Developing a strategy based on weighted mean of vectors (INFO) optimizer for optimal power flow considering uncertainty of renewable energy generation

Abstract

In recent years, more efforts have been exerted to increase the level of renewable energy sources (RESs) in the energy mix in many countries to mitigate the dangerous effects of greenhouse gases emissions. However, because of their stochastic nature, most RESs pose some operational and planning challenges to power systems. One of these challenges is the complexity of solving the optimal power flow (OPF) problem in existing RESs. This study proposes an OPF model that has three different sources of renewable energy: wind, solar, and combined solar and small-hydro sources in addition to the conventional thermal power. Three probability density functions (PDF), namely lognormal, Weibull, and Gumbel, are employed to determine available solar, wind, and small-hydro output powers, respectively. Many meta-heuristic optimization algorithms have been applied for solving OPF problem in the presence of RESs. In this work, a new meta-heuristic algorithm, weighted mean of vectors (INFO), is employed for solving the OPF problem in two adjusted standard IEEE power systems (30 and 57 buses). It is simulated by MATLAB software in different theoretical and practical cases to test its validity in solving the OPF problem of the adjusted power systems. The results of the applied simulation cases in this work show that INFO has better performance results in minimizing total generation cost and reducing convergence time among other algorithms.

Introduction

The optimal power flow (OPF) problem has played a huge role in the planning and operation of the power system since its first inception by Carpentier in 1962 [1]. OPF is applied to attain certain objective functions by adjusting the control variables to their optimal settings without violating the system constraints. OPF methods are classified as conventional and meta-heuristic [2]. The conventional methods, such as mathematical methods and the gradient method, are suitable only for objective functions that are non-convex, non-differentiable, and non-smooth, while the meta-heuristic methods are flexible and powerful search techniques that can successfully tackle complex problems. Solving the OPF problem when considering renewable energy sources (RESs) is one of the more complicated problems due to the intermittent nature of RESs. In the literature, many meta-heuristic methods have been used for solving OPF problem considering RESs. In [3], the authors provided ant lion optimization (ALO) algorithm for solving the OPF with wind energy. White sharks algorithm has been employed in [4] for solving the OPF problem with solar and wind energies. The authors in [5] have provided an economical-environmental-technical dispatch (EETD) model that includes thermal and high penetration of RESs using the coronavirus herd immunity algorithm (CHIA). In [6], an improved equilibrium optimizer (IEO) algorithm has been proposed for solving the OPF problem with the solar and wind sources. In [7], Levy interior search algorithm (LISA) has been employed for solving non-convex, multi-objective OPF with the presence of RESs. Heap optimization algorithm (HOA) has been applied in [8] for three different IEEE standards systems incorporating both wind and solar energy. The authors in [9] have proposed an artificial ecosystem-based optimization (AEO) for optimal placement of FACTS devices and an OPF solution for a power system integrated with stochastic RERs. In [10], a fitness–distance balance-based adaptive guided differential evolution algorithm has been provided for solving the OPF problem in only the IEEE 30 bus power system with wind and solar power. In [11], a hybrid Harris-Hawks optimizer is proposed for modeling RESs considering their stochastic nature. The authors in [12] have proposed a barnacle mating optimizer (BMO) for solving OPF problem without and with RESs. In [13], a new manta ray foraging optimization (MRFO) algorithm is utilized to solve the OPF in the presence and absence of RESs in the power network.

In [14], a grey wolf optimization algorithm (GWO) was used to find the optimal power flow solution using existing solar and wind resources in adapted IEEE-30 and 57 bus power systems. In [15], a new golden ratio optimization method (GROM) algorithm has been employed for solving the OPF in a power network including solar and wind plants. The authors in [16] have provided a modified JAYA (MJAYA) algorithm for solving OPF with different RESs. In [17], the authors have employed the flower pollination algorithm (FPA) to solve an OPF problem in the IEEE-30 bus power system that have been adjusted by three different types of RESs. In [18], the success history-based adaptive differential evolution (SHADE) method and the superiority of feasible solutions (SF) method has been combined to develop SHADE-SF for solving the OPF with existing solar and wind energies. In [19], the authors have developed a particle swarm optimization (PSO) method for solving the OPF in a power network containing storage systems and RESs. In [20], a modified bacteria foraging algorithm has been proposed to solve the OPF problem of a mixture power system consisting of thermal and wind powers, in addition to the static synchronous compensator (STATCOM). In [21], a point estimation method for solving the probabilistic OPF problem with wind energy is proposed. The main contributions of this paper are as follows:

1. A recent meta-heuristic optimization algorithm, weighted mean of vectors (INFO) [22], is developed for solving many objective functions of the OPF problem in mixed (thermal, wind, solar, and hydro) power system.

2. Two power systems (IEEE-30 bus and IEEE-57 bus) have been modified to test the INFO algorithm’s validity.

3. Studying and analyzing the impact of the prohibited operating zones of thermal power generators (TPGs) on the OPF problem.

4. Studying and analyzing the impact of the ramp rate limits of TPGs on the OPF problem.

5. Studying and analyzing the impact of the uncertainty of RESs on the OPF problem.

6. Studying and analyzing the impact of the uncertainty of load demand on the OPF problem.

7. Developing and applying the genetic algorithm (GA) [23], slime mould algorithm (SMA) [24], and hunger games search (HGS) algorithm [25] for solving the OPF problem under the same conditions to test the validity of INFO algorithms compared to other standard and recent optimization algorithms.

In addition to that, the results of some cases of the INFO algorithm are compared with the results of multi-objective evolutionary algorithm based on decomposition (MOEA/D-SF) and summation-based multi-objective differential evolution (SMODE-SF) that are provided in [26]. The other sections of this research are organized as follows: Sect. 2 deals with the identification of the OPF problem. In Sect. 3, the stochastic models of RESs have been provided. Then, the proposed optimization algorithm is presented in Sect. 4. The obtained simulation results are recorded and analyzed in Sect. 5. Finally, the conclusion of this research is provided in Sect. 6.

Mathematical models

Table 1 provides the parameters of the adapted IEEE 30-bus power system. It has three thermal power generators (TPGs) linked at buses 1, 2, and 8. The system is modified by replacing the thermal generators at buses 5, 11, and 13 with three different types of RESs: wind power generator (WPG), a solar PV generator (SPG), and a combined solar PV–small-hydro power generator (SHPG), respectively.

Table 1.

Parameters of the adapted IEEE 30-bus system

Element	Number	Description
Buses	30	In [27]
Branches	41	In [27]
TPGs	3	At buses 1 (swing), 2, and 8
WPG	1	At bus 5
SPG	1	At bus 11
SHPG		At bus13
Control variables	11	The scheduled power produced from: TPG2, TPG3, WPG, SPG, and SHPG, and the voltages of all generator buses
System load	–	283.4 (MW), 126.2 (MVAr)
Accepted range of load buses voltage (p.u)	24	0.95–1.05

Cost of generation from TPGs

The total cost of generation (in $/h) from TPGs can be formed based on a quadratic function with the output power in MW, taking the valve-point effect into consideration as follows [18]:

$$C_{\text{FF}}\left(P_{\text{TG}}\right)=\sum _{i=1}^{N_{\text{TG}}}{a}_i+b_i{P}_{\text{TG},i}+c_i{P}_{\text{TG},i}^2+\left|d_i\text{*sin}\left(e_i\ast \left(P_{\text{TG},i}^{\text{min}}-P_{\text{TG},i}\right)\right)\right|$$ 1

Table 2 provides the cost coefficients related to TPGs [18].

Table 2.

