43

5. Discussion

Discussion of results obtained from using the proposed MSA for solving the DCEED problem for the two test systems considered in this study is presented in this section.

5.1. Test System 1

A summary of the results obtained using the proposed MSA for solving the DCEED problem for this test system is presented in Table 1. It can be seen from Table 1 that the proposed method produced the lowest best total cost of 82059.71$, the lowest average total cost of 82633.07$, and the lowest worst total cost of 83436.31$ over the thirty simulations in comparison with MFO, ALO, WOA, and TSA. Analysis of the total cost standard deviation over thirty simulations shows that the proposed MSA had a lower standard deviation of 406.93 than all the other methods it was compared with except the ALO with a standard deviation of 256.56. Despite the standard deviation of ALO suggesting that the total costs found by the method could be more similar, it is important to note that the worst total cost found by the proposed MSA outperforms the best total cost found by the ALO. A comparison of the best total costs found by the methods being considered is presented in Fig. 2. It can be seen that the MSA with the best total cost of 82059.71$ outperformed the MFO, ALO, WOA, and TSA with the best total costs of 83287.18$, 83574.22$, 84107.96$ and 84247.83$ respectively.

Table 1.

Summary of DCEED results across thirty simulations for test system 1.

Method	Best Total Cost ($)	Average Total Cost ($)	Worst Total Cost ($)	Total Cost Standard Deviation
MSA	82059.71	82633.07	83436.31	406.93
MFO	83287.18	84378.86	85607.28	666.34
ALO	83574.22	84089.74	84473.04	256.56
WOA	84107.96	84852.76	85676.72	480.58
TSA	84247.83	84975.59	85827.63	459.81

Figure 2.

Cost comparison with other methods for test system 1.

The optimal electricity production levels for the 6 thermal generators which best help to meet the load demand at each hour of the day as found by the proposed MSA are presented in Table 2. The spinning reserve allocation is also presented in Table 2 and it can be observed that the proposed MSA recommends that thermal Generator 1 provides the entire required spinning reserve capacity of 14.9131MW. This makes logical sense as that is the thermal generator with the lowest cost coefficient in the system and the power output from the solar PV plants is intermittent as it largely depends on the solar irradiation.

Table 2.

Thermal generation and spinning reserve allocation per hour in MW for test system 1.

Hour	$P_1$	$S{R}_1$	$P_2$	$S{R}_2$	$P_3$	$S{R}_3$	$P_4$	$S{R}_4$	$P_5$	$S{R}_5$	$P_6$	$S{R}_6$	Total Thermal Generation
1	92.8191	14.9131	34.8182	0	24.0517	0	15.5342	0	27.7775	0	18.9384	0	213.939113
2	94.6539	14.9131	30.1392	0	25.0908	0	10.7383	0	28.2111	0	20.2513	0	209.084612
3	91.6764	14.9131	32.3563	0	26.2126	0	19.9800	0	25.7089	0	14.0474	0	209.981476
4	95.6771	14.9131	31.8595	0	20.4481	0	19.9744	0	30.5528	0	12.6539	0	211.165687
5	86.1179	14.9131	31.5754	0	25.8111	0	24.7033	0	24.6300	0	18.8259	0	211.663583
6	90.4668	14.9131	24.1120	0	22.7730	0	18.0890	0	26.3857	0	10.4170	0	192.243369
7	60.6329	14.9131	14.3770	0	23.7117	0	17.4688	0	25.0817	0	10.7511	0	152.023132
8	46.8786	14.9131	16.1982	0	15.7481	0	14.7256	0	8.4489	0	11.0024	0	113.001836
9	15.0396	14.9131	11.9669	0	18.7650	0	20.5102	0	1.3252	0	13.9344	0	81.5412457
10	44.0059	14.9131	1.5430	0	2.5620	0	2.6318	0	9.2758	0	12.1224	0	72.1407959
11	32.2066	14.9131	10.1331	0	1.5446	0	0.1117	0	7.8934	0	0.1127	0	52.0021759
12	12.6403	14.9131	15.6816	0	14.4744	0	10.1573	0	4.2085	0	1.4750	0	58.6371375
13	42.8575	14.9131	12.6551	0	16.3112	0	2.2658	0	3.4055	0	13.6198	0	91.1149136
14	42.4428	14.9131	0.5487	0	2.9800	0	11.6161	0	10.8742	0	9.6467	0	78.1084402
15	39.4387	14.9131	11.4179	0	16.8981	0	16.7771	0	23.7592	0	22.7184	0	131.009396
16	75.1394	14.9131	23.7152	0	13.0504	0	12.6089	0	20.4656	0	19.4585	0	164.438025
17	80.1364	14.9131	23.4575	0	21.9487	0	13.1856	0	28.1523	0	19.1746	0	186.055124
18	88.7069	14.9131	37.2886	0	23.0958	0	20.3783	0	24.7800	0	20.7531	0	215.002643
19	101.0567	14.9131	29.2838	0	25.7764	0	28.8751	0	36.2167	0	23.6648	0	244.873646
20	107.3366	14.9131	28.4044	0	29.3383	0	26.6645	0	33.7600	0	17.0712	0	242.574941
21	86.8621	14.9131	30.7084	0	25.2501	0	25.5276	0	36.2912	0	22.1016	0	226.74094
22	100.6491	14.9131	25.5525	0	25.7710	0	26.0421	0	22.7146	0	17.9048	0	218.63407
23	101.7658	14.9131	26.8613	0	27.1988	0	11.5619	0	26.1531	0	23.9998	0	217.540742
24	90.1780	14.9131	21.2236	0	23.4834	0	25.9205	0	27.4790	0	23.5123	0	211.796712

