Maximization of Total Profit for Hybrid Hydro-Thermal-Wind-Solar Power Systems Considering Pumped Storage, Cascaded Systems, and Renewable Energy Uncertainty in a Real Zone, Vietnam

Abstract

The study maximizes the total profit of a hybrid power system with cascaded hydropower plants, thermal power plants, pumped storage hydropower plants, and wind and solar power plants over one operation day, considering the uncertainty of wind speed and solar radiation. Wind speed and solar radiation in a specific zone in Vietnam are collected using the wind and solar global atlases, and the maximum data are then supposed to be 120% of the collection for uncertainty consideration. The metaheuristic algorithms, including the original Slime mould algorithm (SMA), Equilibrium optimizer, and improved Slime mould algorithm (ISMA), are implemented for the system. ISMA is a developed version of SMA that cancels old methods and proposes new methods of updating new solutions. In the first stage, the cascaded system with four hydropower plants is optimally operated by simulating two cases: simultaneous optimization and individual optimization. ISMA is better than EO and SMA for the two cases, and the results of ISMA from the simultaneous optimization reach greater energy than individual optimization by 154.8 MW, equivalent to 4.11% of the individual optimization. For the whole system, ISMA can reach a greater total profit than EO and SMA over one operating day by USD 6007.5 and USD 650.5, equivalent to 0.12% and 0.013%. The results indicate that the optimization operation of cascaded hydropower plants and hybrid power systems can reach a huge benefit in electricity sales

1. Introduction

In recent years, there has been a growing need for cheaper, more sustainable, and cleaner power sources due to the high cost and environmental impact of fossil fuels [1]. Wind power plants (WPs) and solar power plants (SPs) are promising solutions, but they face challenges due to the uncertain nature of wind and solar radiation [2]. This has led to a focus on energy storage, with pumped storage hydropower plants (PSHPs) emerging as a popular option. PSHPs consist of upper and lower reservoirs connected by a tunnel, with turbines that can function as both generators and pumps [3]. They play a significant role in reducing the total electric generation cost of Thermal Power Plants (ThPs) in a hybrid power system. The study also explores the optimal scheduling of a hybrid power system comprising various power plants, including PSHPs, cascaded hydropower plants (CasHPs), WPs, SPs, and ThPs, to maximize the total electricity sale profit from these power plants. Hourly electric price, load demand, wind speed, and solar radiation over one day are given. Then, the maximum data of wind and solar radiations are collected for an uncertainty case with 120% of the maximum given data, which allows for maximizing the total electric sale profit. PSHPs use the power from other plants to pump water back to the upper reservoirs and the stored water volume can be used for many hours with a high price. The study applies metaheuristic algorithms to determine power of WPs, SPs, and ThPs, discharge and volume of CasHPs, and pumping hours, generation hours, and discharge of generation hours for PSHPs. So, the importance of the study is to reach optimal operation solutions for the system that can supply enough electricity to loads and reach a high total electric sale profit. In addition, the use of suitable energy storage can produce more energy for power systems, and ThPs can continuously operate.

The earliest studies have applied deterministic algorithms, such as the linear programming algorithm [4] and Gradient algorithm (GRA) [5], for simple systems and have reached the same solutions for different runs. On the contrary, metaheuristic algorithms, such as improved particle swarm optimization (MPSO) [6] and Evolutionary Programming (EP) [7], have found different solutions for different runs. In general, the goal of these studies was to compare costs and demonstrate the contribution of PSHPs in meeting load demand, even without inflows to the reservoir. In recent years, studies have focused on complex problems related to PSHPs in various countries, including Turkey [8], China [9], Spain [10], Portugal [11], Iran [12], and Saudi Arabia [13]. These studies have integrated PSHPs into other power sources to compensate for power shortages and reduce peak load demand. They have also examined the effectiveness of PSHPs in hybrid systems, and their potential for increasing total revenue. Additionally, this research has been conducted on optimizing the operation of PSHP to alleviate peak shaving in power systems. However, the studies did not address the cost of constructing PSHPs in these power systems. Several studies have investigated the profitability and benefits of pumped storage hydropower (PSHP) in various electric markets in Turkey [14], China [15], and benchmark systems [16]. While some studies [17,18] found that PSHP could not bring enough profit for the power systems, others suggested that it could compensate for insufficient power and consume surplus power from renewable plants. Additionally, studies have proposed transforming existing Conventional Hydropower Plants (CoHPs) into PSHPs and showed that PSHP could reduce generation costs and cut emissions in hybrid systems [19].

