ESMA-OPF: Enhanced Slime Mould Algorithm for Solving Optimal Power Flow Problem

Abstract

In this work, an enhanced slime mould algorithm (ESMA) based on neighborhood dimension learning (NDL) search strategy is proposed for solving the optimal power flow (OPF) problem. Before using the proposed ESMA for solving the OPF problem, its validity is verified by an experiment using 23 benchmark functions and compared with the original SMA, and three other recent optimization algorithms. Consequently, the ESMA is used to solve a modified power flow model including both conventional energy, represented by thermal power generators (TPGs), and renewable energy represented by wind power generators (WPGs) and solar photovoltaic generators (SPGs). Despite the important role of WPGs and SPGs in reducing CO₂ emissions, they represent a big challenge for the OPF problem due to their intermittent output powers. To forecast the intermittent output powers from SPGs and WPGs, Lognormal and Weibull probability density functions (PDFs) are used, respectively. The objective function of the OPF has two extra costs, penalty cost and reserve cost. The penalty cost is added to formulate the underestimation of the produced power from the WPGs and SPGs, while the reserve cost is added to formulate the case of overestimation. Moreover, to decrease CO₂ emissions from TPGs, a direct carbon tax is added to the objective function in some cases. The uncertainty of load demand represents also another challenge for the OPF that must be taken into consideration while solving it. In this study, the uncertainty of load demand is represented by the normal PDF. Simulation results of ESMA for solving the OPF are compared with the results of the conventional SMA and two further optimization methods. The simulation results obtained in this research show that ESMA is more effective in finding the optimal solution of the OPF problem with regard to minimizing the total power cost and the convergence of solution.

1. Introduction

Optimal power flow (OPF) has been a crucial tool for power systems’ operation in an efficient and secured way since its inception in 1962 by Carpentier [1]. OPF is employed for many objectives in power system such as minimizing total generation cost, reducing CO₂ emissions, minimizing power losses, and maintaining voltage stability, as well as maintaining the optimal settings of control variables. This process is commonly constrained by set of restrictions which have to be satisfied, such as power generator capability, bus voltage, transmission line capacity, and any other technical constraints. Conventional OPF problem deals with fossil fuels power stations. When the penetration level of solar photovoltaic (PV) and wind power into the grid increases, it is necessary to study OPF problem taking into consideration the uncertain nature of these sources. In addition to the uncertainty in generation side, there is also uncertainty in the load side that can occur due to several factors such as the induction of electric vehicles [2,3]. Without taking this uncertainty into our consideration, realized load values will differ from the forecast values of load demand, and consequently, the planned operational state through the OPF might be non-optimal.

In the literature, different methods have been provided to model renewable energy sources (RESs) and load demand uncertainties. G-best guided artificial bee colony is provided in [4] to enhance OPF results that are verified in the past literature with similar experimental conditions. In [5], the authors proposed a modified bacteria foraging algorithm to find the best solution of OPF problem that incorporates with a mixture model for thermal power, wind power, and static synchronous compensator (STATCOM). In [6], the authors studied the OPF problem with and without wind power using the particle swarm optimization algorithm, and the Weibull distribution function was used to model the uncertain wind power generation. Reference [7] provided an OPF problem with comprehensive modeling of uncertainty in solar and wind power depending on a combination between success history-based adaptation of differential evolution and the superiority of feasible solution (SF) that is a methodology for handling constraints. The authors in [8] provided a full review of the OPF problem in its diverse formulas. OPF with traditional sources, RESs, and energy storage systems was studied. In [9], a genetic algorithm, the two-point estimation method, and Monte Carlo simulation were applied to solve the OPF problem including RESs and an energy storage system, and the uncertain nature of wind, solar, and load demand was handled by a new strategy. In [10], the authors presented a novel heap optimization algorithm (HOA) to find the optimal solution of the power flow. HOA-based OPF methodology is flexible and applicable compared with that achieved by using the genetic algorithm. A security constrained OPF (SCOPF) problem was studied in [11] incorporating wind energy and responsive demands. Gaussian distribution and Beta distribution were chosen for modeling the load demand and wind power, respectively. The authors assume operating costs, emission, and security indices as single and multi-objective functions. In [12], the OPF problem was studied incorporating several flexible AC transmission system (FACTS) devices with uncertain wind power and uncertain load demand. The uncertain output power of RESs and load demand are handled in [13] by providing uncertainty probability distribution to optimize the conventional generation. In [14], the authors developed an OPF model with stochastic wind power depending on the Lévy coyote optimization algorithm.

The Slime mould algorithm (SMA) is used in many research studies to solve the OPF problem and the economic dispatch. Like many metaheuristic optimization algorithms, SMA suffers from premature convergence during solving complex nonlinear problems. It also suffers from the insufficient precision of optimization and slow optimization speed, which may lead to easily falling into a local optima [15,16,17,18].

In this paper, an enhanced slime mould algorithm (ESMA) is proposed to overcome the drawbacks of the original SMA for solving the OPF problem. The ESMA is proposed by modifying the original SMA based on the neighborhood dimension learning (NDL) search strategy. The validity of the ESMA was tested experimentally through 23 benchmark functions. It was compared with the original (SMA) [19] and three other recent optimization algorithms, including the colony predation algorithm (CPA) [20], the aquila optimizer (AO) [21], and Runge Kutta (RUN) algorithm [22]. A modified IEEE 30–bus system incorporating one SPG and two WPGs was used for testing the validity of the proposed algorithm in reaching the optimal solution for OPF problem with uncertain renewable energy sources (RESs), variable load demand and ramp rate effect of TPGs, in addition to some theoretical cases. Two other optimization algorithms: black widow optimization algorithm (BWOA) [23], and moth swarm algorithm (MSA) [24], in addition to SMA [19] were applied for solving the OPF problem with the same conditions to verify the validity of ESMA comparing to other algorithms. The remaining sections of this research are organized as follows: Section 2 presents the mathematical modeling and the associated system constraints that are applied on the objective functions of the OPF problem. Then, Section 3 provides the modeling of the uncertain output of WPG and SPG. Section 4 presents the ESMA to solve the OPF problem incorporating the uncertain RESs. In Section 5, simulation results are provided including several practical case studies with the four applied algorithms. Finally, conclusions of this paper are presented in Section 6.

2. Problem Formulation

TPGs have constant power outputs, while WPGs and SPG have variable power outputs that must be balanced with the aid of the energy mix of all involved generators and the reserve power, thus the total power cost consists of the operation costs for all the involved power generators, penalty cost, and reserve cost. In this subsection, the cost models of all power generators involved in the objective function of the OPF is performed as follows.

2.1. Cost Model of TPGs

Fossil fuel is used to operate TPGs. Equation (1) shows how to calculate the fossil fuel cost ($/h) of all TPGs.

$${\mathrm{C}}_{\mathrm{FF}}={\displaystyle \sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{TPG}}}{\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^2$$

(1)

The valve-point loading has an impact on the total fuel cost, so taking it into consideration will give more realistic and precise cost as shown in (2).

$${\mathrm{C}}_{\mathrm{FF}}={\sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{TPG}}}{\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^2+\left|{\mathrm{d}}_{\mathrm{i}}\ast \sin \left({\mathrm{e}}_{\mathrm{i}}\ast \left({\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\min }-{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}\right)\right)\right|$$

(2)

All coefficients of cost and emission for the TPGs that are used in cost calculations are recorded in

Table 1. Cost and emission Coefficients for TPGs [25].

Gen.	Bus	a	B	c	d	e	α	β	$\mathbf{\gamma }$	ω	μ	${\mathbf{P}}_{\mathbf{T}\mathbf{P}\mathbf{G}\mathbf{i}}^0 \left(\mathbf{MW}\right)$	$\mathbf{D}{\mathbf{r}}_{\mathbf{i}} \left(\mathbf{MW}\right)$	$\mathbf{U}{\mathbf{r}}_{\mathbf{i}} \left(\mathbf{MW}\right)$
TPG1	1	0	2	0.00375	18	0.037	4.091	−5.554	6.49	0.0002	6.667	99.211	20	15
TPG2	2	0	1.75	0.0175	16	0.038	2.543	−6.047	5.638	0.0005	3.333	80	15	10
TPG3	8	0	3.25	0.00834	12	0.045	5.326	−3.55	3.38	0.002	2	20	8	4

.

2.2. Direct Power Cost of SPG and WPGs

When private parties have the ownership of SPG or WPG, independent system operators (ISO) will pay an amount of money proportional to the contractually scheduled output powers.

In this concept, the direct cost of wind power produced from the jth WPG in proportion to its scheduled power has been provided in (3).

$${\mathrm{C}}_{\mathrm{dw},\mathrm{j}}={\mathrm{P}}_{\mathrm{sw},\mathrm{j}}{\ast \mathrm{h}}_{\mathrm{j}}$$

(3)

Following the same method, the direct cost of the kth SPG can be modelled by

$${\mathrm{C}}_{\mathrm{ds},\mathrm{k}}={\mathrm{P}}_{\mathrm{ssl},\mathrm{k}}{\ast \mathrm{g}}_{\mathrm{k}}$$

(4)

2.3. Cost Assessment for Uncertain Wind Power Output

One of the most challenging characteristics of wind energy is its intermittent output, so two situations might arise. The first one happens if the actual output power from the WPG is lower than the anticipated power output. This means that there is an overestimation of the output of WPGs. In this case, ISO uses the spinning power reserved for maintaining the reliability of the supplied power to its consumers.