Cost coefficients of TPGs

TPG	Bus	a	b	c	d	e	ϕ	Ψ	$\gamma$	τ	ζ	${\mathrm{P}}_{\mathrm{TPGi}}^0$ (MW)	${\mathrm{Dr}}_{\mathrm{i}}$ (MW)	${\mathrm{Ur}}_{\mathrm{i}}$ (MW)
1	1	30	2	0.00375	18	0.037	4.091	− 5.554	6.49	0.0002	6.667	99.211	20	15
2	2	25	1.75	0.0175	16	0.038	2.543	− 6.047	5.638	0.0005	3.333	80	15	10
3	8	20	3.25	0.00834	12	0.045	5.326	− 3.55	3.38	0.002	2	20	8	4

Cost of generation from RESs

The total cost of generation from RESs consists of three components: direct cost in relation to the scheduled output power, reserve cost in case of overestimation in output power from RESs, and penalty cost in case of underestimation.

Direct cost of RESs

The direct cost associated with the power induced by WPG in relation to the scheduled power is calculated by:

$$C_{\text{wd}}=g_{\text{w}}\ast P_{\text{ws}}$$ 2

Similarly, the direct cost from SPG is given by:

$$C_{\text{sd}}=h_{\text{s}}\ast P_{\text{ss}}$$ 3

While the direct cost associated with the power induced by the SHPG is calculated by:

$$C_{\text{shd}}={\text{h}}_{\text{s}}{P}_{\text{ssh},\text{s}}+m_{\text{h}}{P}_{\text{ssh},\text{h}}$$ 4

Cost evaluation of uncertain wind power

WPG may produce less power than what has been scheduled (overestimation) due to its uncertain nature. In this case, ISO has to mitigate the power demand using an adequate spinning reserve, with the reserve cost expressed as follows:

$$C_{\text{rw}}\left(P_{\text{ws}}-P_{\text{wav}}\right)=K_{\text{rw}}\left(P_{\text{ws}}-P_{\text{wav}}\right)=K_{\text{rw}}\underset{0}{\overset{P_{\text{ws}}}{\int }}(P_{\text{ws}}-P_{\text{w}})\times f_{\text{w}}\left(P_{\text{w}}\right){\text{dP}}_{\text{w}}$$ 5

On contrast, the output power from the WPG may be more than what has been scheduled (underestimation). In this case, ISO has to pay a penalty cost that is given:

$$C_{\text{pw}}\left(P_{\text{wav}}-P_{\text{ws}}\right)=K_{\text{pw}}\left(P_{\text{wav}}-P_{\text{ws}}\right)=K_{\text{pw}}\underset{P_{\text{ws}}}{\overset{P_{\text{wr}}}{\int }}(P_{\text{w}}-P_{\text{ws}})\times f_{\text{w}}\left(P_{\text{w}}\right){\text{dP}}_{\text{w}}$$ 6

Cost evaluation of uncertain solar PV power

Due to the uncertain nature of SPG’s output power, under- and overestimation situations may occur, as with WPG. Consequently, the reserve cost associated with the power produced by SPG in case of overestimation is given by:

$$\begin{array}{cc}\hfill C_{\text{rs}}\left(P_{\text{ss}}-P_{\text{sav}}\right)=& K_{\text{rs}}\left(P_{\text{ss}}-P_{\text{sav}}\right)\hfill \\ \hfill =& K_{\text{rs}}\times f_{\text{s}}(P_{\text{sav}}<P_{\text{ss}})\times [(P_{\text{ss}}-\text{Ex}(P_{\text{sav}}<P_{\text{ss}})]\hfill \\ \hfill \end{array}$$ 7

while, in the case of underestimation, the penalty cost is given by:

$$\begin{array}{cc}\hfill C_{\text{ps}}\left(P_{\text{sav}}-P_{\text{ss}}\right)=& K_{\text{ps}}\left(P_{\text{sav}}-P_{\text{ss}}\right)\hfill \\ \hfill =& K_{\text{ps}}\times f_{\text{s}}(P_{\text{sav}}>P_{\text{ss}})\times [(\text{Ex}(P_{\text{sav}}>P_{\text{ss}})-P_{\text{ss}}]\hfill \\ \hfill \end{array}$$ 8

Cost evaluation of uncertain power from SHPG

In this combined system, the reserve and penalty costs have to be added only to the cost of power produced by the solar PV unit due to its stochastic nature, as explained previously. However, the penalty and reserve costs have been considered on the cost of the total power produced from the combined system due to the small contribution of small-hydro to the total power produced from the combined system.

Thus, the reserve cost of the overestimation of total power produced from the SHPG is given by:

$$C_{\text{rsh}}\left(P_{\text{ssh}}-P_{\text{shav}}\right)=K_{\text{rsh}}\left(P_{\text{ssh}}-P_{\text{shav}}\right)=K_{\text{rsh}}\times f_{\text{sh}}(P_{\text{shav}}<P_{\text{ssh}})\times [P_{\text{ssh}}-(\text{Ex}(P_{\text{shav}}<P_{\text{ssh}})]$$ 9

while the penalty cost due the underestimation of the total power produced by the SHPG is given by:

$$C_{\text{psh}}\left(P_{\text{shav}}-P_{\text{ssh}}\right)=K_{\text{psh}}\left(P_{\text{shav}}-P_{\text{ssh}}\right)=K_{\text{psh}}\times f_{\text{sh}}(P_{\text{shav}}>P_{\text{ssh}})\times [(\text{Ex}(P_{\text{shav}}>P_{\text{ssh}})-P_{\text{ssh}}]$$ 10

Emission

The CO₂ emission in ton per hour (t/h) associated with the power produced from the TPGs in (MW) is determined by:

$$E=\sum _{i=1}^{N_{\mathit{TG}}}[({\phi }_i+{\mathrm{\Psi }}_i{P}_{TPG,i}+{\omega }_i{P}_{TG,i}^2)+{\tau }_i{e}^{\left({\zeta }_i{P}_{TG,i}\right)}]$$ 11

The emission coefficients are provided in Table 2. The emission cost ($/h) is calculated by multiplying the amount of emissions by the carbon tax forced by the electricity regulator as follows:

$$\text{CE}=C_{\text{tax}}E$$ 12

OPF objective functions

In this study, the objective functions of the OPF problem are as follows:

• F1 includes all cost models that were previously presented in the equations from (1) to (10). It disregards emissions cost as expressed by (13).

• F2, expressed by (14) includes the same models that make up F1 in addition to the emissions cost given in (11) and (12).