On the other hand, cascaded hydropower plants and pumped storage hydropower plants could be combined in the same system by adding a pumping function into the cascaded hydropower plants or letting them work separately [20,21,22]. The cooperation among cascaded hydropower plants is driven by the goal of maximizing energy production or profit [23]. Coordinated operations of these plants can enhance the benefits of the entire river system [24]. However, allocating these benefits fairly and efficiently presents a significant challenge for river system operators or managers [25]. The operation combination of PSHPs and CasHPs is very complicated, but the effectiveness of the successful cooperation can lead to a huge benefit. In [20], researchers investigated a larger electric power grid comprising two PSHPs, twenty-nine TPs, and four cascaded power plants. The study aimed to achieve a balance between demand and supply, taking into consideration energy purchases from external power plants. The results revealed that all system constraints were met, but the efficacy of the PSHPs was inconclusive. Another study [21] focused on optimizing the generation over five days for three cascaded hydroelectric units, with the lowest unit capable of running pumps for water storage. The study demonstrated the system’s potential to achieve the highest energy output, but it lacked comparisons with other studies or different methods. In the study [22], researchers determined the optimal location for constructing a PSHP in a six-cascaded hydroelectric power plant system. In general, each previous study had significant contribution in power systems; however, most of the studies have not proved their solution method and metaheuristic algorithms to be highly effective. In addition, data of test systems were almost neglected for replication. All types of power plants were not combined, and the uncertainty of renewable energies was not considered for the large integrated system.

In two early studies regarding renewable energies, real power output can be obtained by substituting certain data into mathematical equations for WPs [26] and SPs [27]. Considering the uncertainty of wind, the cumulative distribution function (cdf) and probability density function (pdf) were proposed to calculate the power output and the costs [28]. Similarly, the electricity generation costs of SPs were also calculated by using the probability function [29]. The costs of wind and solar power considering uncertainty were studied for small systems with wind, solar, and small hydro plants [30], a large system considering Electric Vehicles [31], and transmission power systems [32]. In the study, we investigate the positive impact of pumped storage hydropower plants (PSHPs) on maximizing the total electric sale profit of a hybrid power system with PSHPs, CasHPs, WPs, SPs, and ThPs. Real data of wind speed and solar radiations in a specific zone are collected by using the wind global atlas [33] and solar global atlas [34]. On the other hand, their uncertain characteristics are also considered by using probability functions in calculating generation costs as shown in previous studies [30,31,32]. Three metaheuristic algorithms, including the original Slime mould algorithm (SMA) [35], Equilibrium optimizer [36], and a Slime mould algorithm (ISMA), are applied to find the optimal scheduling of the system in two cases: certain and uncertain wind and solar. The novelty of the paper can be summarized as follows:

• Develop an improved version of SMA to improve the search performance of SMA.
• Select a real zone in Vietnam and then access the wind and solar global atlases to collect real wind speed and solar radiation. In addition, the maximum possible data are supposed to be 120% of the collected data to run uncertainty case of wind and solar.
• Apply a variety of major power plants, such as CasHP, PSHP, WPs, SPs, and ThPs, which are considered as power sources. Their operation principles and characteristics are obtained from previous studies.

The main contributions of the whole paper can be clarified as follows:

• The PSMA can find better solutions and provide more stable searchability than SMA and EO for two cases: considering certain and uncertain characteristics for wind and solar power plants.
• Optimal operation solutions for the PSHP can lead to higher profits for hybrid systems with renewable energies. The total profit can be greater for the system with the PSHP when considering both certain and uncertain wind and solar power.
• For the PSHP, optimal solutions can lead to significant profits after accounting for electric purchases. The PSHP needs to buy electricity to run pumps for storing water, and then it can sell the energy generated by discharging the stored water. The difference between the sale and purchase is the profit.
• CasHPs can reach the maximum energy and it can lead to a great profit for the whole system with many power plant types.

Other parts of the study are as follows: Section 2 presents the problem formulation with objective functions and constraints. Section 3 shows an improved version of SMA. Section 4 presents numerical results to show the contributions. Finally, Section 5 summarizes the whole study with obtained results, contributions, shortcomings, and future work.