To overcome this situation, reserve cost is need to committee reserve power units. This cost can be determined for the jth WPG as follows:

$${\mathrm{C}}_{\mathrm{rw},\mathrm{j}}\left({\mathrm{P}}_{\mathrm{ws},\mathrm{j}}-{\mathrm{P}}_{\mathrm{wav},\mathrm{j}}\right)={\mathrm{K}}_{\mathrm{rw},\mathrm{j}}\left({\mathrm{P}}_{\mathrm{ws},\mathrm{j}}-{\mathrm{P}}_{\mathrm{wav},\mathrm{j}}\right)={\mathrm{K}}_{\mathrm{rw},\mathrm{j}}{{\displaystyle \int }}_0^{{\mathrm{P}}_{\mathrm{ws},\mathrm{j}}}({\mathrm{P}}_{\mathrm{ws},\mathrm{j}}-{\mathrm{P}}_{\mathrm{w},\mathrm{j}})\times {\mathrm{f}}_{\mathrm{w}}\left({\mathrm{P}}_{\mathrm{w},\mathrm{j}}\right){\mathrm{dP}}_{\mathrm{w},\mathrm{j}}$$

(5)

The other situation happens if the actual output of the WPG has a value greater than expected. This means that there is an underestimation in the output power. Then, there will be a remaining power that will be dissipated if there is not any possibility to decrease the output of TPGs. In this regard, ISO should pay a penalty cost according to the rest power. This cost can be determined for the jth WPG by

$${\mathrm{C}}_{\mathrm{pw},\mathrm{j}}\left({\mathrm{P}}_{\mathrm{wav},\mathrm{j}}-{\mathrm{P}}_{\mathrm{ws},\mathrm{j}}\right)={\mathrm{K}}_{\mathrm{pw},\mathrm{j}}\left({\mathrm{P}}_{\mathrm{wav},\mathrm{j}}-{\mathrm{P}}_{\mathrm{ws},\mathrm{j}}\right)={\mathrm{K}}_{\mathrm{pw},\mathrm{j}}{{\displaystyle \int }}_{{\mathrm{P}}_{\mathrm{ws},\mathrm{j}}}^{{\mathrm{P}}_{\mathrm{wr},\mathrm{j}}}({\mathrm{P}}_{\mathrm{w},\mathrm{j}}-{\mathrm{P}}_{\mathrm{ws},\mathrm{j}})\times {\mathrm{f}}_{\mathrm{w}}\left({\mathrm{P}}_{\mathrm{w},\mathrm{j}}\right){\mathrm{dP}}_{\mathrm{w},\mathrm{j}}$$

(6)

2.4. Cost Assessment for Uncertain Solar PV Power

Solar output power has also intermittent and uncertain nature, so the same concept followed to deal with underestimation and overestimation scenarios of output power of WPGs will be followed with solar energy. In this work, solar radiation will follow lognormal probability density functions (PDF) [26], which differs from the Weibull PDF followed by wind power.

Thus, for making calculations simpler, the models of penalty cost and reserve cost are designed in a similar concept used in [27].

Section 3 provides more details for stochastic wind and solar power calculation.

For the kth SPG, the reserve cost is given as follows:

$$\begin{gathered} {\mathrm{C}}_{\mathrm{rs},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{ss},\mathrm{k}}-{\mathrm{P}}_{\mathrm{sav},\mathrm{k}}\right)={\mathrm{K}}_{\mathrm{rs},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{ss},\mathrm{k}}-{\mathrm{P}}_{\mathrm{sav},\mathrm{k}}\right) \\ ={\mathrm{K}}_{\mathrm{rs},\mathrm{k}}\times {\mathrm{f}}_{\mathrm{s}}({\mathrm{P}}_{\mathrm{sav},\mathrm{k}}<{\mathrm{P}}_{\mathrm{ss},\mathrm{k}})\times [({\mathrm{P}}_{\mathrm{ss},\mathrm{k}}-\mathrm{Ex}({\mathrm{P}}_{\mathrm{sav},\mathrm{k}}<{\mathrm{P}}_{\mathrm{ss},\mathrm{k}})] \end{gathered}$$

(7)

Equation (8) shows the formulation of the penalty cost corresponding to the kth SPG.

$$\begin{gathered} {\mathrm{C}}_{\mathrm{ps},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{sav},\mathrm{k}}-{\mathrm{P}}_{\mathrm{ss},\mathrm{k}}\right)={\mathrm{K}}_{\mathrm{ps},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{sav},\mathrm{k}}-{\mathrm{P}}_{\mathrm{ss},\mathrm{k}}\right) \\ ={\mathrm{K}}_{\mathrm{ps},\mathrm{k}}\times {\mathrm{f}}_{\mathrm{s}}({\mathrm{P}}_{\mathrm{sav},\mathrm{k}}>{\mathrm{P}}_{\mathrm{ss},\mathrm{k}})\times [(\mathrm{Ex}({\mathrm{P}}_{\mathrm{sav},\mathrm{k}}>{\mathrm{P}}_{\mathrm{ss},\mathrm{k}})-{\mathrm{P}}_{\mathrm{ss},\mathrm{k}}] \end{gathered}$$

(8)

2.5. Emissions and Carbon Tax

Producing electricity through burning fossil fuels is one of the most emitting sources of greenhouse gases into the surrounding environment. Emissions are determined in (t/h) as follows:

$$\mathrm{E}={\sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{TPG}}}[({\mathsf{\alpha }}_{\mathrm{i}}+{\mathsf{\beta }}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}+{\mathsf{\gamma }}_{\mathrm{i}}{\mathrm{P}}^2{}_{\mathrm{TPG},\mathrm{i}})\times 0.01+{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{e}}^{\left({\mathsf{\mu }}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}\right)}]$$

(9)

The emission coefficients values of the TPGs provided in Table 1 are the same as in [25] with only small modification in the value of μ of the TPG connected at bus 1.

Due to the dangerous effects associated with climate change phenomena, regulators in many countries tend to set severe rules on power sectors to diminish the carbon emission (carbon footprint) [28] by applying a carbon tax which is one of the most useful methods for pushing investors towards renewable energy projects. The emission cost ($/h) is determined by

$${\mathrm{C}}_{\mathrm{E}}={\mathrm{C}}_{\mathrm{tax}}\mathrm{E}$$

(10)

2.6. Objective Functions of the OPF

The objective function (F1) of the OPF include all the previously presented cost models from (2) to (8). F1 neglects the cost of emissions, while another objective function (F2) is proposed with considering the cost of emissions to study its impact on the generation scheduling. F2 is expressed with the same cost models used in F1 in addition to the cost of emissions provided in (9) and (10).

Hence, the objective is to minimize F1 shown in (11):

$$\begin{gathered} \mathrm{F}1={\mathrm{C}}_{\mathrm{FF}}\left({\mathrm{P}}_{\mathrm{TPG}}\right)+{\displaystyle \sum }_{\mathrm{j}=1}^{{\mathrm{N}}_{\mathrm{WPG}}}\left[{\mathrm{h}}_{\mathrm{j}}{\mathrm{P}}_{\mathrm{ws},\mathrm{j}}+{\mathrm{K}}_{\mathrm{rw},\mathrm{j}}\left({\mathrm{P}}_{\mathrm{ws},\mathrm{j}}-{\mathrm{P}}_{\mathrm{wav},\mathrm{j}}\right)+{\mathrm{K}}_{\mathrm{pw},\mathrm{j}}\left({\mathrm{P}}_{\mathrm{wav},\mathrm{j}}-{\mathrm{P}}_{\mathrm{ws},\mathrm{j}}\right)\right] \\ +{\displaystyle \sum }_{\mathrm{j}=1}^{{\mathrm{N}}_{\mathrm{SPG}}}\left[{\mathrm{g}}_{\mathrm{k}}{\mathrm{P}}_{\mathrm{ss},\mathrm{k}}+\left({\mathrm{P}}_{\mathrm{ss},\mathrm{k}}-{\mathrm{P}}_{\mathrm{sav},\mathrm{k}}\right)+{\mathrm{K}}_{\mathrm{ps},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{slav},\mathrm{k}}-{\mathrm{P}}_{\mathrm{ssl},\mathrm{k}}\right)\right] \end{gathered}$$

(11)

To minimize F2 that is formulated as follows:

$$\mathrm{F}2=\mathrm{F}1+{\mathrm{C}}_{\mathrm{tax}}\mathrm{E}$$

(12)

Both system equality and inequality constraints are imposed on the above mentioned OPF objective functions as shown below:

2.6.1. Equality Constraints

To provide an active and reactive power balance in the system, equality constraints should be applied for the system. Equality means that these active and reactive powers should be equivalent to summation of total load and total power loss in the network. Consequently, equality constraints have been given as follows [7]:

$${\mathrm{P}}_{\mathrm{Gi}}={\mathrm{P}}_{\mathrm{Di}}+{\mathrm{V}}_{\mathrm{i}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{NB}}{\mathrm{V}}_{\mathrm{j}}\left[{\mathrm{G}}_{\mathrm{ij}}\cos \left({\mathsf{\delta }}_{\mathrm{ij}}\right)+{\mathrm{B}}_{\mathrm{ij}}\sin \left({\mathsf{\delta }}_{\mathrm{ij}}\right)\right] {⩝} \mathrm{i}\in \mathrm{NB}$$

(13)

$${\mathrm{Q}}_{\mathrm{Gi}}={\mathrm{Q}}_{\mathrm{Di}}+{\mathrm{V}}_{\mathrm{i}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{NB}}{\mathrm{V}}_{\mathrm{j}}\left[{\mathrm{G}}_{\mathrm{ij}}\sin \left({\mathsf{\delta }}_{\mathrm{ij}}\right)-{\mathrm{B}}_{\mathrm{ij}}\cos \left({\mathsf{\delta }}_{\mathrm{ij}}\right)\right] {⩝} \mathrm{i}\in \mathrm{NB}$$

(14)

2.6.2. Inequality Constraints

Equipment and components existing in any power system operate within operating limits which are called inequality constraints. These constraints can be expressed as follows:

1.