• F3 is used to minimize the total emissions (E).

$$\begin{array}{cc}\hfill \text{F}1=& \text{Minimize}[C_{\text{FF}}\left(P_{\text{TG}}\right)+\left(g_{\text{w}}\ast P_{\text{ws}}+K_{\text{rw}}\left(P_{\text{ws}}-P_{\text{wav}}\right)+K_{\text{pw}}\left(P_{\text{wav}}-P_{\text{ws}}\right)\right)\hfill \\ \hfill +& \left(h_{\text{s}}\ast P_{\text{ss}}+K_{\text{rs}}\left(P_{\text{ss}}-P_{\text{sav}}\right)+K_{\text{ps}}\left(P_{\text{sav}}-P_{\text{ss}}\right)\right)\hfill \\ \hfill +& \left(h_{\text{s}}{P}_{\text{ssh},\text{s}}+m_{\text{h}}{P}_{\text{ssh},\text{h}}+K_{\text{rsh}}\left(P_{\text{ssh}}-P_{\text{shav}}\right)+K_{\text{psh}}\left(P_{\text{shav}}-P_{\text{ssh}}\right)\right)]\hfill \\ \hfill \end{array}$$ 13

$$\text{F}2=\text{F}1+C_{\text{tax}}E$$ 14

Equality constraints

This type of constraint is responsible for ensuring an instantaneous balance between the produced and consumed powers. The produced active and reactive powers, plus the power losses in the system, should equal the demand active and reactive powers, as expressed in (15) and (16), respectively.

$$P_{\text{Gi}}=P_{\text{Di}}+V_i\sum _{i=1}^{\text{NB}}{V}_j\left[G_{\text{ij}}\text{cos}\left({\delta }_{\text{ij}}\right)+B_{\text{ij}}\text{sin}\left({\delta }_{\text{ij}}\right)\right]\forall \text{i}\in \text{NB}$$ 15

$$Q_{\text{Gi}}=Q_{\text{Di}}+V_i\sum _{i=1}^{\text{NB}}{V}_j\left[G_{\text{ij}}\text{sin}\left({\delta }_{\text{ij}}\right)-B_{\text{ij}}\text{cos}\left({\delta }_{\text{ij}}\right)\right]\forall i\in \text{NB}$$ 16

Inequality constraints

The OPF problem is constrained by a set of inequality constraints. In this study, the inequality constraints are as follows:

1. Generator constraints

The generator constraints include the operational limits of all power generators in the modified power systems, the prohibited operating zones (POZs), and ramp rate limits for TPGs [26, 28].

• Operational limits

$$P_{\text{TGi}}^{\text{min}}\le P_{\text{TGi}}\le P_{\text{TGi}}^{\text{max}},i=1,\dots .\dots \dots ,N_{\text{TG}}$$ 17

$$P_{\text{ws}}^{\text{min}}\le P_{\text{ws}}\le P_{\text{ws}}^{\text{max}}$$ 18

$$P_{\text{ss}}^{\text{min}}\le P_{\text{ss}}\le P_{\text{ss}}^{\text{max}}$$ 19

$$P_{\text{ssh}}^{\text{min}}\le P_{\text{ssh}}\le P_{\text{ssh}}^{\text{max}}$$ 20

$$Q_{\text{TG},\text{i}}^{\text{min}}\le Q_{\text{TG},\text{i}}\le Q_{\text{TG},\text{i}}^{\text{max}},i=1,\dots .\dots .,N_{\text{TG}}$$ 21

$$Q_{\text{ws}}^{\text{min}}\le Q_{\text{ws}}\le Q_{\text{ws}}^{\text{max}}$$ 22

$$Q_{\text{ss}}^{\text{min}}\le Q_{\text{ss}}\le Q_{\text{ss}}^{\text{max}}$$ 23

$$Q_{\text{ssh}}^{\text{min}}\le Q_{\text{ssh}}\le Q_{\text{ssh}}^{\text{max}}$$ 24

• Prohibited operating zones (POZs) for TPGs

POZs are operational zones where TPGs are not permitted to operate due to technical issues such as vibration in shaft bearings or a fault in the machine or any component such as boilers, pumps, and so on. The range of POZs can be represented as follows:

$$P_{\text{TGi}}^{\text{minpoz},\text{j}}\le {\text{POZ}}_{\text{TGi}}^{\text{j}}\le P_{\text{TGi}}^{\text{maxpoz},\text{j}}$$ 25

• Ramp rate constraints for TPGs

In OPF study, the constraints on the ramp-rate of TPGs are expressed as follows:

$$P_{\text{TG},\text{i}}-P_{\text{TG},\text{i}}^0\le {\text{Ur}}_{\text{i}},\text{if power generation rises}$$ 26

$$P_{\text{TG},\text{i}}^0-P_{\text{TG},\text{i}}\le {\text{Dr}}_{\text{i}},\text{if power generation reduces}$$ 27

• (b) Security constraints

The security constrains can be expressed as follows [26].

$$V_{\text{Gi}}^{\text{min}}\le V_{\text{Gi}}\le V_{\text{Gi}}^{\text{max}},i=1,\dots .\dots \dots ,N_{\text{G}}$$ 28

$$V_{L_p}^{min}\le V_{L_p}\le V_{{\text{L}}_{\text{p}}}^{\text{max}},p=1,\dots .\dots \dots ,N_L$$ 29

$$S_{{\text{L}}_{\text{q}}}\le S_{{\text{L}}_{\text{q}}}^{\text{max}},q=1,\dots .\dots \dots ,n_{\text{l}}$$ 30

where (28) gives the voltage limits of all generator buses with N_G that represents the number of generator buses. The voltage limits of load buses are defined by (29), while the constraint of line capacity is provided by (30). n_l and N_L represent the numbers of transmission lines and load buses in the power network, respectively.

Voltage deviation

Voltage deviation ($V_D$) specifies the cumulative load buses voltage deviation from 1.0 (p.u.) as shown by (31).

$$V_{\text{D}}=\sum _{p=1}^{N_{\text{L}}}\left|V_{\text{Lp}}-1\right|$$ 31

Power losses

The active power losses ($P_{\text{loss}}$) in transmission lines are unavoidable losses due to their inherent resistances. The active power losses of the network are determined as follows:

$$P_{\text{loss}}=\sum _{q=1}^{\text{nl}}\left(G_{q\left(\text{ij}\right)}\times \left[V_{\text{i}}^2+V_{\text{j}}^2-2V_{\text{i}}{V}_{\text{j}}cos\left({\delta }_{\text{ij}}\right)\right]\right)$$ 32

Stochastic power calculation for RESs

Probability distribution of RESs

Weibull PDF is frequently employed to provide the distribution of wind speed [20, 29]. The probability of wind speed based on the Weibull PDF has been calculated as follows:

$$f_v\left(v\right)=\left(\frac{\beta }{\alpha }\right)+{\left(\frac{v}{\alpha }\right)}^{\left(\beta -1\right)}{e}^{-{\left(\frac{v}{\alpha }\right)}^{\beta }}for0<v<\infty$$ 33

The values of Weibull shape (β) and scale (α) are given in Table 3. Figure 1 illustrates the frequency distributions of wind speed and Weibull fitting. This distribution has been obtained after running 8000 scenarios of Monte Carlo simulation.

Table 3.

Parameters of PDFs for stochastic models of RESs in IEEE 30 bus system

WPG			SPG		SHPG
#Turbines	Rated power (Pwr)	Weibull PDF parameters	Rated power, Psr	Lognormal PDF parameters	Solar PV rated power, Psr	Lognormal PDF parameters	Small-hydro rated power, Phr	Gumbel PDF parameters
25	75 MW	β = 2, α = 9	50 MW (at bus 11)	δ = 0.6, µ = 5.2	45 MW (at bus 13)	μ = 5.0 σ = 0.6	5 MW	λ = 15 γ = 1.2

Fig. 1.

Frequency distribution of wind speed

Lognormal PDF accurately represents the distribution of solar irradiance (Gs) [30, 31]. The probability of solar irradiance following a lognormal PDF with standard deviation (σ) and mean (μ) is given via:

$$f_G\left(G_s\right)=\frac{1}{G_s\sigma \sqrt{2\mathrm{\pi }}}\text{exp}\left\{\frac{-{\left(\text{ln}{G}_{\text{s}}-\mu \right)}^2}{2{\sigma }^2}\right\}\text{for}{G}_s>0$$ 34

The frequency distribution of solar irradiance and lognormal fitting are provided in Fig. 2 after running 8000 samples of Monte Carlo simulation.

Fig. 2.