2. The Problem Description

In this paper, the contribution of pumped storage hydropower plants to the cost reduction for thermal power plants is investigated. A typical power system is considered with the presence of power plant types, including $N_{ThP}$ thermal power plants, $N_{PSHP}$ pumped storage hydropower plants, $N_{WP}$ wind power plants, $N_{SP}$ solar power plants, and $N_{CasHP}$ cascaded hydropower plants. The system is plotted in Figure 1. In the figure, CasHP1, CasHP2, CasHP3, and CasHP4 are the four cascaded hydropower plants, in which CasHP1 and CasHP2 are, respectively, the upper plants of CasHP4 and CasHP3. In addition, CasHP3 is the upper plant of CasHP4. Discharges 1–4 and flows 1–4 are, respectively, the discharge and inflow of the four CasHPs. ThP1 and ThP2 are two thermal power plants.

2.1. The Total Profit Maximization

The objective function of the study is to maximize the total electric sales profit for hybrid power systems throughout one day. Basically, ThPs have a cost function modeled as a second-order function based on active power output and specific coefficients; meanwhile, the cost function of hydropower plants is supposed to be zero due to the free water. Regarding the costs of WPs and SPs, generation costs are considered in two aspects: neglecting generation costs and taking them into account. In this paper, we explore both cases of renewable power plants. This objective function is calculated as the total revenue minus the total costs. The objective is mathematically presented as follows:

$$Maximize Profi{t}_{sys}=Sal{e}_{Load}-Cos{t}_{all}$$

(1)

$$Sal{e}_{Load}={\displaystyle \sum }_{h=1}^{N_h}\left(Pric{e}_h.Loa{d}_h\right)$$

(2)

$$Reduce Cos{t}_{all}=C_{ThP}+C_{WP}+C_{SP}$$

(3)

where $G{C}_{TP}$, $G{C}_{WP}$, and $G{C}_{PVP}$ are the total generation costs of all THPPs, WPPs, and PVPPs in the considered power system. The costs of thermal power plants [37], wind, and solar power plants [29] are respectively obtained by

$$C_{ThP}={\displaystyle {\displaystyle \sum }_{h=1}^{N_h}}{\displaystyle {\displaystyle \sum }_{th=1}^{N_{ThP}}}\left({\delta }_{1th}+{\delta }_{2th}{P}_{th,h}+{\delta }_{3th}{P}_{th,h}^2\right)$$

(4)

$$C_{WP}={\displaystyle \sum }_{h=1}^{N_h}{\displaystyle \sum }_{w=1}^{N_{wp}}\left(D{C}_{w,h}+P{C}_{w,h}+R{C}_{w,h}\right)$$

(5)

$$C_{SP}={\displaystyle \sum }_{h=1}^{N_h}{\displaystyle \sum }_{pv=1}^{N_{SP}}\left(D{C}_{pv,h}+P{C}_{pv,h}+R{C}_{pv,h}\right)$$

(6)

where ${DC}_{w,h}$, ${RC}_{w,h}$, and ${PC}_{w,h}$ are the direction, reserve, and penalty costs of the wth WP at the hth hour; and $D{C}_{pv,h}$, $R{C}_{pv,h}$, and $P{C}_{pv,h}$ are the direction, reserve, and penalty costs of the pvth SP at the hth hour.

Here, the total electric sale profit belongs to the whole power plants in the system. It is supposed that all power plants working in the power system belong to one generation company and the sharing profit is in charge of the CEO of the company. The study acts as an optimization operation expert and proposes the optimal generation schedule for the total profit maximization.

2.2. Constraints

2.2.1. Active Power Balance

The power systems need to balance total generated power and total consumed power, as expressed in the following model [38]:

$${\displaystyle \sum }_{th=1}^{N_{Thp}}{P}_{th,h}+{\displaystyle \sum }_{sp=1}^{N_{SP}}{P}_{sp,h }+{\displaystyle \sum }_{w=1}^{N_{WP}}{P}_{w,h }+{\displaystyle \sum }_{cas=1}^{N_{CasHP}}{P}_{cas,h}+{\displaystyle \sum }_{ps=1}^{N_{PSHP}}\left(1-K_{ps,h }\right)P_{ps,h}-{\displaystyle \sum }_{ps=1}^{N_{PSHP}}{K}_{ps,h }.Pum{p}_{ps,h}-Loa{d}_h=0;h=1,\dots , N_h$$

(7)

where $K_{ps,h }$is the operating status of the psth PSHP at the hth hour, it has two values: 1 for the pumping action, and 0 for the generating action. Its values and actions are summarized as follows:

$$K_{ps,h }=\left\{\begin{array}{c}0 for generating action \\ 1 for pumping action \end{array}\right.$$