Constraints of Generators:

Equations (15)–(17) represent the limits of active power generation for TPGs, WPGs, and SPGs, respectively.

$${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\min }\le {\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}\le {\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\max },\mathrm{i}=1,\dots .\dots \dots ,{\mathrm{N}}_{\mathrm{TPG}}$$

(15)

$${\mathrm{P}}_{\mathrm{ws},\mathrm{j}}^{\min }\le {\mathrm{P}}_{\mathrm{ws},\mathrm{j}}\le {\mathrm{P}}_{\mathrm{ws},\mathrm{j}}^{\max },\mathrm{j}=1,\dots .\dots .,{\mathrm{N}}_{\mathrm{WPG}}$$

(16)

$${\mathrm{P}}_{\mathrm{ssl},\mathrm{k}}^{\min }\le {\mathrm{P}}_{\mathrm{ssl},\mathrm{k}}\le {\mathrm{P}}_{\mathrm{ssl},\mathrm{k}}^{\max },\mathrm{k}=1,\dots .\dots \dots ,{\mathrm{N}}_{\mathrm{SPG}}$$

(17)

Reactive power limits of all generators are showed by (18) to (20) with the same arrangement in the previous active power limits equations.

$${\mathrm{Q}}_{\mathrm{TPG},\mathrm{i}}^{\min }\le {\mathrm{Q}}_{\mathrm{TPG},\mathrm{i}}\le {\mathrm{Q}}_{\mathrm{TPG},\mathrm{i}}^{\max },\mathrm{i}=1,\dots .\dots .,{\mathrm{N}}_{\mathrm{TPG}}$$

(18)

$${\mathrm{Q}}_{\mathrm{ws},\mathrm{j}}^{\min }\le {\mathrm{Q}}_{\mathrm{ws},\mathrm{j}}\le {\mathrm{Q}}_{\mathrm{ws},\mathrm{j}}^{\max },\mathrm{j}=1,\dots ,{\mathrm{N}}_{\mathrm{WPG}}$$

(19)

$${\mathrm{Q}}_{\mathrm{ssl},\mathrm{k}}^{\min }\le {\mathrm{Q}}_{\mathrm{ssl},\mathrm{k}}\le {\mathrm{Q}}_{\mathrm{ssl},\mathrm{k}}^{\max }, \mathrm{k}=1,\dots .\dots .,{\mathrm{N}}_{\mathrm{SPG}}$$

(20)

Equation (21) shows the constraints applied on generator buses voltage.

$${\mathrm{V}}_{\mathrm{Gi}}^{\min }\le {\mathrm{V}}_{\mathrm{Gi}}\le {\mathrm{V}}_{\mathrm{Gi}}^{\max }, \mathrm{i}=1,\dots .\dots \dots ,{\mathrm{N}}_{\mathrm{G}}$$

(21)

2.

Security constraints:

The voltage limits applied to load buses are shown by (22), while (23) gives the capacity constraints of transmission lines.

$${\mathrm{V}}_{{\mathrm{L}}_{\mathrm{p}}}^{\min }\le {\mathrm{V}}_{{\mathrm{L}}_{\mathrm{p}}}\le {\mathrm{V}}_{{\mathrm{L}}_{\mathrm{p}}}^{\max }, \mathrm{p}=1,\dots .\dots \dots ,{\mathrm{N}}_{\mathrm{B}}$$

(22)

$${\mathrm{S}}_{{\mathrm{L}}_{\mathrm{q}}}\le {\mathrm{S}}_{{\mathrm{L}}_{\mathrm{q}}}^{\max }, \mathrm{q}=1,\dots .\dots \dots ,{\mathrm{N}}_{\mathrm{L}}$$

(23)

The active power losses (P_loss) in transmission lines and voltage deviation (V_d) represent also essential parameters when studying OPF problem. It is not allowable to avoid the active power loss in transmission lines because of their inherent resistances. P_loss of the grid is given by

$${\mathrm{P}}_{\mathrm{loss}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{NL}}{\displaystyle \sum }_{\mathrm{j}\ne 1}^{\mathrm{NL}}\left({\mathrm{G}}_{\mathrm{ij}}\times \left[{\mathrm{V}}_{\mathrm{i}}^2+{\mathrm{V}}_{\mathrm{j}}^2-2{\mathrm{V}}_{\mathrm{i}}{ \mathrm{V}}_{\mathrm{j}}\cos \left({\mathsf{\delta }}_{\mathrm{ij}}\right)\right]\right)$$

(24)

Equation (25) refers to the voltage deviation (V_d) which indicates how much load buses voltages, in a power system, deviate from a nominal voltage value which usually is 1 p.u. V_d gives a sign for the voltage quality of the power system.

$${\mathrm{V}}_{\mathrm{d}}={{\displaystyle \sum }}_{\mathrm{p}=1}^{{\mathrm{N}}_{\mathrm{L}}}\left|{\mathrm{V}}_{\mathrm{Lp}}-1\right|$$

(25)

3.

TPGs Ramp Rate Constraints:

In practical conditions, the produced power from a thermal power generator should follow ramp rates because of the physical limitations of thermal generators. This means that to rise the generated power from a thermal generator, it follows the up-ramp rate. In contrast, to reduce the generated power, it follows the down-ramp rate. In this regard, the limits of ramp rate can be formed as follows:

$$\max ({\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\min },{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^0-{\mathrm{Dr}}_{\mathrm{i}})\le {\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}\le \min \left({\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\max },{ \mathrm{P}}_{\mathrm{Gt},\mathrm{i}}^0+{\mathrm{Ur}}_{\mathrm{i}}\right), \mathrm{i}=1,\dots ,{\mathrm{N}}_{\mathrm{TPG}}$$

(26)

The constraints of ramp-rate in OPF study are formulated as following:

$${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}-{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^0\le {\mathrm{Ur}}_{\mathrm{i}}, \mathrm{if} \mathrm{generation} \mathrm{increases}$$

$${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^0-{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}\le {\mathrm{Dr}}_{\mathrm{i}}, \mathrm{if} \mathrm{generation} \mathrm{decreases}$$

It is worth pointing out that the presence of ramp-rate limits will increase the cost of generation due to changing the new operating points of TPGs from their original operating points without ramp rate constraints. The new limits of a TPG with limits of ramp-rate should not outdo the limits (${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\mathrm{low}}$) and (${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\mathrm{high}}$) which are provided in (27) and (28), respectively.

$${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\mathrm{low}}=\max ({\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\min },{\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^0-{\mathrm{Dr}}_{\mathrm{i}})$$

(27)

$${\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\mathrm{high}}=\min ({\mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^{\max },{ \mathrm{P}}_{\mathrm{TPG},\mathrm{i}}^0+{\mathrm{Ur}}_{\mathrm{i}})$$

(28)

3. Stochastic Solar Power, Wind Power, and Models of Uncertainty

This Weibull PDF is followed by wind speed distribution for calculations of mean power of wind turbines as presented in [5,27]. The probability (${\mathrm{f}}_{{\mathrm{v}}_{\mathrm{w}}}$) of wind speed (${\mathrm{v}}_{\mathrm{w}}$) in (m/s) based on Weibull PDF has been provided as follows:

$${\mathrm{f}}_{{\mathrm{v}}_{\mathrm{w}}}\left({\mathrm{v}}_{\mathrm{w}}\right)=\left(\frac{\mathrm{k}}{\mathrm{c}}\right)+{\left(\frac{{\mathrm{v}}_{\mathrm{w}}}{\mathrm{c}}\right)}^{\left(\mathrm{k}-1\right)}{\mathrm{e}}^{-{\left(\frac{{\mathrm{v}}_{\mathrm{w}}}{\mathrm{c}}\right)}^{\mathrm{k}}} \mathrm{for} 0<{\mathrm{v}}_{\mathrm{w}}<\infty$$

(29)

In (30), the mean of Weibull PDF is defined as follows:

$${\mathrm{M}}_{\mathrm{wbl}}={\mathrm{c}\ast \mathsf{\Gamma }}_{\mathrm{w}}\left(1+{\mathrm{k}}^{-1}\right)$$

(30)

where, ${\mathsf{\Gamma }}_{\mathrm{w}}$ indicates gamma function which is defined as

$${\mathsf{\Gamma }}_{\mathrm{w}}\left(\mathrm{x}\right)={{\displaystyle \int }}_0^{\infty }{\mathrm{e}}^{-\mathrm{t}}{\mathrm{t}}^{\mathrm{x}-1}\mathrm{dt}$$

(31)

In this study, the IEEE-30 bus system is modified via replacing two TPGs at bus 5 and bus 11 with two WPGs in addition to replacing TPG at bus 13 with SPG, as shown in Figure 1. Other system characteristics are provided in

Table 2. Characteristics of the modified IEEE-30 bus system [7].

Item	Number	Details
Buses	30	[29]
Branches	41	[29]
TPGs (TPG1, TPG2, and TPG3)	3	Connected at bus 1 (swing), bus 2, and bus 8
WPGs (WPG1 and WPG2)	2	Connected at bus 5 and bus 11
SPG	1	Connected at bus 13
Control variables	11	The scheduled output power of: TPG2, TPG3, WPG1, WPG2, SPG, and generator buses voltages (6 generators)
Connected load	-	283.4 MW, 126.2 MVAR
Allowable voltage range for load buses	24	[0.95–1.05] p.u.

. The selected values of Weibull scale ($\mathrm{c}$) and shape ($\mathrm{k}$) parameters are mentioned in

Table 3. Weibull and lognormal PDF Parameters for WPGs and SPG.

WPGs					SPG
WPG	No. of Turbines	Rated Power (Pwr)	Weibull PDF Parameters	Weibull Mean, Mwbl	Rated Power, Psr	Lognormal PDF Parameters	Lognormal Mean, Mlogn
1 (at bus 5)	25	75 MW	k = 2, c = 9	v = 7.976 (m/s)
2 (at bus 11)	20	60 MW	k = 2, c = 10	v = 8.862 (m/s)	50 MW (at bus 13)	δ = 0.6, μ = 6	I = 483 (W/m2)

. Weibull fitting and distributions of wind frequency are provided in Figure 2. They are obtained through Monte Carlo simulation running with 8000 iterations [7].

Standard [30] regulates the design requirements of wind turbines and determines wind turbine class IA which is the highest turbulent and proper for operating at the maximum annual average wind speed of 10 m/s at hub height. Parameters of wind farm scale ($\mathrm{c}$) and shape ($\mathrm{k}$), have been chosen sensibly in order to keep the mean of Weibull PDF at a maximum value nearby 10. This will help in reaching both diverse and realistic geographic places for the two WPGs.

The output power from the SPG is in a direct relationship with the irradiance (I) that tracks lognormal PDF [26]. The solar irradiance probability ${\mathrm{f}}_{\mathrm{I}}$ with standard deviation ($\mathsf{\sigma }$) and mean ($\mathsf{\mu }$), is determined by

$${\mathrm{f}}_{\mathrm{I}}\left(\mathrm{I}\right)=\frac{1}{\mathrm{I}\mathsf{\sigma }\sqrt{2\mathsf{\pi }}}\exp \left\{\frac{-{\left(\ln \mathrm{x}-\mathsf{\mu }\right)}^2}{2{\mathsf{\sigma }}^2}\right\}\mathrm{for} \mathrm{I}>0$$

(32)

Equation (33) presents the mean parameter of the lognormal PDF (${\mathrm{M}}_{\mathrm{logn}}$).

$${\mathrm{M}}_{\mathrm{logn}}=\exp (\mathsf{\mu }+\frac{{\mathsf{\sigma }}^2}{2})$$

(33)

The distribution of frequency and lognormal probability for the solar irradiance obtained by running Monte-Carlo simulation with a sample size of 8000 iterations are shown in Figure 3. The selected values of lognormal PDF parameters are provided in Table 3.