Distribution of solar irradiance at bus 11

A Gumbel distribution model is frequently used to distribute river flow rate [32, 33]. The river flow rate probability (Q_w) based on Gumbel distribution with scale parameter (γ) and location parameter (λ) is given by:

$$f_Q\left(Q_w\right)=\left(\frac{1}{\gamma }\right)exp\left(\frac{Q_w-\lambda }{\gamma }\right)exp\left[-exp\left(\frac{Q_w-\lambda }{\gamma }\right)\right]$$ 35

The combined SHPG is connected at bus 13; thus, there are two PDFs: the first one is for the solar irradiance with lognormal fitting as shown in Fig. 3, and the other one is for the river flow rate of the small-hydro unit based on Gumbel fitting as illustrated in Fig. 4. These two diagrams were obtained after 8000 scenarios of Monte Carlo simulation with the values of Gumbel parameters listed in Table 3.

Fig. 3.

Distribution of solar irradiance for solar PV at bus 13

Fig. 4.

Distribution of river flow rate for small-hydro at bus 13

Power models for WPG, SPG, and SHPG

The wind plant connected at bus 5 has 25 turbines, and the rated power of each turbine is 3 MW. Consequently, the total power capacity of the wind plant is 75 MW. The power produced from the turbine differs based on the speed of the wind. The correlation between the power produced and wind speed (v) is given as:

$$P_{\text{w}}\left(v\right)=\left\{\begin{array}{ccc}\begin{array}{c}0\\ P_{\text{wr}}\\ P_{\text{wr}}\\ \end{array}& \begin{array}{c}\text{for}\\ \text{for}\\ \text{for}\\ \end{array}& \begin{array}{c}v<v_{\text{in}}andv>v_{\text{out}}\\ v_{\text{in}}\le v\le v_{\text{r}}\\ v_{\text{r}}<v\le v_{\text{out}}\\ \end{array}\end{array}\right)$$ 36

where pwr denotes the rated power of wind turbine. v_in represents the cut-in speed with value of (v_in = 3 m/s), v_r represents the rated speed with value of (v_r = 16 m/s), and v_out denotes the cut-out speed of the wind turbine with value of (v_out = 25 m/s).

The solar output power in relation to solar irradiance for solar PV is determined by:

$$P_{\text{w}}\left(v\right)=\left\{\begin{array}{c}0\text{for}v〈v_{\text{in}}\text{and}v〉v_{\text{out}}\\ P_{\text{wr}}\left(\frac{(v-v_{\text{in}}}{(v_{\text{r}}-v_{\text{in}}}\right)\\ P_{\text{wr}}\text{for}{v}_{\text{r}}<v\le v_{\text{out}}\\ \end{array}\right)\text{for}{v}_{\text{in}}\le v\le v_{\text{r}}$$ 37

where P_sr denotes the rated power of the solar PV generator, G_std denotes the standard solar irradiance which equals to 1000 W/m², and R_c refers to a certain point of irradiance which equals to 120 W/m².

The power produced from the small-hydro generator is determined as [34]:

$$P_H(Q_w)=\eta \rho g{Q}_w{H}_w$$ 38

where H_w represents the effective pressure head with value of (25 m), η refers to the efficiency of turbine-generator assembly with value of (0.85), ρ denotes the density of water with value of (1000 kg/m³), and g represents the acceleration due to gravity with value of (9.81 m/s²).

Probabilities assessment for wind power

The wind power may be discrete in some zones or continuous in other zones. For discrete zones, the probabilities of wind power are given via:

$$f_w\left(P_w\right)\left\{P_w=0\right\}=1-\text{exp}\left[-{\left(\frac{v_{\text{in}}}{\alpha }\right)}^{\beta }\right]+\text{exp}\left[-{\left(\frac{v_{\text{out}}}{\alpha }\right)}^{\beta }\right]$$ 39

$$f_w\left(P_w\right)\left\{P_w=P_{\mathit{wr}}\right\}=exp\left[-{\left(\frac{v_r}{\alpha }\right)}^{\beta }\right]-exp\left[-{\left(\frac{v_{\text{out}}}{\alpha }\right)}^{\beta }\right]$$ 40

while the probability for the continuous region is calculated as:

$$f_w\left(P_w\right)=\frac{\beta \left(v_r-v_{\text{in}}\right)}{{\alpha }^{\beta }\ast P_{\text{wr}}}{\left[v_{\text{in}}+\frac{P_w}{P_{\text{wr}}}\left(v_r-v_{\text{in}}\right)\right]}^{\beta -1}\times exp\left[-{\left(\frac{v_{\text{in}}+\frac{P_w}{P_{\text{wr}}}\left(v_r-v_{\text{in}}\right)}{\alpha }\right)}^{\beta }\right]$$ 41

Solar PV power over or underestimation cost calculation

Figure 5 illustrates the histogram of active power that can be produced from the SPG at bus 11. From Fig. 5, the overestimation cost in (7) can be calculated as follows:

$$C_{\text{Rs}}\left(P_{\text{ss}}-P_{\text{sav}}\right)=K_{\text{Rs}}\left(P_{\text{ss}}-P_{\text{sav}}\right)=K_{\text{Rs}}\sum _{n=1}^{N_b^{-}}\left[P_{\text{ss}}-P_{\text{sn}-}\right]\ast f_{\text{sn}-}$$ 42

where P_sn- denotes the actual power lower than the planned power P_ss, on left-half plane of P_ss in Fig. 5. f_sn- refers to the incident relative frequency of P_sn-. N_b− denotes the discrete bins number on left-half of P_ss, while the penalty cost provided in (8) can be gotten by:

$$C_{\text{Ps}}\left(P_{\text{sav}}-P_{\text{ss}}\right)=K_{\text{Ps}}\left(P_{\text{sav}}-P_{\text{ss}}\right)=K_{\text{Ps}}\sum _{n=1}^{N_b^{+}}\left[P_{\text{sn}+}-P_{\text{ss}}\right]\ast f_{\text{sn}+}$$ 43

where P_sn+ denotes the actual power larger than the planned power P_ss, on right-half plane of P_ss in Fig. 5. f_sn+ denotes the relative frequency of incident of P_sn+. N_b+ refers to the discrete bins number on right-half of P_ss.

Fig. 5.

Available solar active power (MW) at bus 11

Calculation of over or underestimation cost for combined SHPG output power

At bus 13, the powers from both solar photovoltaic and small-hydro are totaled and represented in Fig. 6. Similar to (42), the cost of overestimation for the SHPG is:

$$C_{\text{Rsh}}\left(P_{\text{ssh}}-P_{\text{shav}}\right)=K_{\text{Rsh}}\left(P_{\text{ssh}}-P_{\text{shav}}\right)=K_{\text{Rsh}}\sum _{n=1}^{N_b^{-}}\left[P_{\text{ssh}}-P_{\text{shn}-}\right]\ast f_{\text{shn}-}$$ 44

where P_shn- denotes the actual power lower than the planned power P_ssh, on left-half plane of P_ssh in Fig. 6. f_shn- is the incident relative frequency of P_shn-. ${\text{N}}_{\text{b}}^{-}$ refers to the discrete bins number on left-half of P_ssh. Like the same manner in (43), penalty cost for underestimation of the SHPG is gotten by:

$$C_{\text{Psh}}\left(P_{\text{shav}}-P_{\text{ssh}}\right)=K_{\text{Psh}}\left(P_{\text{shav}}-P_{\text{ssh}}\right)=K_{\text{Psh}}\sum _{n=1}^{N_b^{+}}\left[P_{\text{shn}+}-P_{\text{ssh}}\right]\ast f_{\text{shn}+}$$ 45

where P_shn+ denotes the actual power higher than the planned power P_ssh, on right-half plane of P_ssh in Fig. 6. f_shn+ is the relative frequency of incident of P_shn+. ${\text{N}}_{\text{b}}^{+}$ refers to the discrete bins number on right-half of P_ssh.