(8)

2.2.2. Generation Limits of Power Plants

All power plants have limits of generating capacity so that their generators can work stably and effectively. The power output must be within the minimum and maximum limits as follows:

$$P_{th}^{Min}\le P_{th,h}\le P_{th}^{Max}$$

(9)

$$P_{sp}^{Min}\le P_{sp,h }\le P_{sp}^{Max}$$

(10)

$$P_w^{Min}\le P_{w,h }\le P_w^{Max}$$

(11)

$$P_{cas}^{Min}\le P_{cas,h}\le P_{cas}^{Max}$$

(12)

$$P_{ps}^{Min}\le P_{ps,h}\le P_{ps}^{Max}$$

(13)

$$Pum{p}_{ps,h}=\left\{\begin{array}{c}0 for generating action \\ P_{ps}^{Max} for pumping action \end{array}\right.$$

(14)

2.2.3. Pumped Storage Hydropower Plant’s Hydraulic Constraints

Inequality constraints: the upper reservoir and hydro turbines of the power plants have an operating range, called the minimum and maximum limits as the following model [39]:

$$V_{ps}^{Min}\le V_{ps,h}\le V_{ps}^{Max}$$

(15)

$$Q_{ps}^{Min}\le Q_{ps,h}\le Q_{ps}^{Max}$$

(16)

Here, $Q_{ps,h}$ is the psth PSHP’s discharge at the hth hour, and it is function of power output and discharge coefficients as follows [16]:

$$Q_{ps,h}=m_{1ps}{\left(P_{ps,h}\right)}^2+m_{2ps}{P}_{ps,h}+m_{3ps}$$

(17)

Equality constraints: Equality constraints are about the continuity of water in reservoirs, the relationship between discharge and pump flow, and the reservoir volumes at different hours. These constraints are as follows [40]:

$$\begin{gathered} V_{ps,h-1}+I_{ps,h}-V_{ps,h}-\left(K_{ps,h }-1\right).Q_{ps,h}+K_{ps,h }.Q_{pumps}=0; \\ k=1, \dots ,N_5;h=1, \dots , N_1 \end{gathered}$$

(18)

$$Q_{pumps}={\eta }_{ps}.Q_{ps}^{Max}$$

(19)

$$V_{ps,0}=V_{ps}^{Initial}$$

(20)

$$V_{ps,N_h}=V_{ps}^{Finall}$$

(21)

where $V_{ps,0}$ and $V_{ps}^{Initial}$ are the psth PSHP’s reservoir at the 0th hour and the beginning of the scheduled day. $V_{ps,N_h}$ and $V_{ps}^{Finall}$ are the psth PSHP’s reservoir at the last hour and the end of the scheduled day. $V_{ps,0}$ is the special value of $V_{ps,h-1}$ when considering h = 1.

2.2.4. Cascaded Hydropower Plants’ Constraints

The cascaded hydropower plant system has a diagram as shown in Figure 2. The cascaded system has the constraints as follows:

Equality constraint: each CasHP is subject to the water continuity constraint as follows [41]:

$$V_{cas,h-1}+I_{cas,h}-V_{cas,h}-Q_{cas,h}+{\displaystyle \sum }_{cas\text{'}=1}^{N{U}_{cas}}\left(Q_{cas\text{'},h-T_{cas\text{'},cas}}\right)=0$$

(22)

where $T_{cas\text{'},cas}$ is the travelling water time from the cas’th to the casth CasHP; $Q_{cas\text{'},h-T_{cas\text{'},cas}}$ is the discharge of the cas’th upper CasHP of the casth CasHP at the $\left(h-T_{cas\text{'},cas}\right)th$ hour; and $N{U}_{cas}$ is the casth CasHP’s upstream CasHP number.

In addition, the CasHPs are also constrained by the initial volume and end volume as indicated in Equations (20) and (21) of the PSHPs.

For each CasHP, power output is a function of discharge, volume, and other coefficients as follows:

$$P_{cas,h}={\gamma }_{1cas}{\left(V_{cas,h}\right)}^2+{\gamma }_{2cas}{\left(Q_{cas,h}\right)}^2+{\gamma }_{3cas}{V}_{cas,h}{Q}_{cas,h}+{\gamma }_{4cas}{V}_{cas,h}+{\gamma }_{5cas}{Q}_{cas,h}+{\gamma }_{6cas}$$

(23)

Inequality constraints: For each CasHP, volume and discharge also have their operating range [42] as shown in Equations (15) and (16).