3.1. Models of Solar and Wind Powers

As previously demonstrated, there are two WPGs in the new IEEE-30 bus power system. WPG 1 is linked to bus 5, and its cumulative active power is 75 MW. WPG 2 is linked to bus 11, and its cumulative active power equals 60 MW.

The output power of a wind turbine (${\mathrm{P}}_{\mathrm{w}}$) depends on the speed of wind (${\mathrm{v}}_{\mathrm{w}}$) as described below [7]:

$${\mathrm{P}}_{\mathrm{w}}\left({\mathrm{v}}_{\mathrm{w}}\right)=\left\{\begin{array}{c}0 \mathrm{for} {\mathrm{v}}_{\mathrm{w}}<{\mathrm{v}}_{\mathrm{in},\mathrm{w}} \mathrm{and} {\mathrm{v}}_{\mathrm{w}}>{\mathrm{v}}_{\mathrm{out},\mathrm{w}}\\ {\mathrm{P}}_{\mathrm{wr}}\left(\frac{{\mathrm{v}}_{\mathrm{w}}-{\mathrm{v}}_{\mathrm{in},\mathrm{w}}}{{\mathrm{v}}_{\mathrm{r},\mathrm{w}}-{\mathrm{v}}_{\mathrm{in},\mathrm{w}}}\right) \mathrm{for} {\mathrm{v}}_{\mathrm{in},\mathrm{w}}\le {\mathrm{v}}_{\mathrm{w}}\le {\mathrm{v}}_{\mathrm{r},\mathrm{w}}\\ {\mathrm{P}}_{\mathrm{wr}} \mathrm{for} {\mathrm{v}}_{\mathrm{r},\mathrm{w}}\le {\mathrm{v}}_{\mathrm{w}}\le {\mathrm{v}}_{\mathrm{out},\mathrm{w}}\end{array}\right.$$

(34)

Enercon E82-E4, which is a product datasheet of wind turbine, states the different speeds of a wind turbine with rated power 3 MW as follows: ${\mathrm{v}}_{\mathrm{in},\mathrm{w}}$ = 3 m/s, ${\mathrm{v}}_{\mathrm{out},\mathrm{w}}$ = 25 m/s, and ${\mathrm{v}}_{\mathrm{r},\mathrm{w}}$ = 16 m/s.

Likewise, the output power of SPG, depending on solar irradiance (I), as stated below [31]:

$${\mathrm{P}}_{\mathrm{s}}\left(\mathrm{I}\right)=\left\{\begin{array}{c}{\mathrm{P}}_{\mathrm{sr}}\left(\frac{{\mathrm{I}}^2}{{\mathrm{I}}_{\mathrm{std}}{\mathrm{R}}_{\mathrm{c}}}\right) \mathrm{for} 0<\mathrm{I}<{\mathrm{R}}_{\mathrm{c}}\\ {\mathrm{P}}_{\mathrm{sr}}\left(\frac{\mathrm{I}}{{\mathrm{I}}_{\mathrm{std}}}\right) \mathrm{for} \mathrm{I}\ge {\mathrm{R}}_{\mathrm{c}}\end{array}\right.$$

(35)

3.2. Probability Calculation of Wind Power

The output power of WPG, according to (34), has a discrete nature in certain wind speed regions. The value of power output will be zero when the value of ${\mathrm{v}}_{\mathrm{w}}$ is below ${\mathrm{v}}_{\mathrm{in},\mathrm{w}}$ and above ${\mathrm{v}}_{\mathrm{out},\mathrm{w}}$, while the wind turbine will produce the rated value ${\mathrm{P}}_{\mathrm{wr}}$ when the value of ${\mathrm{v}}_{\mathrm{w}}$ is between ${\mathrm{v}}_{\mathrm{r},\mathrm{w}}$ and ${\mathrm{v}}_{\mathrm{out},\mathrm{w}}$. For the illustrated discrete zones, the probabilities of wind turbine output power are provided by [32]:

$${\mathrm{f}}_{\mathrm{w}}\left({\mathrm{P}}_{\mathrm{w}}\right)\left\{{\mathrm{P}}_{\mathrm{w}}=0\right\}=1-\exp \left[-{\left(\frac{{\mathrm{v}}_{\mathrm{in},\mathrm{w}}}{\mathrm{c}}\right)}^{\mathrm{k}}\right]+ \exp \left[-{\left(\frac{{\mathrm{v}}_{\mathrm{out},\mathrm{w}}}{\mathrm{c}}\right)}^{\mathrm{k}}\right]$$

(36)

$${\mathrm{f}}_{\mathrm{w}}\left({\mathrm{P}}_{\mathrm{w}}\right)\left\{{\mathrm{P}}_{\mathrm{w}}={\mathrm{P}}_{\mathrm{wr}}\right\}=\exp \left[-{\left(\frac{{\mathrm{v}}_{\mathrm{r},\mathrm{w}}}{\mathrm{c}}\right)}^{\mathrm{k}}\right]-\exp \left[-{\left(\frac{{\mathrm{v}}_{\mathrm{out},\mathrm{w}}}{\mathrm{c}}\right)}^{\mathrm{k}}\right]$$

(37)

The wind turbine has a continuous output power between ${\mathrm{v}}_{\mathrm{in},\mathrm{w}}$ and ${\mathrm{v}}_{\mathrm{r},\mathrm{w}}$. Thus, for the continuous zone, the probabilities of wind turbine output power are provided by [33]

$${\mathrm{f}}_{\mathrm{w}}\left({\mathrm{P}}_{\mathrm{w}}\right)=\frac{\mathrm{k}\left({\mathrm{v}}_{\mathrm{r},\mathrm{w}}-{\mathrm{v}}_{\mathrm{in},\mathrm{w}}\right)}{{\mathrm{c}}^{\mathrm{k}}\ast {\mathrm{P}}_{\mathrm{Wr}}}{\left[{\mathrm{v}}_{\mathrm{in},\mathrm{w}}+\frac{{\mathrm{P}}_{\mathrm{w}}}{{\mathrm{P}}_{\mathrm{wr}}}\left({\mathrm{v}}_{\mathrm{r},\mathrm{w}}-{\mathrm{v}}_{\mathrm{in},\mathrm{w}}\right)\right]}^{\mathrm{k}-1}\times \exp \left[-{\left(\frac{{\mathrm{v}}_{\mathrm{in},\mathrm{w}}+\frac{{\mathrm{P}}_{\mathrm{w}}}{{\mathrm{P}}_{\mathrm{wr}}}\left({\mathrm{v}}_{\mathrm{r},\mathrm{w}}-{\mathrm{v}}_{\mathrm{in},\mathrm{w}}\right)}{\mathrm{c}}\right)}^{\mathrm{k}}\right]$$

(38)

3.3. Over/Underestimation Cost of Solar Power

The SPG produces a power that has a stochastic nature. Figure 4 verifies this nature, and it also defines the scheduled output power of the SPG to be transferred to the grid through the dotted line. It is important to notice that the scheduled power of SPG is an inconstant value, thus the owner of the SPG and the ISO should sign a jointly power agreement. Equations (39) and (40) are developed for calculating the cost of under- and overestimation (${\mathrm{C}}_{\mathrm{sp},\mathrm{k}},{ \mathrm{C}}_{\mathrm{sr},\mathrm{k}}$) in the output power of the SPG, respectively.

$${\mathrm{C}}_{\mathrm{sp},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{sav}}-{\mathrm{P}}_{\mathrm{ss}}\right)={\mathrm{K}}_{\mathrm{ps}}\left({\mathrm{P}}_{\mathrm{sav}}-{\mathrm{P}}_{\mathrm{ss}}\right)={\mathrm{K}}_{\mathrm{ps}}{\displaystyle \sum }_{\mathrm{n}=1}^{{\mathrm{N}}_{\mathrm{s}}^{+}}\left[{\mathrm{P}}_{\mathrm{sn}+}-{\mathrm{P}}_{\mathrm{ss}}\right] \ast {\mathrm{f}}_{\mathrm{sn}+}$$

(39)

$${\mathrm{C}}_{\mathrm{sr},\mathrm{k}}\left({\mathrm{P}}_{\mathrm{ss}}-{\mathrm{P}}_{\mathrm{sav}}\right)={\mathrm{K}}_{\mathrm{rs}}\left({\mathrm{P}}_{\mathrm{ss}}-{\mathrm{P}}_{\mathrm{sav}}\right)={\mathrm{K}}_{\mathrm{rs}}{\displaystyle \sum }_{\mathrm{n}=1}^{{\mathrm{N}}_{\mathrm{s}}^{-}}\left[{\mathrm{P}}_{\mathrm{ss}}-{\mathrm{P}}_{\mathrm{sn}-}\right] \ast {\mathrm{f}}_{\mathrm{sn}-}$$

(40)

where ${\mathrm{P}}_{\mathrm{sn}+}$ and ${\mathrm{P}}_{\mathrm{sn}-}$ indicate the surplus and shortage powers which lie on the left and right half plane of schedule power ${\mathrm{P}}_{\mathrm{ss}}$ in Figure 4, respectively. Similarly, ${\mathrm{f}}_{\mathrm{sn}+} \mathrm{and} {\mathrm{f}}_{\mathrm{sn}-}$ represent relative frequencies of the happening of ${\mathrm{P}}_{\mathrm{sn}+} \mathrm{and} {\mathrm{P}}_{\mathrm{sn}-}$. While ${\mathrm{N}}_{\mathrm{s}}^{+} \mathrm{and} {\mathrm{N}}_{\mathrm{s}}^{-}$ are the numbers of discrete bins lie on the right and left plane of ${\mathrm{P}}_{\mathrm{ss}}$ for the generation of PDF, respectively.

4. Optimization Algorithm

4.1. Original SMA

The slime mould optimization algorithm is developed in [19]. It generally denotes the physarum polycephalum and it has been named by this name because its first classification was under category of fungus [34].