Fig. 6.

Available total active power (MW) from the combined SHPG at bus 13

The direct, reserve, and penalty cost coefficient values for stochastic solar, wind, and small-hydro powers are recorded in Table 4. The coefficients of direct cost are identified in such a way that wind power cost has the highest value, followed by solar power cost, and finally hydro power cost [18, 35].

Table 4.

Cost coefficients in $/MW for stochastic RESs

Direct			Reserve			Penalty
Wind (bus 5)	Solar (bus 11 and bus 13)	Small hydro (bus 13)	Wind (bus 5)	Solar (bus 11)	Combined solar-small hydro (bus 13)	Wind (bus 5)	Solar (bus 11)	Combined solar-small hydro (bus 13)
gw = 1.7	hs = 1.6	mh = 1.5	KRw = 3	KRs = 3	KRsh = 3	KPw = 1.4	KPs = 1.4	KPsh = 1.4

Optimization algorithm

INFO is a new population-based optimization algorithm that determines the weighted mean for a group of vectors in the search space [22]. It is an efficient algorithm based on the concept of the weighted mean of vectors. INFO avoids the basis of nature’s inspiration, thus reducing the challenges of other population-based optimization algorithms (GA, SMA, HGS, and DE) that are represented in the large number of function evaluations required during optimization and high computational costs. To update the vectors’ positions in each generation, INFO uses three operators as follows:

• Stage 1: Updating rule

• Stage 2: Vector combining

• Stage 3: Local search

Updating rule stage

INFO uses the mean-based rule (MeanRule) to update the position of vectors, which is extracted from the weighted mean for a set of random vectors. The main formulation of the updating rule is defined as,

$$\begin{array}{c}\mathit{if}rand<0.5\\ z{1}_l^g=x_l^g+{\sigma }_I\times MeanRule+randn\times \frac{\left(x_{\text{bs}}-x_{a1}^g\right)}{\left(f\left(x_{\text{bs}}\right)-f\left(x_{a1}^g\right)+1\right)}\\ z{2}_l^g=x_{\mathit{bs}}+{\sigma }_I\times MeanRule+randn\times \frac{\left(x_{a1}^g-x_b^g\right)}{\left(f\left(x_{a1}^g\right)-f\left(x_{a2}^g\right)+1\right)}\\ \mathit{else}\\ z{1}_l^g=x_a^g+{\sigma }_I\times MeanRule+randn\times \frac{\left(x_{a2}^g-x_{a3}^g\right)}{\left(f\left(x_{a2}^g\right)-f\left(x_{a3}^g\right)+1\right)}\\ z{2}_l^g=x_{\mathit{bt}}+{\sigma }_I\times MeanRule+randn\times \frac{\left(x_{a1}^g-x_{a2}^g\right)}{\left(f\left(x_{a1}^g\right)-f\left(x_{a2}^g\right)+1\right)}\\ \mathit{end}\\ \end{array}$$ 46

where $z{1}_l^g$ and $z{2}_l^g$ represent the new vectors in the g-th generation and ${\sigma }_I$ refers to the scaling rate of a vector. $a1\ne a2\ne a3\ne l$ represent different integers randomly selected from the range [1, N_P]; $\mathit{randn}$ denotes a normally distributed random value. It is worthy to be mentioned that in (47), ${\alpha }_I$ can be changed depending on an exponential function provided in (48):

$${\sigma }_I=2{\alpha }_I\times rand-{\alpha }_I$$ 47

$${\alpha }_I=2\times \text{exp}\left(-4.\frac{g}{\mathit{Maxg}}\right)$$ 48

In (46), the MeanRule is formulated as,

$$MeanRule=r\times WM{1}_l^g+\left(1-r\right)\times WM{2}_l^{g}l=1,2,...,Np$$ 49

where r denotes a random number in the range from 0 to 0.5 and $WM{1}_l^g$ and $WM{2}_l^g$ are provided in [22]. In addition, the convergence acceleration part (CA) is also inserted to the updating rule operator in (46) to promote global search ability. CA can be defined as,

$$\begin{array}{c}\text{CA}=randn\times \frac{\left(x_{\text{bs}}-x_{a1}^g\right)}{\left(f\left(x_{\text{bs}}\right)-f\left(x_{a1}^g\right)+1\right)},\text{or as}\\ \text{CA}=randn\times \frac{\left(x_{a2}^g-x_{a3}^g\right)}{\left(f\left(x_{a2}^g\right)-f\left(x_{a3}^g\right)+1\right)},\text{or as}\\ CA=randn\times \frac{\left(x_{a1}^g-x_{a2}^g\right)}{\left(f\left(x_{a1}^g\right)-f\left(x_{a2}^g\right)+1\right)},\text{or as}\\ \end{array}$$ 50

Vector combining stage

According to (51–53), INFO combines the two vectors ($z{1}_l^g$ and $z{2}_l^g$) with vector $x_l^g$ regarding the condition $rand<0.5$ to generate the new vector $u_l^g$. Indeed, the purpose of this operator is to support the ability of local search to provide a new and promising vector:

$$\begin{array}{c}\mathit{if}rand<0.5\\ \mathit{if}ra0nd<0.5\\ u_l^g=z{1}_l^g+{\mu }_I.\left|z{1}_l^g-z{2}_l^g\right|\\ \end{array}$$ 51

$$\begin{array}{c}\mathit{else}\\ u_l^g=z{2}_l^g+{\mu }_I.\left|z{1}_l^g-z{2}_l^g\right|\\ \mathit{end}\\ \end{array}$$ 52

$$\begin{array}{c}\mathit{else}\\ u_l^g=x_l^g\\ \mathit{end}\\ \end{array}$$ 53

where $u_l^g$ represents the obtained vector using the vector combining in g-th generation and ${\mu }_I$ =$0.05\times randn$.

Local search stage

INFO uses the local search stage to prevent from deception and dropping into locally optimal solutions. According to this operator, a new vector can be generated around $x_{\mathit{bs}}^g$, if r < 0.5, where rand represents a random value from 0 to 1:

$$\begin{array}{c}\mathit{if}rand<0.5\\ \mathit{if}rand<0.5\\ u_l^g=x_{\mathit{bs}}+randn\times \left(MeanRule+randn\times \left(x_{\mathit{bs}}^g-x_{a1}^g\right)\right)\\ \end{array}$$ 54

$$\begin{array}{c}\mathit{else}\\ u_l^g=x_{\mathit{rnd}}+randn\times \left(MeanRule+randn\times \left({\upsilon }_1\times x_{\mathit{bs}}-{\upsilon }_2\times x_{\mathit{rnd}}\right)\right)\\ \mathit{end}\\ \mathit{end}\\ \end{array}$$ 55

in which

$$x_{\mathit{rnd}}=\varphi \times x_{\text{avg}}+\left(1-\varphi \right)\times \left(\varphi \times x_{\text{bt}}+\left(1-\varphi \right)\times x_{\text{bs}}\right)$$ 56

$$x_{\text{avg}}=\frac{x_a+x_b+x_c}{3}$$ 57

where $\varphi$ represents a random number in the range of [0–1] and $x_{\text{rnd}}$ denotes a new solution, which combines the components of the solutions, $x_{\text{avg}}$, $x_{\text{bt}}$, and $x_{\text{bs}}$, randomly. This rises the nature of randomness of the proposed algorithm to better search in the solution space. ${\upsilon }_1$ and ${\upsilon }_2$ denote two random numbers given as:

$${\upsilon }_1=\left\{\begin{array}{c}2\times randifp>0.5\\ 1otherwise\end{array}\right)$$ 58

$${\upsilon }_2=\left\{\begin{array}{c}randifp<0.5\\ 1otherwise\end{array}\right)$$ 59

where $p$ refers to a random number from 0 to 1. The random numbers ${\upsilon }_1$ and ${\upsilon }_2$ can rise the influence of the best position on the vector. The pseudo-code of the proposed INFO algorithm is provided in Table 5, while the flowchart of INFO is illustrated in Fig. 7.