3. Improved Slime Mould Algorithm

3.1. Slime Mould Algorithm

In the Slime mould algorithm (SMA), each control variable has two ways of updating new values depending on the comparison result between a random number and the fitness computation. The two ways are summarized in the following equation [35]:

$$c{v}_{x,y}^{new}=\left\{\begin{array}{c}{r}_1.c{v}_{x,y} if rd\ge tanh\left(F{N}_y-F{N}_{b1}\right) \\ c{v}_{x,b1}+r_2\left(r_3.c{v}_{x,r1}-c{v}_{x,r2}\right) else \end{array}\right.$$

(24)

$$Rang{e}_{r1}=\left[{\displaystyle \frac{G_c}{G_M}}-1; 1-{\displaystyle \frac{G_c}{G_M}}\right]$$

(25)

$$Rang{e}_{r2}=\left[-arctanh\left(1-{\displaystyle \frac{G_c}{G_M}}\right); arctanh\left(1-{\displaystyle \frac{G_c}{G_M}}\right)\right]$$

(26)

$$r_3=\left\{\begin{array}{c}1-rd.\log \left(1+{\displaystyle \frac{F{N}_{b1}-F{N}_y}{F{N}_{b1}-F{N}_{w1}}}\right) if y>0.5N_s\\ 1+rd.\log \left(1+{\displaystyle \frac{F{N}_{b1}-F{N}_y}{F{N}_{b1}-F{N}_{w1}}}\right) else\end{array}\right.$$

(27)

3.2. Improved Slime Mould Algorithm

The upper model in Equation (24) is a shortcoming of SMA in searching for new values for the xth control variable. In fact, the upper model uses a random number between 0 and 1 to multiply the old value, this result can lead to a zero solution if the random value is zero. This model can cause an ineffective search process if the condition of [$rd\ge tanh\left(F{N}_y-F{N}_{b1}\right)]$ happens. So, in the ISMA, the upper model is replaced with another model, which searches around the current value of the xth control variable. The change can avoid finding the zero solutions with all zero control variables. This is the first modification, and it is expressed as follows:

$$c{v}_{x,y}^{new}=c{v}_{x,y}+r_2\left(r_3.c{v}_{x,b1-4}-c{v}_{x,y}\right) if rd\ge tanh\left(F{N}_y-F{N}_{b1}\right)$$

(28)

The lower model in Equation (24) finds new values of the xth control variable by finding a close value of the xth control variable in the best solution. On the other hand, the lower model uses an increased interval by using two random solutions. The search cannot lead to the best control variable because one interval cannot enlarge the search space, and using two random solutions cannot find the most effective increased interval. So, the lower model is modified as follows:

$$c{v}_{x,y}^{new}=\left\{\begin{array}{c}c{v}_{x,b1}+r_2\left(r_3.c{v}_{x,b1-4}-c{v}_{x,b1}\right) if 0.5\le rd<tanh\left(F{N}_y-F{N}_{b1}\right)\\ c{v}_{x,b1}+r_2\left(r_3.c{v}_{x,b1-4}-c{v}_{x,b1}\right)+r_2\left(r_3.c{v}_{x,r3}-c{v}_{x,b1}\right) if rd<0.5 \& tanh\left(F{N}_y-F{N}_{b1}\right)\end{array}\right.$$

(29)

In Equation (29), two models exploit the search space around the best solution with different scales: one increased interval in the upper model and two increased intervals in the lower model. The condition is the comparison of the random number and 0.5 to balance the increased steps in one or two intervals.

4. Numerical Results

4.1. Data, Parameter Settings, and Simulation Scebnario

In this section, the total profit of one hybrid power system with four thermal power plants, four cascaded hydropower plants, one solar power plant, one wind power plant, and one pumped storage hydropower plant is maximized under the consideration of renewable power source uncertainty. In the first simulation scenario, three algorithms, SMA [35], EO [36], and ISMA are run to reach the maximum energy of the cascaded hydropower plant system. Then, the most effective solution of the system is used to maximize the total profit of the whole system in the second simulation scenario. The whole study is summarized as follows:

(1)

The first simulation scenario: energy maximization of the cascaded hydropower plants:

Case 1: Optimization operation of each plant;

Case 2: Simultaneous optimization operation of the whole plants.

(2)

The second simulation scenario: maximization of the total profit of the whole system.