The mathematical model of SMA is as follows [19]:

4.1.1. Food Approach

Based on the smell in the surrounding air, slime mould can approach food [19]. Equations (41)–(44) is provided to imitate the contraction method:

$$\overrightarrow{\mathrm{X}\left(\mathrm{t}+1\right)}=\left\{\begin{array}{c}\overrightarrow{{\mathrm{X}}_{\mathrm{b}}\left(\mathrm{t}\right)}+\overrightarrow{\mathrm{vb}}\times (\overrightarrow{\mathrm{W}}\times \overrightarrow{{\mathrm{X}}_{\mathrm{A}}\left(\mathrm{t}\right)}-\overrightarrow{{\mathrm{X}}_{\mathrm{B}}\left(\mathrm{t}\right)}), \mathrm{r}<\mathrm{p}\\ \overrightarrow{\mathrm{vc}}\times \overrightarrow{\mathrm{X}\left(\mathrm{t}\right)}, \mathrm{r}\ge \mathrm{p}\end{array}\right.$$

(41)

$$\overrightarrow{\mathrm{vb}}=\left[-\mathrm{a},\mathrm{a}\right]$$

(42)

$$\mathrm{a}=\mathrm{arctanh}\left(-\left(\frac{\mathrm{t}}{\max \_\mathrm{iter}}\right)+1\right)$$

(43)

$$\mathrm{p}=\tan \mathrm{h}\left|\mathrm{S}\left(\mathrm{i}\right)-\mathrm{DF}\right|$$

(44)

The slime mould weight ($\overrightarrow{\mathrm{W}}$) is calculated as follows:

$$\overrightarrow{\mathrm{W}\left(\mathrm{SmellIndex}\left(\mathrm{i}\right)\right)}=\left\{\begin{array}{c}1+\mathrm{r}\ast \log \left(\frac{\mathrm{bF}-\mathrm{S}\left(\mathrm{i}\right)}{\mathrm{bF}-\mathrm{wF}}+1\right), \mathrm{condition}\\ 1-\mathrm{r}\ast \log \left(\frac{\mathrm{bF}-\mathrm{S}\left(\mathrm{i}\right)}{\mathrm{bF}-\mathrm{wF}}+1\right), \mathrm{others}\end{array}\right.$$

(45)

$$\mathrm{SmellIndex} = \mathrm{sort}\left(\mathrm{S}\right)$$

(46)

In (45), condition means that S(i) ranks the first half of population.

4.1.2. Wrap Food

The mathematical approach followed by slime mould for updating its location is as follows:

$$\overrightarrow{\mathrm{W}\left(\mathrm{SmellIndex}\left(\mathrm{i}\right)\right)}=\left\{\begin{array}{c}1+\mathrm{r}\ast \log \left(\frac{\mathrm{bF}-\mathrm{S}\left(\mathrm{i}\right)}{\mathrm{bF}-\mathrm{wF}}+1\right), \mathrm{condition}\\ 1-\mathrm{r}\ast \log \left(\frac{\mathrm{bF}-\mathrm{S}\left(\mathrm{i}\right)}{\mathrm{bF}-\mathrm{wF}}+1\right), \mathrm{others}\end{array}\right.$$

(47)

4.1.3. Grabble Food

This mathematical approach means that if the concentration of food is high, the weight close to the region is bigger. On the other hand, if the concentration of food is little, the weight close to the region will decrease and hence it decides to turn to explore other areas.

$\overrightarrow{\mathrm{vb}}$ represents a vector that has random values oscillates in the range of [−a, a] and gradually accesses zero with the iterations increment. Values of $\overrightarrow{\mathrm{vc}}$ have been taken in the range of [−1, 1] and eventually tend towards zero. Synergistic interaction that takes place between $\overrightarrow{\mathrm{vb}}$ and $\overrightarrow{\mathrm{vc}}$ provides a simulation for the selective behavior of slime mould. It will keep distributing organic material for exploring other regions, even if it found a better food source, trying to find a food source with higher quality instead of using all of it in one source.

4.2. Proposed ESMA

The ESMA is developed by the neighborhood dimension learning (NDL) search strategy to improve the search abilities of the ESMA and to make the exploration-exploitation more balanced [35]. In this strategy, firstly, a radius (${\mathrm{R}}_{\mathrm{i}}^{\mathrm{t}}$) is defined by Euclidean distance (${\mathrm{D}}_{\mathrm{i}}$) between the current location of ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{t}}$ and the candidate location ${\mathrm{X}}_{\mathrm{i}\_\mathrm{SMA}}^{\mathrm{t}+1}$ from the following equation:

$${\mathrm{R}}_{\mathrm{i}}^{\mathrm{t}}=\Vert {\mathrm{X}}_{\mathrm{i}}^{\mathrm{t}}-{\mathrm{X}}_{\mathrm{i}\_\mathrm{SMA}}^{\mathrm{t}+1}\Vert$$

(48)

Then, the neighbors of ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{t}}$ denoted by ${\mathrm{N}}_{\mathrm{i}}^{\mathrm{t}}$ is formed by (49) considered to ${\mathrm{R}}_{\mathrm{i}}^{\mathrm{t}}$:

$${\mathrm{N}}_{\mathrm{i}}^{\mathrm{t}}=\left\{{\mathrm{X}}_{\mathrm{j}}^{\mathrm{t}}{|\mathrm{D}}_{\mathrm{i}}\left({\mathrm{X}}_{\mathrm{i}}^{\mathrm{t}},{\mathrm{X}}_{\mathrm{j}}^{\mathrm{t}}\right)\le {\mathrm{R}}_{\mathrm{i}}^{\mathrm{t}},{\mathrm{X}}_{\mathrm{j}}^{\mathrm{t}}\in \mathrm{Pop}\right\}$$

(49)

As soon as the neighborhood of ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{t}}$ is formed, multi neighbors learning are performed using this equation:

$${\mathrm{X}}_{\mathrm{i}\_\mathrm{NDL}}^{\mathrm{t}+1}={\mathrm{X}}_{\mathrm{i},\mathrm{d}}^{\mathrm{t}}+\mathrm{rand}\times \left({\mathrm{X}}_{\mathrm{n},\mathrm{d}}^{\mathrm{t}}-{\mathrm{X}}_{\mathrm{r},\mathrm{d}}^{\mathrm{t}}\right)$$

(50)

where d-th dimension of ${\mathrm{X}}_{\mathrm{i}\_\mathrm{NDL}}^{\mathrm{t}+1}$ is computed by the d-th dimension of a random of neighbor ${\mathrm{X}}_{\mathrm{n},\mathrm{d}}^{\mathrm{t}}$ chosen from ${\mathrm{N}}_{\mathrm{i}}^{\mathrm{t}}$, and a random position ${\mathrm{X}}_{\mathrm{r},\mathrm{d}}^{\mathrm{t}}$ from the population of slime mould.

Finally, selecting the best position is given using (51).

$${\mathrm{X}}_{\mathrm{i}}^{\mathrm{t}+1}=\left\{\begin{array}{c}{\mathrm{X}}_{\mathrm{i}\_\mathrm{SMA}}^{\mathrm{t}+1}, \mathrm{if} \mathrm{f}\left({\mathrm{X}}_{{\mathrm{i}}_{\mathrm{SMA}}}^{\mathrm{t}+1}\right)<\mathrm{f}\left({\mathrm{X}}_{{\mathrm{i}}_{\mathrm{NDL}}}^{\mathrm{t}+1}\right)\\ {\mathrm{X}}_{\mathrm{i}\_\mathrm{NDL}}^{\mathrm{t}+1}, \mathrm{otherwise}\end{array}\right.$$

(51)

The flow chart of ESMA is shown in Figure 5.

5. Results

5.1. Performance of the Proposed ESMA Algorithm

In this subsection, the effectiveness and performance of the proposed ESMA algorithm are evaluated on many benchmark functions including the best values, mean values, worst values, and standard deviation (STD) for the solutions obtained by the ESMA technique, the original SMA algorithm, and the three recent optimization algorithms, including CPA, AO, and RUN. The results of the proposed ESMA algorithm are compared with these well known optimization algorithms. These algorithms have been performed in 20 independent runs. The maximum number of iterations for each run is 200 and the population size is 50. All simulations were executed using MATLAB R2016a on a laptop including Core i5-4210U [email protected] GHz of speed, and 8 GB of RAM. Figure 6 illustrates the qualitative metrics using ESMA algorithm for 11 benchmark functions including 2D views of the functions, search history, average fitness history, and convergence curve.

The statistical results of the proposed ESMA algorithm, the original SMA algorithm, and three recent algorithms when applied for unimodal benchmark functions, multimodal benchmark functions, and composite benchmark functions are presented in

Table 4. The statistical results of unimodal benchmark functions using the proposed ESMA algorithm and other recent algorithms ¹.

Function		ESMA	SMA	CPA	AO	RUN
F1	Best	0.00	1.8 × 10−293	6.5 × 10−190	7.15 × 10−76	9.07 × 10−86
	Worst	5.4 × 10−295	1.1 × 10−156	1.41 × 10−70	4.17 × 10−57	1 × 10−75
	Mean	2.7 × 10−296	5.3 × 10−158	7.07 × 10−72	2.09 × 10−58	5.02 × 10−77
	Std	0.00	2.4 × 10−157	3.16 × 10−71	9.33 × 10−58	2.24 × 10−76
F2	Best	7.4 × 10−162	2.9 × 10−132	4.8 × 10−106	1.3 × 10−36	1.04 × 10−47
	Worst	1.9 × 10−148	1.02 × 10−64	1.07 × 10−14	9.5 × 10−32	2.3 × 10−42
	Mean	1.8 × 10−149	5.1 × 10−66	5.36 × 10−16	1.37 × 10−32	1.87 × 10−43
	Std	5.1 × 10−149	2.28 × 10−65	2.4 × 10−15	3.28 × 10−32	5.35 × 10−43
F3	Best	0.00	1.1 × 10−254	2.2 × 10−186	3.39 × 10−71	5.75 × 10−77
	Worst	2.1 × 10−298	1.4 × 10−140	2.51 × 10−26	1.58 × 10−62	4.54 × 10−61
	Mean	1.8 × 10−299	6.8 × 10−142	1.26 × 10−27	1.28 × 10−63	2.33 × 10−62
	Std	0.00	3 × 10−141	5.61 × 10−27	3.7 × 10−63	1.01 × 10−61
F4	Best	8.9 × 10−159	5.7 × 10−137	3.78 × 10−97	8.91 × 10−40	1.35 × 10−42
	Worst	3.8 × 10−146	9.02 × 10−51	3.54 × 10−19	1.67 × 10−31	4.16 × 10−34
	Mean	1.9 × 10−147	4.51 × 10−52	1.77 × 10−20	1.26 × 10−32	2.09 × 10−35
	Std	8.5 × 10−147	2.02 × 10−51	7.93 × 10−20	4.05 × 10−32	9.3 × 10−35
F5	Best	26.08305	25.34758	28.12882	24.13219	23.34961
	Worst	28.94341	28.98832	28.88188	29.26575	26.52757
	Mean	28.24751	28.57338	28.53022	26.85075	24.73586
	Std	0.854929	0.807162	0.196671	1.924605	0.832917
F6	Best	0.205897	0.01122	0.000936	1.61 × 10−8	7.23 × 10−9
	Worst	4.076296	3.74817	1.069296	0.000556	2.24 × 10−8
	Mean	1.054545	0.810915	0.293309	0.000145	1.19 × 10−8
	Std	0.980102	0.967813	0.368271	0.000182	3.7 × 10−9
F7	Best	3.22 × 10−5	1.84 × 10−5	3.23 × 10−5	6.04 × 10−6	3.74 × 10−5
	Worst	0.003883	0.00744	0.009786	0.000818	0.000865
	Mean	0.001199	0.001253	0.00206	0.00021	0.00042
	Std	0.001113	0.00176	0.002655	0.000197	0.000268

¹ The best values obtained are in bold.