The optimal solar PV generation profile of the thirteen plants per hour found by the proposed MSA is presented in Table 3. It can be seen in Table 3 that during the first and last four hours of the day, the solar plants are unable to generate any power and this is because the solar energy resource is not available at those times of the day. However, it can be seen from Fig. 3 that as the day progresses, the solar generation begins to pick up till it peaks around hour 12 and then starts to decline. The impact of the increased solar PV generation on the thermal generation during hours of the day when there is adequate solar irradiation is shown in Fig. 3. It can be deduced that an inverse relationship exists between the amounts of power produced by both the thermal and solar PV plants. As solar PV generation increases from around hour 6 till it peaks around hour 11, the thermal generation decreases during those hours, and from hour 12 onwards, the thermal generation increases steadily to complement the declining solar PV generation. A detailed presentation of the total amount of electricity generated from thermal plants, solar PV plants, spinning reserve allocation, and their associated costs can be found in Table 4. It is observable that the total costs obtained for mid-hours of the day are much lower than the total costs for earlier and later hours. Comparing the generation profile with the total cost profile for the 24-hour period, it can be concluded that the total energy costs reduced with increasing solar PV share, showing that the proposed MSA recognized solar PV generation as the cheaper alternative to thermal generation.

Table 3.

Solar PV generation per hour in MW for test system 1.

Hour	Pgs1	Pgs2	Pgs3	Pgs4	Pgs5	Pgs6	Pgs7	Pgs8	Pgs9	Pgs10	Pgs11	Pgs12	Pgs13	Total Solar PV Generation
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0.0516	0.0646	0.0646	0	0.07747	0	0	0.1033	0.1033	0	0.1033	0.1033	0.1033	0.77473
6	1.04	1.30	1.30	1.56	1.56	1.82	1.82	2.07	2.07	2.07	2.07	2.07	2.07	22.81767
7	2.92	3.65	3.65	4.38	4.38	5.11	5.11	5.84	5.84	5.84	5.84	5.84	5.84	64.22807
8	5.19	6.49	6.49	7.78	7.78	0	9.08	10.38	10.38	10.38	10.38	10.38	10.38	105.09555
9	0	9.28	9.28	11.14	11.14	12.99	12.99	0	14.85	14.85	14.85	14.85	14.85	141.07874
10	8.95	11.19	11.19	13.43	13.43	15.67	15.67	17.91	0	0	17.91	17.91	17.91	161.17233
11	9.63	12.04	12.04	14.45	0	16.86	16.86	19.26	19.26	19.26	19.26	19.26	19.26	197.44714
12	9.70	12.13	12.13	14.56	14.56	16.98	16.98	19.41	19.41	19.41	0	19.41	19.41	194.07265
13	9.00	11.25	0	13.49	13.49	0	15.74	0	17.99	17.99	17.99	17.99	17.99	152.93285
14	7.66	9.58	9.58	11.49	11.49	13.41	13.41	15.32	15.32	15.32	15.32	15.32	15.32	168.54910
15	5.90	0	7.37	8.85	8.85	10.32	10.32	11.79	11.79	11.79	11.79	11.79	0	110.57719
16	4.10	5.13	5.13	6.16	6.16	7.18	7.18	8.21	8.21	8.21	8.21	8.21	8.21	90.28802
17	2.49	3.11	3.11	3.74	3.74	4.36	4.36	4.98	4.98	4.98	4.98	4.98	4.98	54.79751
18	1.24	1.55	1.55	1.86	1.86	2.17	2.17	2.48	2.48	2.48	2.48	2.48	2.48	27.24038
19	0.29	0	0.36	0.43	0.43	0.50	0.50	0.57	0.57	0.57	0.57	0.57	0.57	5.93823
20	0	0	0.0024	0	0.0029	0.0034	0.0034	0	0.0039	0	0.0039	0.0039	0.0039	0.02764
21	0	0	0	0	0	0	0	0	0	0	0	0	0	0
22	0	0	0	0	0	0	0	0	0	0	0	0	0	0
23	0	0	0	0	0	0	0	0	0	0	0	0	0	0
24	0	0	0	0	0	0	0	0	0	0	0	0	0	0