The whole study is performed on a personal computer with the central processing unit (CPU) Core i7, 2.6 GHz, and 8 GB of random accessing memory. (RAM). MATLAB software with the version of R2018a is used to code the program of algorithms. For running the three algorithms, the iteration number and population size are set to 50 and 200 for each plant of Case 1, 50, and 1000 for the whole cascaded system of Case 2 in the first scenario, and 200 and 2000 in the second scenario.

The whole system is plotted in Figure 1. The system consists of four THPPs, one PSHP, one WPP, and one PVPP. The ThPs’ cost function and generation limits are given in

Table A1. The four ThPs’ cost function’s coefficients.

th	${\mathit{\delta }}_{1\mathit{t}\mathit{h}}$	${\mathit{\delta }}_{2\mathit{t}\mathit{h}}$	${\mathit{\delta }}_{3\mathit{t}\mathit{h}}$	${\mathit{P}}_{\mathit{t}\mathit{h}}^{\mathit{M}\mathit{i}\mathit{n}}$	${\mathit{P}}_{\mathit{t}\mathit{h}}^{\mathit{M}\mathit{a}\mathit{x}}$
1	38.5	7.959	0.0127	50	1000
2	39	7.8	0.0135	50	1000
3	35	7.4	0.0142	50	1000
4	36	7.6	0.0143	50	1000

in the Appendix A. The PSHP’s whole data, including discharge function, generation and pump limits, and pump efficiency, are collected from [5]. Three costs of the WP and SP, including direct cost, penalty cost, and reserve cost, in the second simulation scenario are reported in Figure A1 and Figure A2 in the Appendix A. In the system, a location with geographical coordinates of 11°12′39″ and 108°40′53″ in Binh Thuan province is selected to collect wind and solar data by referring to the websites for the wind global atlas [29] and solar global atlas [37]. This location is shown in Figure 3. The wind power plant has fifty turbines, and each turbine has a rated power of 2.5 MW. The hourly wind speed and power are plotted in Figure 4, and it is supposed that the wind speed can be greater than the maximum index by 20% for uncertainty consideration. Finally, the wind power plant’s maximum power is given in Figure 5.

The solar power plant’s rated power is selected to be 150,000-kWp using Tilt and Azimuth of solar panels with 12° and 180°. The maximum solar power is given in Figure 6. The maximum power of wind and solar plants for uncertainty and the electric price are reported in Figure 7.

4.2. Results of the First Simulation Scenario

The one-day energy of the four cascaded power plants is given in Figure 8. Three algorithms can reach greater one-day energy in Case 2 than in Case 1. The greatest energy for each case is found by ISMA, which is 3767.86 MWh for Case 1 and 3922.66 MWh for Case 2. So, Case 2 finds a greater energy than Case 1 by (3922.66–3767.86) = 154.81 MWh, which is about 4.1% of Case 1’s energy. The comparison indicates that the simultaneous optimization operation of the whole system can reach greater energy than planning the optimal operation of each plant. In addition, ISMA is the most suitable algorithm for the first scenario.

The hourly power of each cascaded power plant is given in Figure 9. The sum of hourly energy from Case 2 is greater than Case 1 by 6.89 MW for the first CasHP (CasHP1), 42.09 MW for the second CasHP (CasHP2), −81.76 MW for the third CasHP (CasHP3), and 187.59 MW for the fourth CasHP (CasHP4). Clearly, CasHP3 can reach greater energy in Case 1 than in Case 2, which is different from other plants.

4.3. Results of the Second Simulation Scenario

The results obtained from three algorithms are reported in Figure 10. The maximum profit of ISMA is USD 5,196,221.4, while that of EO and SMA is USD 5,190,213.9 and USD 5,195,570.9, respectively. The exact computation indicates that ISMA can reach a greater total profit than EO and SMA over one operating day by USD 6007.5 and USD 650.5, equivalent to 0.12% and 0.013%, respectively. The lowest total profit of EO, SMA, and ISMA is, respectively, USD 5,176,953.8, USD 5,174,086.4, and USD 5,176,953.8. EO and ISMA have the same lowest profit. The comparisons indicate that ISMA is more suitable than EO and SMA for the study.