,

Table 5. The statistical Results of multimodal benchmark functions using the proposed ESMA algorithm and other recent algorithms ¹.

Function		ESMA	SMA	CPA	AO	RUN
F8	Best	−1850.29	−1909.03	−1909.03	−1901.64	−1660.17
	Worst	−1460.88	−941.222	−841.176	−718.461	−1397.78
	Mean	−1653.2	−1854.84	−1665.13	−1061.31	−1580.75
	Std	102.5683	215.5892	266.4826	353.9877	78.84949
F9	Best	0.00	0.00	0.00	0.00	0.00
	Worst	0.00	0.00	0.00	0.00	0.00
	Mean	0.00	0.00	0.00	0.00	0.00
	Std	0.00	0.00	0.00	0.00	0.00
F10	Best	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	Worst	8.88 × 10−16	8.88 × 10−16	1.87 × 10−14	8.88 × 10−16	8.88 × 10−16
	Mean	8.88 × 10−16	8.88 × 10−16	1.78 × 10−15	8.88 × 10−16	8.88 × 10−16
	Std	0.00	0.00	0.00	3.97 × 10−15	0.00
F11	Best	0.00	0.00	0.00	0.00	0.00
	Worst	0.00	0.00	0.00	0.00	0.00
	Mean	0.00	0.00	0.00	0.00	0.00
	Std	0.00	0.00	0.00	0.00	0.00
F12	Best	0.019339	0.000221	3.04 × 10−5	1.05 × 10−10	2.99 × 10−9
	Worst	0.224734	0.883664	0.073697	5.46 × 10−6	1.57 × 10−8
	Mean	0.055819	0.053203	0.012284	6.23 × 10−7	6.93 × 10−9
	Std	0.046062	0.195804	0.01766	1.2 × 10−6	3.43 × 10−9
F13	Best	0.997042	0.007141	0.003875	1.82 × 10−7	4.07 × 10−8
	Worst	−1850.29	−1909.03	−1909.03	−1901.64	−1660.17
	Mean	−1460.88	−941.222	−841.176	−718.461	−1397.78
	Std	−1653.2	−1854.84	−1665.13	−1061.31	−1580.75

¹ The best values obtained are in bold.

and

Table 6. The statistical results of composite benchmark functions using the proposed ESMA algorithm and other recent algorithms ¹.

Function		ESMA	SMA	CPA	AO	RUN
F14	Best	0.998004	0.998004	0.998004	0.998004	0.998004
	Worst	7.873993	12.67051	12.67051	12.67051	10.76318
	Mean	2.528537	2.517909	2.953424	2.573682	2.081493
	Std	2.107136	3.145931	3.84665	2.574285	2.241941
F15	Best	0.00	1.1 × 10−254	2.2 × 10−186	0.000325	0.000307
	Worst	2.1 × 10−298	1.4 × 10−140	2.51 × 10−26	0.001155	0.001223
	Mean	1.8 × 10−299	6.8 × 10−142	1.26 × 10−27	0.00053	0.00072
	Std	0.00	3 × 10−141	5.61 × 10−27	0.00018	0.000467
F16	Best	−1.03163	−1.03163	−1.03163	−1.03163	−1.03163
	Worst	−1.03161	−1.03013	−1.03163	−1.02995	−1.03163
	Mean	−1.03163	−1.03155	−1.03163	−1.03104	−1.03163
	Std	4.12 × 10−6	0.000335	9.56 × 10−10	0.000547	8.13 × 10−12
F17	Best	0.397887	0.397887	0.397887	0.397893	0.397887
	Worst	0.397887	0.397894	0.399219	0.399082	0.397887
	Mean	0.397887	0.397889	0.397954	0.398149	0.397887
	Std	2.12 × 10−10	1.94 × 10−6	0.000298	0.000274	4.27 × 10−12
F18	Best	3	3	3	3.00648	3
	Worst	3	30.00163	30	3.100674	3
	Mean	3	7.050142	4.663726	3.042457	3
	Std	2.37 × 10−14	9.89173	6.125915	0.02737	5.41 × 10−13
F19	Best	−0.30048	−0.30048	−0.30048	−0.30048	−0.30048
	Worst	−0.30048	−0.30048	−0.30038	−0.30048	−0.30048
	Mean	−0.30048	−0.30048	−0.30047	−0.30048	−0.30048
	Std	1.14 × 10−16	1.14 × 10−16	2.19 × 10−5	1.14 × 10−16	1.14 × 10−16
F20	Best	−3.322	−3.32198	−3.322	−3.2947	−3.322
	Worst	−3.2031	−0.19258	−3.14355	−2.67041	−3.20309
	Mean	−3.28612	−3.06081	−3.29758	−3.1067	−3.28038
	Std	0.055765	0.747204	0.050586	0.143442	0.058183
F21	Best	−10.1532	−10.1531	−5.0552	−10.1527	−10.1532
	Worst	−5.05074	−5.05489	−4.20446	−10.1019	−5.0552
	Mean	−5.81968	−8.87754	−5.00614	−10.1365	−8.3689
	Std	1.867737	2.25068	0.190098	0.016429	2.494761
F22	Best	−10.4029	−10.4029	−5.08767	−10.4028	−10.4029
	Worst	−5.0865	−5.12737	−4.97896	−10.3216	−5.08767
	Mean	−6.94796	−9.8648	−5.08015	−10.3889	−9.07412
	Std	2.601126	1.620212	0.025303	0.019955	2.36137
F23	Best	−10.5364	−10.5363	−10.5364	−10.5362	−10.5364
	Worst	−4.63728	−5.12744	−4.82737	−10.4701	−5.12848
	Mean	−6.7263	−9.99261	−5.6062	−10.5212	−9.18443
	Std	2.561402	1.655735	1.540624	0.019921	2.402536

¹ The best values obtained are in bold.

, respectively. The best values were achieved with the proposed ESMA, SMA, CPA, AO, and RUN algorithms shown in bold. It is obviously seen that the ESMA algorithm reaches to the best solution for many benchmark functions. The convergence curves of all algorithms for the unimodal benchmark functions are shown in Figure 7 while the boxplots for these techniques for this type of functions are displayed in Figure 8. In addition, Figure 9 shows the convergence curves of these algorithms for the multimodal benchmark functions while Figure 10 presents the boxplots for each algorithm for these functions. Finally, the convergence curves of all algorithms for the composite benchmark functions are displayed in Figure 11 while the boxplots for all algorithms for this type of benchmark functions are presented Figure 12. From these figures, it is clearly seen that the proposed ESMA technique reaches a stable point for all functions and the boxplots of the ESMA technique are very narrow for most functions compared to the other well known techniques.

5.2. Real-World Application

In this subsection, several case studies are implemented on the modified IEEE-30 bus power system to solve the OPF problem. The results of applying the enhanced ESMA for all cases are provided and explained. Conventional SMA and two different optimization algorithms, BWOA and MSA, are also applied to confirm the validity of ESMA.

The case studies are organized as follows: Cases 1 and 2 are dedicated to minimizing the total generation cost without and with carbon tax respectively, Cases 3 and 4 are performed for studying the impact of reserve and penalty cost variation on the optimized cost respectively, Case 5 is implemented for studying of ramp rate effect of TPGs, and finally, Case 6 is performed for studying the impact of load demand uncertainties.

The simulation of optimization techniques is performed based on MATLAB using a computer with the following properties: Intel^® Core™ i3-380M Processor 3M Cache, 2.53 GHz and 4 GB RAM.

In every case, the objective function reaches its optimum value after running the algorithm 5 times, and a maximum of 1000 iterations are set as the ending criteria of the algorithms. The settings of control variables are also listed.

5.2.1. Case 1: Minimization of Generation Cost

This case is performed to optimize scheduled output powers from all different types of energy sources used in the modified power system to reach the possible minimum value of the total power cost based on (11).

Figure 13 provides the convergences of applied OPF optimization techniques. The optimally obtained results of overall cost, control variables, reactive powers, and other parameters have been provided in

Table 7. Case 1 simulation results.

Control Variables	Min.	Max.	ESMA	SMA	BWOA	MSA
PTG1 (MW)	50	140	134.9143078	134.9126665	134.9318602	134.9080709
PTG2 (MW)	20	80	27.68805837	28.41701	30.52148	29.91749
PTG3 (MW)	10	35	10.01257102	10	10.02768	41.42875
Pws,1 (MW)	0	75	43.57823017	43.97604	43.11587	10.15632
Pws,2 (MW)	0	60	37.45089	36.74404	37.82475	37.68851
Pss (MW)	0	50	35.5275	35.11807	33.36969	35.20921
V1 (p.u.)	0.95	1.1	1.069956	1.072577	1.069029	1.071379
V2 (p.u.)	0.95	1.1	1.056832	1.057642	1.099901	1.057197
V5 (p.u.)	0.95	1.1	1.033471	1.037017	1.095503	1.079234
V8 (p.u.)	0.95	1.1	1.088021	1.040221	1.02164	1.075681
V11 (p.u.)	0.95	1.1	1.097041	1.097474	1.00625	1.07746
V13 (p.u.)	0.95	1.1	1.052377	1.048536	1.067481	1.047355
QTG1 (MVAr)	−20	150	−6.58886	−1.73162	−20	−3.94493
QTG2 (MVAr)	−20	60	16.43639	12.89954	60	7.538948
QTG3 (MVAr)	−15	40	40	35.9758	15.65167	40
Qws,1 (MVAr)	−30	35	21.18113	24.64654	35	35
Qws,2 (MVAr)	−25	30	29.5482	30	2.883899	22.88516
Qss (MVAr)	−20	25	16.47241	15.1557	25	15.4901
Total power cost ($/h)			781.9375819	781.9580658	784.3423563	782.4230422
Emissions (t/h)			1.762973771	1.76261913	1.764219911	1.76173491
Carbon tax, Ctax ($/h)			0	0	0	0
Ploss (MW)			5.771556781	5.76783193	6.401297765	5.908352008
Vd (p.u.)			0.458684989	0.451488019	0.53484521	0.443429768

.