Figure 3.

MSA Hourly Total Generation Profile for test system 1.

The ramp-up and down profiles of the thermal generators based on their optimal operating levels as determined by the proposed MSA are shown in Fig. 4. It can be observed that the proposed MSA did not violate the ramping constraints of the thermal generators. The optimal fuel cost curve found by the MSA is shown in Fig. 5 and it can be seen that the curve has a similar pattern to the thermal generation profile shown in Fig. 3. It can also be seen from Fig. 6 that as the solar PV cost rises at mid-hours of the day due to increased solar PV generation, the emission cost significantly reduces at those hours.

Figure 4.

MSA Generator Up and Down Ramping Profile for test system 1.

Figure 5.

MSA Hourly Fuel Cost Profile of Thermal Generators for test system 1.

Figure 6.

MSA Hourly Emission Cost, Solar PV Cost and Spinning Reserve Cost Profiles for test system 1.

Since the proposed MSA recommended that Generator 1 provides all the required SR capacity, the SR cost therefore remains static at all hours of the day.

The impact of the increased solar PV generation at hours of the day when the energy resource is available on the overall energy cost can be deduced from Fig. 7. As the solar PV generation increases during the mid-hours of the day, the overall cost significantly reduces. The cost convergence curves of the proposed MSA for solving the DCEED problem for 24 hours of the day are presented in Fig. 8. It can be seen for all hours of the day that the solver converged towards the best solution it found while satisfying all equality and inequality constraints.

Figure 7.

MSA Hourly Total Cost Profile for test system 1.

Figure 8.

MSA Total Cost Minimization Curves for test system 1.

5.2. Test System 2

Table 5 presents a summary of the results obtained from using the proposed MSA for solving the DCEED problem for test system 2. It can be noted from Table 5 that the proposed MSA outperformed the MFO, TSA, ALO, and WOA in finding the least total cost of 201444.22$, the least average total cost of 201790.16$ and the lowest worst total cost of 202233.23$ over thirty runs. Analysis of the total cost standard deviations obtained for each method also shows that the proposed MSA with a total cost standard deviation of 281.73 performed second best in comparison to the other methods in that regard, only being outperformed by the MFO with a total cost standard deviation of 245.93. Regardless of the relatively impressive standard deviation obtained for the MFO in this test system, the proposed MSA outperformed the MFO in minimizing the total energy cost which is the main goal of the objective function. Fig. 9 highlights a comparison of the best total cost obtained from using the methods considered for solving the DCEED problem for test system 2 and it can be observed that the proposed MSA had a superior performance in relation to the others. The best total costs obtained for the WOA, ALO, TSA, MFO, and MSA are 203381.46$, 202098.8$, 201950.98$, 201681.24$, and 201444.22$ respectively.

Table 5.

Summary of DCEED results across thirty simulations for test system 2.

Method	Best Total Cost ($)	Average Total Cost ($)	Worst Total Cost ($)	Total Cost Standard Deviation
MSA	201444.22	201790.16	202233.23	281.73
MFO	201681.24	202045.78	202506.95	245.93
TSA	201950.98	202679.06	204516.71	752.15
ALO	202098.80	202752.44	203571.90	436.49
WOA	203381.46	204644.79	206447.40	845.16

Figure 9.

Cost comparison with other methods for test system 2.

Table 6 presents the hourly breakdown of the power output levels for the three thermal generators in the second test system alongside the amount of spinning reserve that each unit caters to. Likewise, the power output of each solar PV unit in the system is presented in Table 7. The hourly power output due to the thermal plants and solar PV plants are thus graphically represented in Fig. 10. It can be observed from Fig. 10 that the solar PV plants only started producing a significant amount of power between hours 5 – 19, and this is due to the availability of solar irradiation during those hours. Fig. 10 also shows an increase in the amount of power produced by the solar PV plants between hours 5 – 12 while the solar generation share gradually dropped with declining solar irradiation levels during the later hours of the day. The largest solar share was recorded at the time of the day with the largest solar irradiation – hour 12, with a total solar PV generation of 121.30MW as shown in Table 7. The ramp-up and down rates of the three thermal generators are presented in Fig. 11 and it can be noted that the ramp-up and ramp-down rate constraints have been effectively satisfied by the model in solving the DCEED problem for this test system.