Table 5.

Pseudo-code of the INFO algorithm

Fig. 7.

INFO Flowchart

Case studies

In this part, eight case studies are applied, where cases 1–7 are dedicated to the reformed IEEE 30 bus network, and case 8 is dedicated to the reformed IEEE 57 bus network as follows:

Case 1: minimization of total generation cost

The purpose of Case 1 is to minimize the total generation cost of the adapted IEEE-30 bus power system based on (13). The PDF parameters of this case are illustrated in Table 3, while the cost coefficients are provided in Table 4. The optimal results obtained by INFO, HGS, SMA, and GA are recorded in Table 6, in addition to results of MOEA/D-SF and SMODE-SF provided by [26]. The control variables are the voltages of all power generators in the system and the scheduled real powers of all power generators except the scheduled real power of TPG1 (slack bus). The permissible ranges of control variables are almost similar to the values in [18]. However, POZs for TPG2 linked at bus 2 are applied in this study. The ranges of the two POZs are (30, 40) and (55, 65) MW. The range of scheduled power of SHPG at bus 13 is sensibly set based on the cumulative rated power of the combined system. The dependent variables are the real power of TPG1 and the reactive power produced by all generators. Along with the results of the objective function in Table 6, the values of the cumulative voltage deviation of load buses using (31) and system power loss using (32) are also recorded.

Table 6.

Simulation results (Case 1)

Control variables	Min	Max	INFO	MOEA/D-SF [26]	SMODE-SF [26]	HGS	SMA	GA
PTG1 (MW)	50	140	139.439	139.048	139.848	139.998	139.733	136.04
PTG2 (MW) with POZ: [30, 40]; [55,65]	20	80	54.053	53.763	55	54.362	55	54.442
PTG3 (MW)	10	35	11.200	11.558	10	10	10.059	51.529
Pws (MW)	0	75	52.352	52.616	53.391	53.117	52.745	14.824
Pss (MW)	0	50	17.606	17.593	16.818	17.592	17.514	17.518
Pssh (MW)	0	50	15.284	15.319	14.989	14.919	14.962	15.422
V1 (p.u.)	0.95	1.1	1.0788	1.0785	1.0823	1.0777	1.0793	1.0745
V2 (p.u.)	0.95	1.1	1.0647	1.0644	1.0672	1.0625	1.0637	1.0665
V5 (p.u.)	0.95	1.1	1.0425	1.0436	1.0406	1.0423	1.0678	1.0344
V8 (p.u.)	0.95	1.1	1.0952	1.0398	1.0345	1.1000	1.0416	1.0849
V11 (p.u.)	0.95	1.1	1.0901	1.0876	1.0743	1.0972	1.0948	1.0890
V13 (p.u.)	0.95	1.1	1.0601	1.0622	1.0668	1.0991	1.0538	1.0925

Parameters	Min	Max	INFO	MOEA/D-SF [26]	SMODE-SF [26]	HGS	SMA	GA
QTG1 (MVAr)	− 50	140	2.914	2.736	7.191	4.166	6.026	− 10.284
QTG2 (MVAr)	− 20	60	22.323	21.153	27.547	16.085	13.104	40.154
QTG3 (MVAr)	− 15	40	40	39.396	33.207	40	40	40
Qws (MVAr)	− 30	35	26.354	27.549	24.238	26.946	35	16.934
Qss (MVAr)	− 20	25	25	24.836	20.801	25	25	24.641
Qssh (MVAr)	− 20	25	20.286	21.071	24.052	25	17.986	25
Total cost ($/h)			892.618	892.954	893.314	892.781	892.869	893.0412
Emissions (t/h)			2.3340	2.2772	2.3950	2.4176	2.3777	1.8869
Carbon tax, Ctax ($/h)			0	0	0	0	0	0
Ploss (MW)			6.5341	6.4975	6.6453	6.5877	6.6125	6.3751
VD (p.u.)			0.4514	0.4567	0.4369	0.4967	0.4428	0.4957
Computational time (sec.)			209.99			241.59	211.21	265.43

The results demonstrate the effectiveness of INFO in comparison with other algorithms, as it achieved the lowest total generation cost (892.618 $/h) with the shortest computational time (209.99 s) and the fastest solution convergence as shown in Fig. 8.

Fig. 8.

Convergence characteristics (Case 1)

Case 2: minimization of total generation cost with ramp rate

The objective of Case 2 is to minimize the total cost, like in Case 1, but it differs from Case 1 by taking the ramp rate limits of TPGs into consideration. The power produced at the prior hour for TPGs and their ramp-rate limits are recorded in Table 2 [28]. Table 7 contains the optimal results for this case, and Fig. 9 compares the convergence characteristics of the applied algorithms. The influence of the ramp rate limits of the TPGs on total cost can be clearly observed from the simulation results of Case 2. The total power cost rose owing to shifting the new operating points of TPGs from their original operating points. Similar to Case 1, INFO has proven its validity in terms of achieving the minimum cost of power produced (911.1526 $/h) with low computational time (292.59 s) and fast solution convergence as compared to other implemented algorithms.

Table 7.

Simulation results (Case 2)

Control variables	Min	Max	INFO	HGS	SMA	GA
PTG1 (MW)	79.211	114.211	114.211	114.211	114.2105	114.0792
PTG2 (MW)	65	80	65	65	65	65.0322
PTG3 (MW)	12	24	19.52527	20.01209	18.33185	18.49168
Pws (MW)	0	75	56.27241	53.89438	56.70678	56.17604
Pss (MW)	0	50	17.6584	19.23079	19.2405	18.75047
Pssh (MW)	0	50	16.10761	16.53025	15.2772	16.2888
V1 (p.u.)	0.95	1.1	1.073562	1.083592	1.075078	1.074081
V2 (p.u.)	0.95	1.1	1.062813	1.070699	1.064262	1.063369
V5 (p.u.)	0.95	1.1	1.042646	1.044434	1.046509	1.032497
V8 (p.u.)	0.95	1.1	1.042439	1.1	1.058089	1.090554
V11 (p.u.)	0.95	1.1	1.091227	1.098514	1.098171	1.0861
V13 (p.u.)	0.95	1.1	1.09922	1.010631	1.058238	1.1

Parameters	Min	Max	INFO	HGS	SMA	GA
QTG1 (MVAr)	− 50	140	0.002398	10.65197	1.247066	1.161469
QTG2 (MVAr)	− 20	60	18.10617	29.88095	18.91063	27.09806
QTG3 (MVAr)	− 15	40	39.69713	40	40	40
Qws (MVAr)	− 30	35	25.26442	25.03405	28.51421	15.86332
Qss (MVAr)	− 20	25	25	25	25	24.12699
Qssh (MVAr)	− 20	25	25	3.163938	19.32408	25
Total cost ($/h)			911.1526	911.7756	911.179	911.2717
Emissions (t/h)			0.53247	0.532391	0.532656	0.528982
Carbon tax, Ctax ($/h)			0	0	0	0
Ploss (MW)			5.374687	5.478514	5.366771	5.418361
VD (p.u.)			0.513715	0.476038	0.456517	0.473033
Computational time (sec.)			292.59	396.66	366.88	488.94

Fig. 9.