The voltage of ith bus is denoted by ${\mathrm{V}}_{\mathrm{i}}$ in Table 7. The effectiveness of ESMA is very clear from the simulation results of case 1. Comparing to the conventional SMA and other performed optimization techniques for OPF, ESMA has a fast convergence and better solution quality. Among all applied algorithms, ESMA achieved the minimum total power cost of 781.9376 $/h.

5.2.2. Case 2: Total Power Cost Minimization with Carbon Tax

The objective of this case is to minimize the total power cost, with imposing a fixed carbon tax (C_tax) on the thermal power generators because of their CO₂ emissions according to (12). The forced C_tax is supposed to be 20 ($/ton) [7]. Increasing the penetration level of wind and solar energy into the power grid is expected with forcing the carbon tax; this conception obviously appears in the simulation results. Similarly to Case 1, a comparison between the convergence of ESMA and other performed algorithms is provided through Figure 14, and the optimally obtained results of overall cost, control variables, reactive powers, and other parameters have been provided in

Table 8. Case 2 simulation results.

Control Variables	Min.	Max.	ESMA	SMA	BWOA	MSA
PTG1 (MW)	50	140	123.4182597	123.558571	123.9385776	123.0458701
PTG2 (MW)	20	80	32.8988	32.70381	31.55832	31.94699
PTG3 (MW)	10	35	10.00018	10	10.02575	10.00019
Pws,1 (MW)	0	75	45.87829	46.01788	43.52164	45.40579
Pws,2 (MW)	0	60	38.84621	38.76592	38.18338	38.19805
Pss (MW)	0	50	37.63186	37.63258	41.63565	40.09511
V1 (p.u.)	0.95	1.1	1.070554	1.069455	1.076582	1.067868
V2 (p.u.)	0.95	1.1	1.057246	1.056339	1.060155	1.055309
V5 (p.u.)	0.95	1.1	1.036466	1.034248	1.033567	1.031957
V8 (p.u.)	0.95	1.1	1.040751	1.082863	1.031061	1.040321
V11 (p.u.)	0.95	1.1	1.099149	1.1	1.07291	1.099459
V13 (p.u.)	0.95	1.1	1.053391	1.053591	1.056897	1.1
QTG1 (MVAr)	−20	150	−3.04826	−4.14907	6.942384	−6.10507
QTG2 (MVAr)	−20	60	12.62003	12.03468	18.60405	12.98216
QTG3 (MVAr)	−15	40	35.90165	40	24.44273	34.59923
Qws,1 (MVAr)	−30	35	23.35533	21.07284	22.66046	19.91761
Qws,2 (MVAr)	−25	30	30	30	22.74522	29.40543
Qss (MVAr)	−20	25	16.83568	16.72546	20.55141	25
Total power cost ($/h)			810.3557515	810.3738858	812.0366457	810.8842547
Emissions (t/h)			0.88593426	0.893157637	0.913174063	0.867357442
Carbon tax, Ctax ($/h)			20	20	20	20
Ploss (MW)			5.273603276	5.278762466	5.463313439	5.291996546
Vd (p.u.)			0.464133712	0.467254932	0.450579985	0.538738367

.

The minimum value of total power cost achieved by ESMA is 810.3558 $/h which means that ESMA outdoes SMA and all other used techniques with regard to total power cost minimization and the convergence of the solution.

Load buses voltages have an important role in OPF study. Therefore, it is essential to deal with the constraints of load buses voltages. Operating voltages of all load buses in the IEEE- 30 bus system must be within the range [0.95–1.05] p.u.

In this concern, Figure 15 provides voltage profiles of load buses for Case 1 and Case 2. It is clear from Figure 15 that all voltages of load buses are inside limits after finishing the optimization process of the two cases.

5.2.3. Case 3: Impact of Reserve Cost Variation on the Optimized Cost

This case is applied to study and analyze the impact of variation of reserve cost coefficient on various optimized costs. All relevant parameters were kept as those used in Case 1 except reserve cost coefficients. In this case, SPG and WPGs have reserve cost coefficients started from ${\mathrm{K}}_{\mathrm{rs}}={\mathrm{K}}_{\mathrm{rw},1}={\mathrm{K}}_{\mathrm{rw},2}=\mathrm{Rc}=3$ (Case 1) to ${\mathrm{K}}_{\mathrm{rs}}={\mathrm{K}}_{\mathrm{rw},1}={\mathrm{K}}_{\mathrm{rw},2}=\mathrm{Rc}=4$ until $\mathrm{Rc}=6$ with an increase of 1, and their penalty cost coefficients are ${\mathrm{K}}_{\mathrm{ps}}={\mathrm{K}}_{\mathrm{pw},1}={\mathrm{K}}_{\mathrm{pw},2}=\mathrm{Pc}=1.5$ as the same values used in Case 1. The optimally scheduled powers of all generators are shown in Figure 16. To make Figure 16 clearer, Case 3a denotes $\mathrm{Rc}=4$, Case 3b denotes $\mathrm{Rc}=5$, while Case 3c denotes $\mathrm{Rc}=6$. The optimally scheduled powers from WPGs and SPG decrease with the increase in reserve cost coefficient as reducing the scheduled power requests a smaller quantity from the spinning reserve. When there is a shortage of output power from WPGs or SPG, TPGs compensate it. Therefore, the cost of TPGs rises as noticed from the cost profile of TPGs in Figure 17 while the costs of WPGs and SPG decreases gradually to a certain amount. In total, the entire cost of the produced power rises with the rise in the reserve cost coefficient.

5.2.4. Case 4: Impact of Penalty Cost Variation on the Optimized Cost

The effect of varying the penalty cost coefficient on various optimized costs is studied and analyzed in this case. All relevant parameters were kept as those used for Case 1 except the penalty cost coefficients. In this case, SPG and WPGs have coefficients of penalty cost started from ${\mathrm{K}}_{\mathrm{ps}}={\mathrm{K}}_{\mathrm{pw},1}={\mathrm{K}}_{\mathrm{pw},2}=\mathrm{Pc}=1.5$ (Case 1) to $\mathrm{Pc}=4$ (Case 4a), $\mathrm{Pc}=5$ (Case 4b), and $\mathrm{Pc}=6$ (Case 4c), while reserve cost coefficients remain unchanged from Case 1, ${\mathrm{K}}_{\mathrm{rs}}={\mathrm{K}}_{\mathrm{rw},1}={\mathrm{K}}_{\mathrm{rw},2}=\mathrm{Rc}=3$. Figure 18 illustrates the optimally scheduled power from all generators.

If the penalty cost coefficient rises, the scheduled output powers of WPGs and SPG will increase because the rise in the scheduled output power will help to decrease the penalty cost in case of wind speed or solar irradiance has a high value. This concept is unlike the concept in the previous case because the raises in the scheduled output powers do not sound identical for all RESs due to the extremely non-linear relationships between PDFs and penalty or reserve cost of RESs. From Figure 19, a progressive increment in WPGs cost and a minor fluctuation in the SPG cost are observed. It is also noticed that the cost of TPGs is approximately constant in addition to a steady growth in total power cost.

All generator buses voltages for Cases 3 and 4 are presented in Figure 20. It is noticed that all generator buses voltages lie on the specified range of p.u. voltage (0.95–1.1). The generators’ reactive powers are also illustrated in Figure 21. The reactive power limits mentioned in Table 7 indicate that TPG3 and WPG2 reached their reactive power capability maximum limits for many cases. Accordingly, considering the reactive power constraints when performing any optimization technique is very important.

5.2.5. Case 5: The Impact of Considering Ramp-Rate Limits of TPGs

The impact of considering ramp rate limits of TPGs while minimizing the total power cost based on (11) is studied and analyzed in this case. The produced power at the preceding hour for TPGs and their limits of ramp-rate are provided in Table 1 [36].

Table 9. Case 5 simulation results (considering the effect of ramp rate).

Control Variables	Min.	Max.	ESMA	SMA	BWOA	MSA
PTG1 (MW)	79.211	114.211	92.16880318	92.44231076	90.18995958	91.32417994
PTG2 (MW)	65	80	65.00002	65	65	65
PTG3 (MW)	12	24	12	12	12.26708	12
Pws,1 (MW)	0	75	44.16894	43.88145	43.08437	44.17037
Pws,2 (MW)	0	60	37.37163	37.4186	37.35422	37.33551
Pss (MW)	0	50	37.31031	37.29943	40.20372	38.24819
V1 (p.u.)	0.95	1.1	1.067626	1.066194	1.065552	1.063945
V2 (p.u.)	0.95	1.1	1.058474	1.0575	1.00708	1.055106
V5 (p.u.)	0.95	1.1	1.036813	1.036672	1.071627	1.1
V8 (p.u.)	0.95	1.1	1.041757	1.041546	1.051095	1.095858
V11 (p.u.)	0.95	1.1	1.099689	1.099994	1.094939	1.079155
V13 (p.u.)	0.95	1.1	1.058374	1.067352	1.098482	1.095742
QTG1 (MVAr)	−20	150	−3.79857	−5.54952	6.587242	−6.53175
QTG2 (MVAr)	−20	60	10.15416	9.34017	−20	−2.01028
QTG3 (MVAr)	−15	40	35.62043	34.85711	40	40
Qws,1 (MVAr)	−30	35	23.13793	23.5976	35	35
Qws,2 (MVAr)	−25	30	29.97999	29.61801	27.06355	21.56442
Qss (MVAr)	−20	25	18.33159	21.65564	25	25
Total power cost ($/h)			804.5476	804.5697852	805.1313475	804.8074077
Emissions (t/h)			0.20445652	0.206348589	0.191626873	0.198813008
Ploss (MW)			4.619704547	4.641789739	4.699343938	4.678246985
Vd (p.u.)			0.485876946	0.518261166	0.54920658	0.517553778

provides the optimal results of this case such as total power cost, control variables, reactive powers, and other important parameters.