The optimal generation and pump power of power plants are reported in

Table A2. Optimal generation and pump power of power plants obtained by ISMA.

h	${\mathit{P}}_{\mathit{t}\mathit{h},\mathit{h}}$ (th = 1)	${\mathit{P}}_{\mathit{t}\mathit{h},\mathit{h}}$ (th = 2)	${\mathit{P}}_{\mathit{t}\mathit{h},\mathit{h}}$ (th = 3)	${\mathit{P}}_{\mathit{t}\mathit{h},\mathit{h}}$ (th = 4)	${\mathit{P}}_{\mathit{w},\mathit{h} }$ (w = 1)	${\mathit{P}}_{\mathit{s}\mathit{p},\mathit{h} }$ (sp = 1)	${\mathit{P}}_{\mathit{p}\mathit{s},\mathit{h}}$ (ps = 1)	$\mathit{P}\mathit{u}\mathit{m}{\mathit{p}}_{\mathit{p}\mathit{s},\mathit{h}}$ (ps = 1)
1	236.51	225.24	234.38	217.14	50.42	0.00	0.00	−300.00
2	240.93	227.27	227.87	226.88	46.40	0.00	0.00	−300.00
3	233.84	225.82	238.36	221.12	45.12	0.00	0.00	−300.00
4	235.68	226.29	236.94	217.69	45.12	0.00	0.00	−300.00
5	341.05	327.89	322.39	320.77	46.40	0.00	0.00	−300.00
6	342.51	318.14	324.27	321.35	49.06	1.01	0.00	−300.00
7	843.62	796.28	776.71	765.58	51.81	18.10	0.00	0.00
8	762.84	729.79	699.98	691.31	49.06	49.60	267.68	0.00
9	802.53	757.44	727.04	725.24	53.22	80.82	300.00	0.00
10	866.87	825.57	797.89	781.57	65.46	106.64	0.00	0.00
11	855.24	812.89	783.02	777.58	87.12	125.29	0.00	0.00
12	804.08	767.77	741.10	725.04	110.77	134.31	155.68	0.00
13	762.50	722.87	701.76	688.43	127.84	133.89	299.38	0.00
14	760.98	720.29	703.61	694.09	138.35	123.54	291.76	0.00
15	771.23	729.18	711.64	695.04	143.81	103.45	272.74	0.00
16	927.48	879.32	853.25	839.84	146.59	73.91	0.00	−300.00
17	859.88	815.26	785.28	773.73	146.59	38.81	0.00	0.00
18	763.69	716.82	703.63	688.06	138.35	6.76	0.00	0.00
19	730.23	694.42	676.96	661.29	117.89	0.00	130.98	0.00
20	734.71	688.88	668.91	659.80	95.28	0.00	163.82	0.00
21	721.41	686.83	660.51	647.79	75.77	0.00	217.96	0.00
22	269.25	259.14	258.56	256.86	65.38	0.00	0.00	−300.00
23	274.27	262.64	257.51	259.54	60.67	0.00	0.00	−300.00
24	277.17	264.37	264.72	257.66	56.12	0	0	−300.00

in Appendix A.

The cost, revenue, and profit of each thermal power plant are shown in Figure 11. The four thermal power plants have the same manner in that revenue and profit are approximately the same from hours 1 to 8 and from hours 22 to 24. For the other remaining hours, the total revenue and profit have a bigger deviation. Checking the optimal solution in Table A2 in the Appendix A, the generation of these power plants is under 350 MW for hours 1–6 and 280 MW for hours 22–24, whereas it is much higher than these values for hours 7–21, which are about from 700 to under 1000 MW. Clearly, these hours cannot reach high profit for the power system. Paying attention to the pump power of PSHP in the last column in the table, we can see the values of—300 MW. It indicates that the PSHP runs pumps over the first six hours and the last three hours. So, the system achieves some profit.

The cost, revenue, and profit of the WP and SP are plotted in Figure 12. The optimal solutions have reached great profit and revenue at hours with high solar radiation and wind speed. The optimal generations in Table A2 in the Appendix A indicate that the plants have low generations at hours 1–6 and 22–24, where SP has 0 MW at the hours. At other hours with high load demand, the generations of the plants are highly used. The profit is directly proportional to the generation. The system’s total cost, revenue, and profit are reported in Figure 13. Thanks to the zero cost of PSHP, the total profit and revenue are high at hours 8–17. The generation of PSHP is about 40 to 140 MW in the period and approximately zero MW at others.

4.4. Discussion on the Advantages and Disadvantages

In this section, the advantages and disadvantages of the study are discussed. The study applied three algorithms, and valid and optimal solutions satisfying all constraints could be found. The proposed ISMA could converge to better solutions than EO and SMA. However, the algorithms need a high setting for control parameters, such as 100 for the population size and 2000 for the iteration number. The solution method to handle all constraints is powerful, reaching a 100% success rate. So, the study reached the advantages of powerful applied algorithms and a high-success-rate solution method.