Convergence characteristics (Case 2)

Case 3: minimization of total generation cost with carbon tax

A carbon tax (C_tax) with a rate of $20/ton is assumed in this case. The objective is to minimize the total cost by utilizing (14). With the forcing of the carbon tax, the level of RESs penetration into power system is expected to rise, and this concept is ensured from the simulation results recorded in Table 8. A higher penetration of RESs is achieved as compared to Case 1, when no penalty was imposed on carbon emissions. The extent of RESs penetration in the optimum generation schedule depends solely on the emission volume and rate of the carbon tax imposed. For this scenario, Fig. 10 illustrates the convergence characteristics of INFO and other techniques. INFO has the best performance in terms of total cost minimization, where the minimum overall power cost is achieved by INFO (925.637 $/h) with a low computational time (167.73 s). Figure 11 provides a comparison between the total generation costs in Cases 1–3 using INFO. It implies that a forced thermal ramp rate and a carbon tax on thermal generators will raise total generation costs. Figure 12 provides the voltages of the load buses of the IEEE 30 bus power system in Cases 1–3 using INFO. It illustrates that all buses’ voltages are within the lower and upper limits (0.95 to 1.05 p.u.).

Table 8.

Simulation results (Case 3)

Control variables	Min	Max	INFO	HGS	SMA	GA
PTG1 (MW)	50	140	129.0397	129.1782	129.2923	128.3409
PTG2 (MW) with POZ: [30,40]; [55,65]	20	80	54.99998	54.9996	54.99905	54.86755
PTG3 (MW)	10	35	17.68155	17.45998	17.78784	20.33599
Pws (MW)	0	75	54.47191	54.90071	54.03414	54.13224
Pss (MW)	0	50	17.57203	17.68506	17.70922	16.35268
Pssh (MW)	0	50	15.52201	15.071	15.48369	15.22092
V1 (p.u.)	0.95	1.1	1.077	1.075261	1.076526	1.078515
V2 (p.u.)	0.95	1.1	1.064125	1.062673	1.063016	1.065273
V5 (p.u.)	0.95	1.1	1.043087	1.042073	1.04456	1.038033
V8 (p.u.)	0.95	1.1	1.099971	1.093007	1.1	1.09359
V11 (p.u.)	0.95	1.1	1.1	1.092711	1.098997	1.099995
V13 (p.u.)	0.95	1.1	1.060374	1.072647	1.063278	1.050603

Parameters	Min	Max	INFO	HGS	SMA	GA
QTG1 (MVAr)	− 50	140	2.161844	0.441472	2.985078	4.480408
QTG2 (MVAr)	− 20	60	21.20808	19.27012	17.23751	26.85153
QTG3 (MVAr)	− 15	40	40	40	40	40
Qws (MVAr)	− 30	35	26.11925	25.47406	28.31236	21.19609
Qss (MVAr)	− 20	25	25	25	25	25
Qssh (MVAr)	− 20	25	20.20393	24.67188	21.24579	17.12077
Total cost ($/h)			925.637	926.0098	925.9642	926.2777
Emissions (t/h)			1.230229	1.240533	1.249003	1.179387
Carbon tax, Ctax ($/h)			20	20	20	20
Ploss (MW)			5.887176	5.894565	5.906281	5.850296
VD (p.u.)			0.456535	0.504001	0.467485	0.428755
Computational time (sec.)			167.73	196.42	197.28	287.19

Fig. 10.

Convergence characteristics (Case 3)

Fig. 11.

Comparison of total generation costs in the Cases 1–3 using INFO

Fig. 12.

Voltages of load buses of the IEEE 30 bus power system in the Cases 1–3 using INFO

Case 4: minimization of carbon emission

Case 4 aims at achieving the optimum minimization of the carbon emission in the IEEE 30 bus system based on (15). The results obtained by INFO and other implemented algorithms are provided in Table 9. The results show that the minimum emissions value is achieved by INFO, which is 0.095832 $/h with low computational time of 177.47 s. The results also show that all system constraints are within acceptable limits.

Table 9.

Simulation results of Case 4

Control variables	Min	Max	INFO	MOEA/D-SF [26]	SMODE-SF [26]	HGS	SMA	GA
PTG1 (MW)	50	140	50	60.003	50.047	50	50	50
PTG2 (MW) with POZ: [30,40]; [55,65]	20	80	46.634	65	47.535	50.055	47.463	46.357
PTG3 (MW)	10	35	35	34.89	35	35	35	35
Pws (MW)	0	75	75	74.29	74.282	60.065	67.363	75
Pss (MW)	0	50	48.489	28.529	50	41.182	40.215	46.859
Pssh (MW)	0	50	30.823	23.755	29.336	50.000	47.084	32.934
V1 (p.u.)	0.95	1.1	1.0603	1.0545	1.0738	1.0684	1.0757	1.0248
V2 (p.u.)	0.95	1.1	1.0018	1.0465	1.0596	0.9500	1.0700	1.0489
V5 (p.u.)	0.95	1.1	1.0976	1.0277	1.0393	1.0999	0.9513	1.0326
V8 (p.u.)	0.95	1.1	1.0987	1.0232	1.0196	1.0876	1.0937	1.0672
V11 (p.u.)	0.95	1.1	1.1000	1.0619	1.0576	1.1000	1.0980	1.0829
V13 (p.u.)	0.95	1.1	1.0270	1.0457	1.0481	1.0445	1.0242	1.0489

Parameters	Min	Max	INFO	MOEA/D-SF [26]	SMODE-SF [26]	HGS	SMA	GA
QTG1 (MVAr)	− 50	140	33.518	8.771	28.944	31.422	24.359	− 48.458
QTG2 (MVAr)	− 20	60	− 20	19.405	21.749	− 20.000	60.000	60.000
QTG3 (MVAr)	− 15	40	40	31.835	9.1	40	40.000	40.000
Qws (MVAr)	− 30	35	35	22.165	25.342	35	− 30.000	27.980
Qss (MVAr)	− 20	25	25	22.017	21.241	25.000	25.000	25.000
Qssh (MVAr)	− 20	25	13.176	21.514	20.95	15.643	10.862	22.746
Total cost ($/h)			1019.61	994.342	1020.490	1021.794	1025.666	1020.784
Emissions (t/h)			0.095832	0.1052	0.0959	0.095915	0.095838	0.095833
Carbon tax, Ctax ($/h)			0	0	0	0	0	0
Ploss (MW)			2.5461	3.0671	2.7999	2.9016	3.7258	2.75
VD (p.u.)			0.4852	0.4542	0.468	0.4442	0.487	0.4343
Computation time (sec.)			177.47	–	–	219.23	205.06	343.82

Case 5: considering uncertainty in load demand using INFO

In this case, the effectiveness of INFO is tested considering the uncertainty in load demand. A procedure based on loading level is followed to study the OPF during some different levels (scenarios) of system loading. The determination of load uncertainty is formulated using the normal PDF with standard deviation (σ_d) = 10 and mean (μ_d) = 70 [36]. The loading scenarios are assumed to be 4 scenarios with probabilities and means as shown in Table 10.

Table 10.