Figure 22 provides the convergences of the different applied techniques in Case 5.

The impact of the ramp rate limits of the TPGs is clear from the Case 5 simulation results. If we look at the results of Case 5 in Table 9 and the results of Case 1 in Table 7, we can find that the total power cost is increased as expected as a result of changing the new operating points of TPGs from their original operating points without ramp rate constraints. The effectiveness of ESMA is very clear from the simulation results of Case 5. Compared to the conventional SMA and other applied optimization algorithms, ESMA has the fastest convergence and the best solution quality. The minimum overall power cost achieved using the ESMA is 804.5476 $/h. Thus, ESMA outperforms SMA and all other performed techniques with regard to total power cost minimization and the convergence of solutions for this practical case. There is no violation in system constraints after performing the OPF study as shown in Table 9.

Figure 23 shows also the voltage profile of load buses in Case 5. It is noticed that all voltages of load buses are inside the predefined limits.

5.2.6. Case 6: The Impact of Load Demand Uncertainties

In Case 6, another realistic situation of load demand variation is provided. The load demand uncertainty has been modeled by the normal PDF [37]. A level-based procedure is executed to accomplish the process of optimization during some different levels (scenarios) of the power system load.

The uncertainty of the load using the normal PDF with standard deviation (σ_d) = 10 and mean (μ_d) = 70 is illustrated in Figure 24. There are four-color areas in Figure 24. These areas denote the four considered levels of load demand (P_d).

The occurring probability of the i-th level of load demand (${{\Delta }}_{\mathrm{sc},\mathrm{i}}$) is depicted below [12]:

$${{\Delta }}_{\mathrm{sc},\mathrm{i}}={{\displaystyle \int }}_{{\mathrm{P}}_{\mathrm{d},\mathrm{i}}^{\mathrm{low}}}^{{\mathrm{P}}_{\mathrm{d},\mathrm{i}}^{\mathrm{high}}}\frac{1}{{\mathsf{\sigma }}_{\mathrm{d}}\surd 2\mathsf{\pi }}\exp \left[\frac{{\left({\mathrm{P}}_{\mathrm{d}}-{\mathsf{\mu }}_{\mathrm{d}}\right)}^2}{2{\mathsf{\sigma }}_{\mathrm{d}}{}^2}\right]{\mathrm{dP}}_{\mathrm{d}}$$

(52)

Equation (51) determines the mean of the i-th level of load demand (${\mathrm{P}}_{\mathrm{D},\mathrm{i}}$).

$${\mathrm{P}}_{\mathrm{D},\mathrm{i}}=\frac{1}{{{\Delta }}_{\mathrm{sc},\mathrm{i}}}{{\displaystyle \int }}_{{\mathrm{P}}_{\mathrm{d},\mathrm{i}}^{\mathrm{low}}}^{{\mathrm{P}}_{\mathrm{d},\mathrm{i}}^{\mathrm{high}}}\left(\frac{1}{{\mathsf{\sigma }}_{\mathrm{d}}\surd 2\mathsf{\pi }}\exp \left[\frac{{\left({\mathrm{P}}_{\mathrm{d}}-{\mathsf{\mu }}_{\mathrm{d}}\right)}^2}{2{\mathsf{\sigma }}_{\mathrm{d}}{}^2}\right]\right){\mathrm{dP}}_{\mathrm{d}}$$

(53)

The calculated values of the four means, as a percentage of nominal loading, and the four probabilities for the four loading levels are recorded in

Table 10. Probabilities and means of loading levels [12].

Loading Levels (L)	%Loading, Pd (Mean)	Level Probability, Δsc
L1	54.749	0.15866
L2	65.401	0.34134
L3	74.599	0.34134
L4	85.251	0.15866

. For each scenario, the four performed algorithms try to optimally minimize the total power cost based on (11). In each loading scenario, the scheduled powers of all the system generators are optimized.

For simplicity, only the results of loading levels 1 and 3 (L1 and L3) are recorded in

Table 11. Case 6 simulation results (L1).

Control Variables	Min.	Max.	ESMA	SMA	BWOA	MSA
PTG1 (MW)	50	140	50	50.00162628	50.00110622	50
PTG2 (MW)	20	80	20.01171	20.00039	20.01272	20.00006
PTG3 (MW)	10	35	10.01082	10.00013	10.02353	26.8289
Pws,1 (MW)	0	75	28.15856	28.14932	27.77365	10.00017
Pws,2 (MW)	0	60	24.33794	24.42657	24.38182	22.7425
Pss (MW)	0	50	23.80816	23.76047	24.22643	26.88557
V1 (p.u.)	0.95	1.1	1.053932	1.039198	1.022464	1.032892
V2 (p.u.)	0.95	1.1	1.050299	1.033567	1.022308	1.024873
V5 (p.u.)	0.95	1.1	1.043892	1.018211	1.021484	1.016617
V8 (p.u.)	0.95	1.1	1.045135	1.025042	1.022508	1.02046
V11 (p.u.)	0.95	1.1	1.067695	1.066508	1.065401	1.070566
V13 (p.u.)	0.95	1.1	1.043551	1.033307	1.021062	1.099479
QTG1 (MVAr)	−20	150	−10.0862	−5.23151	−19.4103	−3.53245
QTG2 (MVAr)	−20	60	0.380669	2.905193	−0.16671	−12.781
QTG3 (MVAr)	−15	40	17.16212	12.39216	19.25725	9.219822
Qws,1 (MVAr)	−30	35	12.689	4.636251	16.49788	6.093273
Qws,2 (MVAr)	−25	30	13.09423	18.31955	19.9989	18.19375
Qss (MVAr)	−20	25	6.054028	8.00693	5.814833	25
Total power cost ($/h)			410.3294562	410.3686027	410.7465692	411.4454656
Emissions (t/h)			0.104010026	0.104028661	0.104018561	0.104026511
Ploss (MW)			1.145097577	1.180840787	1.261597597	1.296340501
Vd (p.u.)			0.65909623	0.337539839	0.240641881	0.508944469

and

Table 12. Case 6 simulation results (L3).

Control Variables	Min.	Max.	ESMA	SMA	BWOA	MSA
PTG1 (MW)	50	140	94.72947881	94.79720263	91.11355617	92.34923281
PTG2 (MW)	20	80	20.00008	20.00044	20	20.0001
PTG3 (MW)	10	35	10.0001	10	10	10.00009
Pws,1 (MW)	0	75	32.34282	32.20228	33.97687	32.9448
Pws,2 (MW)	0	60	27.78931	27.92108	28.72745	28.27017
Pss (MW)	0	50	29.52108	29.5243	30.53354	30.87101
V1 (p.u.)	0.95	1.1	1.064273	1.067498	1.061448	1.066858
V2 (p.u.)	0.95	1.1	1.053301	0.973881	1.049094	0.950696
V5 (p.u.)	0.95	1.1	1.03721	1.038962	1.040377	1.1
V8 (p.u.)	0.95	1.1	1.041313	1.042134	1.079471	1.039051
V11 (p.u.)	0.95	1.1	1.097061	1.098575	1.031874	1.084509
V13 (p.u.)	0.95	1.1	1.050823	1.045943	1.070388	1.048776
QTG1 (MVAr)	−20	150	ESMA	SMA	BWOA	MSA
QTG2 (MVAr)	−20	60	−3.8362	13.44399	−2.14165	8.865571
QTG3 (MVAr)	−15	40	3.995005	−20	−8.97172	−20
Qws,1 (MVAr)	−30	35	20.7829	23.94863	40	15.9741
Qws,2 (MVAr)	−25	30	15.0575	19.74814	19.21955	35
Qss (MVAr)	−20	25	25.63449	26.5095	2.694168	21.59284
Total power cost ($/h)			10.81905	9.157136	20.92825	10.8832
Emissions (t/h)			576.2737229	576.4366891	576.8699631	576.6928804
Carbon tax, Ctax ($/h)			0.226220505	0.226766724	0.200169215	0.208418926
Vd (p.u.)			2.970382594	3.032800323	2.938923514	3.022914078

. It is clear from the results of Case 6 that the operation cost and power losses of the system under realistic loading levels are justifiably lesser than their values under theoretical conditions in Case 1. Figure 25 illustrates that ESMA outdoes all other performed methods with regard to overall power cost minimization and the convergence of solution. The voltage profiles of load buses for different loading levels for the ESMA are plotted in Figure 26. The values of voltage and system variables are recorded in Table 11 and Table 12. These values show that all system constraints are satisfied.

6. Conclusions

This research provided an enhanced slime mould algorithm to solve optimally the OPF problem. The performance of the proposed algorithm has been evaluated using 23 benchmark test functions and has been compared with the conventional SMA and three recent optimization algorithms: CPA, AO, and RUN. The results of the proposed ESMA algorithm are compared with these well-known optimization algorithms. The ESMA algorithm reached the best solution for many benchmark functions.

Consequently, the ESMA was used to solve the OPF problem in the IEEE-30 bus power system that modified to include stochastic solar and wind energy sources. The uncertain outputs of solar and wind energy sources were modelled through lognormal and Weibull PDFs, respectively.

The total cost of power produced from all power generators was optimally minimized in two different case studies; one of them neglected the carbon emission and the other considered it. The effect of varying the reserve and penalty cost coefficients of SPG and WPGs on the total power cost were also analyzed.

The impacts of considering ramp rate limits of TPGs and uncertain load demand have been studied to observe the validity of ESMA under more practical cases.

To test ESMA validity in real world applications, conventional SMA and two other optimization techniques, BWOA and MSA, were performed for the adjusted IEEE-30 bus system at the same system conditions. Simulation results obtained by the ESMA in each case study were compared with the results obtained by other applied algorithms.

Simulation results of all case studies prove the effectiveness of ESMA for solving of OPF problem with maintaining the system constraints inside the limits that have been set by the ISO. ESMA outperforms the conventional SMA and other optimization algorithms in terms of total power cost minimization and the convergence of problem solution under the practical and theoretical cases.