On the contrary, the study also copes with disadvantages that need to be improved in future work. The study used data obtained from the wind and solar global atlases, which reported the mean wind speed and mean solar radiation of each hour in every month. So, we collected the mean data for each hour for 12 days, where one was represented for one month. There are twelve months, and each hour has twelve wind speed values. The best value among the twelve values of each hour was selected and reported in Figure 6. Similarly, we have values for 24 h in Figure 6. Clearly, the data details for each hour on each day of each month are not provided in the wind and solar global atlases. For solving the uncertainty case, we have supposed that the maximum value for uncertainty case is 120% of the highest real value of each hour, as shown in Figure 4 and Figure 5. The best way to reach more practical data of wind and solar radiation is to use the best day of the year when the wind speed is practically stable during the day. On the other hand, the worst day should be used to collect the minimum data for the uncertainty case. Thus, the practical data are not 100% accurate.

In addition, the study ignored practical factors regarding control technology for pumps and generators in pumped storage hydropower plants and operation change time from pumping mode to generation mode or from generation mode to pumping mode. The status change time has an impact on the generation of the pumped storage hydropower plants to electric power systems, but the problem is not considered in the paper. In this study, the time was supposed to be 0 s. The problem can influence the fluctuation of power, and the power balance constraint cannot be satisfied exactly. The power systems last a few minutes, fluctuating power throughout the process. About the thermal power plants, we have also supposed that the start-up time and costs or power changing time were zero. So, the exact cost was not obtained by using the assumption. Basically, turbine technology in thermal power plants is a factor related to fuel cost, power changing cost, power changing time, and life cycle. The operation schedule was one day for calculating the total profit. Thus, the payback period of the whole system could not be found, and the decision to build the hybrid system was not concluded in this study.

5. Conclusions

The study optimized the maximum total electric sale profit of hybrid power systems with the different power plant types and real data on renewable energies in a specific zone of Vietnam. The generation costs of these thermal power plants and renewable power plants were taken into account, while those of the hydropower plant were neglected. The wind and solar global atlases were employed to collect wind and solar data, and the maximum data for the uncertainty case was supposed to be 120% of the reported data from the wind and solar global atlases. The study ran a hybrid power system with four cascaded hydropower plants, four thermal power plants, one pumped storage hydropower plant, one wind power plant, and one solar power plant over one day. The inflows of the pumped storage hydropower plant were supposed to be zero, and it must consume power generated by other plants to run the pumps. The operation of the cascaded hydropower plant was complicated, and they needed the highest power when operating in the hybrid power system. The plants were optimally operated under two considerations: simultaneous and separate. Then, the most effective solution was employed to run other power plants. The simulation results were obtained by running three algorithms, including the original Slime mould algorithm (SMA), Equilibrium optimizer, and improved Slime mould algorithm (ISMA). The results and contributions are summarized as follows:

• For the one-day energy of the cascaded hydropower plants, ISMA reached greater energy than EO and SMA by 35.92 MWh (0.96%) and 4.62 MWh (0.122%) for the simultaneous operation, and 6.43 MWh (0.164%) and 1.05 MWh (0.027%) for the separate operation.
• For the whole system’s profit, ISMA could reach a greater total profit than EO and SMA by USD 6007.5 (0.12%) and USD 650.5 (0.013%).

From the results above, ISMA performed more effectively than EO and SMA, and the contribution of the study was to provide a high-performance metaheuristic for the optimization operation of the hybrid power system. In addition, the proposed ISMA has achieved a great profit for the whole system. However, the study has faced limits that need to be improved. The study has ignored the power loss on lines connecting power plants together, and power flows on the transmission power grid where power plants are located. In addition, the electric prices in an electric market one day ahead were not considered, but electric prices in distribution power grids were used to optimize the hybrid power system. The investment cost and other operating costs of power plants were not considered to evaluate the chance to recover the investment cost. In practice, wind speed and solar radiation normally changes every five minutes, and the use of hourly data is not as practical. So, the limitations will be considered in future work. All power plant types in the study or in Vietnam will be integrated in a real transmission power grid in Vietnam or standard IEEE transmission power grid. The electric price of one day ahead will be applied to change the price data. The real transmission power grids, real power plants, and real electric market will form a more practical study that can tackle the shortcomings of the current study.