Table 6.

Thermal generation and spinning reserve allocation per hour in MW for test system 2.

Hour	$P_1$	$S{R}_1$	$P_2$	$S{R}_2$	$P_3$	$S{R}_3$	Total Thermal Generation
1	37.00	14.91313	47.93	0	55.07	0	140.00
2	37.00	14.91313	48.62	0	64.38	0	150.00
3	37.00	14.91313	62.91	0	55.09	0	155.00
4	37.00	14.91313	73.00	0	50.00	0	160.00
5	37.25	14.91313	54.44	0	72.42	0	164.11
6	40.86	14.91313	56.36	0	55.40	0	152.63
7	37.00	14.91313	42.02	0	50.00	0	129.02
8	37.00	14.91313	41.10	0	50.00	0	128.10
9	38.32	14.91313	40.00	0	50.00	0	128.32
10	37.03	14.91313	40.00	0	50.00	0	127.03
11	37.00	14.91313	40.00	0	50.00	0	127.00
12	37.00	14.91313	41.70	0	50.00	0	128.70
13	37.51	14.91313	40.04	0	50.00	0	127.55
14	37.00	14.91313	42.98	0	50.00	0	129.98
15	37.18	14.91313	40.00	0	50.58	0	127.76
16	37.11	14.91313	40.00	0	51.59	0	128.70
17	37.00	14.91313	40.00	0	50.03	0	127.03
18	37.00	14.91313	50.58	0	72.66	0	160.24
19	37.00	14.91313	84.84	0	73.01	0	194.85
20	42.77	14.91313	90.61	0	106.60	0	239.98
21	37.03	14.91313	79.25	0	108.72	0	225.00
22	37.00	14.91313	58.08	0	94.92	0	190.00
23	37.00	14.91313	50.33	0	72.67	0	160.00
24	37.00	14.91313	50.74	0	57.26	0	145.00

Table 7.

Solar PV generation per hour in MW for test system 2.

Hour	Pgs1	Pgs2	Pgs3	Pgs4	Pgs5	Pgs6	Pgs7	Pgs8	Pgs9	Pgs10	Pgs11	Pgs12	Pgs13	Total Solar PV Generation
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0	0.06	0.06	0.08	0.08	0.09	0	0.10	0.10	0.10	0.10	0.10	0	0.89
6	0	1.30	0	0	0	1.82	1.82	2.07	2.07	2.07	2.07	2.07	2.07	17.37
7	0	3.65	3.65	4.38	0	5.11	0	5.84	5.84	5.84	5.84	5.84	0	45.98
8	0	0	0	0	0	0	0	10.38	10.38	10.38	10.38	10.38	0	51.90
9	0	0	0	11.14	0	12.99	12.99	14.85	14.85	0	0	14.85	0	81.68
10	8.95	11.19	0	13.43	0	0	15.67	17.91	17.91	0	17.91	0	0	102.97
11	0	12.04	12.04	0	14.45	16.86	0	19.26	19.26	19.26	0	0	0	113.17
12	0	12.13	0	0	14.56	0	16.98	0	19.41	19.41	19.41	19.41	0	121.30
13	0	0	11.25	0	13.49	0	15.74	0	17.99	17.99	17.99	0	17.99	112.45
14	7.66	9.58	0	11.49	0	0.00	0	15.32	15.32	15.32	0	0	15.32	90.02
15	0	0	7.37	0	8.85	10.32	10.32	11.79	0	11.79	11.79	0	0	72.24
16	4.10	0	0	0	6.16	0	0	8.21	0	8.21	8.21	8.21	8.21	51.30
17	0	3.11	3.11	3.74	3.74	4.36	0	4.98	4.98	4.98	4.98	4.98	0	42.97
18	1.24	1.55	1.55	1.86	1.86	2.17	2.17	2.48	2.48	2.48	2.48	2.48	0	24.76
19	0.29	0.36	0.36	0.43	0.43	0.50	0.50	0	0	0.57	0.57	0.57	0.57	5.15
20	0	0	0.00	0.00	0.00	0	0.00	0	0	0	0.00	0.00	0.00	0.02
21	0	0	0	0	0	0	0	0	0	0	0	0	0	0
22	0	0	0	0	0	0	0	0	0	0	0	0	0	0
23	0	0	0	0	0	0	0	0	0	0	0	0	0	0
24	0	0	0	0	0	0	0	0	0	0	0	0	0	0

Figure 10.

MSA Hourly Total Generation Profile for test system 2.

Figure 11.

MSA Generator Up and Down Ramping Profile for test system 2.

The fuel cost, emission cost, solar cost, and spinning reserve cost incurred hourly in supplying the hourly load demand in the system are presented in Table 8. A graphical representation of the hourly fuel cost for the system is presented in Fig. 12. By comparing the load profile which is essentially represented by Fig. 10 with the fuel cost profile in Fig. 12, it can be seen that although the system load increased steadily from hours 7 – 12 and dropped gradually from hours 12 – 17, the fuel cost for the system between hours 7 – 17 remained approximately the same. The uptake in solar PV generation during those hours where enough solar irradiation was available for power generation is due to the MSA model recognizing solar PV as a better alternative to the thermal generators from a cost and environmental impact perspective. It can also be discerned that the share of power due to solar PV may have been higher if the thermal generators had smaller lower operating limits. Nevertheless, to satisfy the operational limit constraints of the thermal generation units, a minimum total of 127MW must be produced by the thermal generation units.

Figure 12.

MSA Hourly Fuel Cost Profile of Thermal Generators for test system 2.

The emission and solar costs are presented in Fig. 13 and it can be observed that the solar cost is in direct proportionality with the solar generation share in the system, with the highest solar cost of 32.14$ being recorded at hour 12 as shown in Table 8. The emission cost line graph presented in Fig. 13 also shows that the system incurred the least emission costs at hours of the day when solar PV generation was available. An overall representation of the fuel, emission, solar and spinning reserve costs is presented in Fig. 14. It can be seen that the spinning reserve cost remained the same for the system at every hour of the day because the MSA model recommended generator 1 (${\mathit{P}}_{\mathbf{1}}$) which is the thermal generator with the relatively cheapest cost coefficient to provide the required spinning reserve capacity for test system 2. As also shown in Fig. 14, the total overall cost incurred between hours 7 – 17 is much lower compared to other hours of the day, despite having relatively larger system load demands between those hours of the day. The cost minimization curves of the proposed MSA for solving the DCEED problem for this test system are presented in Fig. 15. It can be observed that the MSA converged towards the best solution it found while satisfying all equality and inequality constraints for all hours of the day.

Figure 13.

MSA Hourly Emission Cost and Solar PV Cost Profiles for test system 2.

Figure 14.

MSA Hourly Total Cost Profile for test system 2.

Figure 15.

MSA Total Cost Minimization Curves for test system 2.

6. Conclusion

The work presented in this study proposes the use of Moth Swarm Algorithm for solving the day-ahead economic and emission dispatch problem in a power system made up of a combination of thermal and solar PV units. Two test systems have been considered for the implementation thereof. For the first test system, the power generation system consists of six thermal generators and thirteen solar photovoltaic plants, and the system load considered is the hourly energy demand of a data centre in South Africa. The second test system is made up of a combination of three thermal generators and thirteen solar photovoltaic plants. The model has been implemented for the 24-hour period and results obtained from the study have been compared with state-of-the-art methods like MFO, ALO, WOA, and TSA for both test systems.

Analyses show that the MSA outperformed the other methods in fulfilling the objective of the study which is minimizing the energy cost while obeying all operational constraints for the 24-hour period for the two test systems considered. It is also noticeable from the results obtained that the proposed MSA gives preference to solar PV generation over thermal generation at hours of the day when the resource is available. An increased solar share can also be observed during hours of peak sunshine which suggests that the proposed MSA recognizes solar PV as a better alternative to thermal generation for minimizing the fuel and emission costs in the system.

However, since solar energy resource is not available at every hour of the day, the potential exists to incorporate reliable and cost-effective storage solutions to better utilize the solar energy resource for power generation. The proposed MSA also recommended that the required spinning reserve capacity be provided by the thermal generator with the least cost coefficient at every hour of the day. The dominant performance of the proposed MSA in comparison with the other methods while solving the DCEED problem is a testament to its robust ability to solve even more complex nonlinear and non-convex optimization problems. In future works, the authors hope to investigate the use of other novel metaheuristic algorithms to solve the DCEED problem as well as other optimization problems. The integration of more renewable energy sources into the mix for solving the DCEED problem may also be investigated in subsequent works.

Declarations

Author contribution statement

Oluwafemi Ajayi: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Reolyn Heymann: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data included in article/supp. material/referenced in article

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.