System loading levels [36]

Loading scenarios (S)	%Loading, Pd (mean)	Level probability, Δsc
S1	54.749	0.15866
S2	65.401	0.34134
S3	74.599	0.34134
S4	85.251	0.15866

Table 11 displays the simulation results of INFO during the four different scenarios, which show that the total generation cost increases as the loading scenario increases, as shown in Fig. 13. It also shows that the total power losses (P_loss) and carbon emission increase with increasing the loading scenario, while the voltage deviation (V_D) reduces with increasing the loading scenario, as shown in Fig. 14. All system constraints are also within the predefined limits during all loading scenarios, and the voltages of the load buses during the 4 loading scenarios are within the predefined range (0.95–1.05 p.u.) as illustrated in Fig. 15.

Table 11.

Simulation results of Case 5

Control variables	Min	Max	S1	S2	S3	S4
PTG1 (MW)	50	140	67.29841	114.0169	134.9079	134.9079
PTG2 (MW)	20	80	20.69843	20	20	29.99787
PTG3 (MW)	10	35	10	10	10	10
Pws (MW)	0	75	35.50237	26.16256	29.84953	44.29236
Pss (MW)	0	50	10.89238	9.23767	10.93679	14.31491
Pssh (MW)	0	50	12.25656	9.414471	10.43364	13.23397
V1 (p.u.)	0.95	1.1	1.06221	1.069824	1.073623	1.075053
V2 (p.u.)	0.95	1.1	1.054786	1.056967	1.058363	1.060491
V5 (p.u.)	0.95	1.1	1.045089	1.039918	1.038911	1.041508
V8 (p.u.)	0.95	1.1	1.044612	1.041559	1.040357	1.04087
V11 (p.u.)	0.95	1.1	1.083243	1.092768	1.098566	1.099297
V13 (p.u.)	0.95	1.1	1.050237	1.049353	1.051214	1.053895

Parameters	Min	Max
QTG1 (MVAr)	− 50	140	− 1.16675	0.848446	2.301908	2.130022
QTG2 (MVAr)	− 20	60	5.024958	10.20528	13.52355	16.41787
QTG3 (MVAr)	− 15	40	17.71991	21.37765	25.56229	32.55231
Qws (MVAr)	− 30	35	11.33337	15.10191	18.0101	20.9893
Qss (MVAr)	− 20	25	17.42101	22.54645	25	25
Qssh (MVAr)	− 20	25	7.674512	9.380485	11.74322	14.77
Total cost ($/h)			483.0995	568.9144	634.6492	733.3229
Emissions (t/h)			0.11951	0.5313	1.7646	1.7617
Carbon tax, Ctax ($/h)			0	0	0	0
Ploss (MW)			1.4905	3.4841	4.7154	5.1449
VD (p.u.)			0.7463	0.6306	0.5599	0.4976

Fig. 13.

Total cost versus loading scenarios

Fig. 14.

Impact of loading level on voltage deviation, power losses, and emissions

Fig. 15.

Voltages of the load buses during the 4 loading scenarios

Case 6: optimized cost against reserve cost

All parameters of this case are similar to those of Case 1, excluding the coefficient of reserve cost (KR). To study the impact of varying the values of reserve cost coefficients, three subcases are performed with three values of reserve cost coefficient for both wind and solar power, as follows: Case 6a (KR = 4), Case 6b (KR = 5) and Case 6c (KR = 6), while the penalty cost coefficient for all the RESs, KP = 1.4 is still as its value in Case 1 without any change. The optimized power schedule of all generators is illustrated by the bar chart in Fig. 16. It is noticed that with increasing reserve cost coefficient, the optimum schedule power from WPG, SPG, and SHPG declines. This can be interpreted as a reduction in scheduled power, which necessitates less spinning reserve. Thermal generator cost increases are shown in Fig. 17 because the lower power outputs of the RESs are compensated for by thermal power generators. Therefore, the costs of WPG, SPG, and SHPG power gradually decrease to an extent. The total cost rises as the reserve cost coefficient rises.

Fig. 16.

Scheduled real power versus reserve coefficient

Fig. 17.

Impact of reserve cost coefficient variation on costs of all generators

Case 7: optimized cost against penalty cost

All system parameters in this study case are the same as in Case 1 excluding penalty cost coefficients. To study the impact of varying the values of penalty cost coefficients, three subcases are performed with three values of penalty cost coefficient for both wind and solar power as follows: KP = 3 (Case 7a), KP = 4 (Case 7b), and KP = 5 (Case 7c), while the reserve cost coefficient for all RESs is, KR = 3 as its value in Case 1. The optimized schedule power of all power generators is detailed by bar chart in Fig. 18. With increasing the penalty cost coefficient, the scheduled outputs from RESs have a tendency to increase because raising the planned power will help reduce the penalty cost in case of high wind speed or solar irradiance. The cost of WPG power is found progressively increasing in Fig. 19, with increasing the costs of SPG and SHPG, while the TG power cost decreases, and the overall cost shows a slight rise.

Fig. 18.

Scheduled real power versus penalty coefficient

Fig. 19.

Impact of penalty cost coefficient variation on costs of all generators

Case 8: minimizing total generation cost in IEEE-57 bus system

In this case study, the IEEE-57 bus system has been adapted by replacing the 3 TPGs linked at buses 2, 6, and 9 by WPG, SPG, and SHPG, respectively. This case is employed to assess the ability of INFO to optimally solve power systems with more complexity. The active and reactive power components of system loading are 1250.8 MW and 336.4 MVAR, respectively. Table 12 gives the main information about the IEEE 57 bus network. The performed objective function (F1) and system constraints in this case are the same as previously explained in Sect. 2. For verifying the results obtained by INFO, three other optimization algorithms, HGS, SMA, and GA, are employed for solving the OPF problem in the same situations. The results obtained show the effectiveness of INFO with regard to minimizing total generation cost with low computational time and maintaining the system constraints within acceptable settings, as shown in Table 13 and Fig. 20, while the convergences of INFO and other algorithms are provided in Fig. 21.

Table 12.

Parameters of the modified IEEE 57-bus system

Item	Number	Details
Buses	57	[37]
Branches	80	[37]
TPGs	4	At bus 1 (swing), bus 5, bus 8, and bus 12
WPG	1	At bus 2
SPG	1	At bus 6
SHPG	1	At bus 9
Control variables	13	The scheduled power output from: TPG2, TPG3, TPG4, WPG, SPG, and SHPG, generator buses voltages (7 generators)
System load	–	1250.8 (MW), 336.4 (MVAr)
Permissible load buses voltage range (p.u.)	50	0.94–1.06

Table 13.

Simulation results of Case 8

Control variables	Min	Max	INFO	HGS	SMA	GA
PTG1 (MW)	0	576	99.17419	84.90879	169.8347	83.73183
PTG2 (MW)	40	140	116.6247	131.5992	120.6323	124.7215
PTG3 (MW)	100	550	335.6194	335.6193	261.3185	344.2851
PTG4 (MW)	100	410	410	410	409.9993	409.826
Pws (MW)	30	100	100	99.99994	99.99998	100
Pss (MW)	30	100	100	100	100	100
Pssh (MW)	30	100	100	100	99.99994	100
V1 (p.u.)	0.95	1.1	1.068027	1.041218	1.076445	1.059716
V2 (p.u.)	0.95	1.1	1.067912	1.043056	1.074899	1.060651
V3 (p.u.)	0.95	1.1	1.065202	1.047076	1.067004	1.062033
V6 (p.u.)	0.95	1.1	1.064254	1.058902	1.057475	1.062128
V8 (p.u.)	0.95	1.1	1.068951	1.076876	1.05653	1.06553
V9 (p.u.)	0.95	1.1	1.048491	1.046721	1.042432	1.046303
V12 (p.u.)	0.95	1.1	1.053144	1.045035	1.054339	1.060932

Parameters	Min	Max	INFO	HGS	SMA	GA
QTG1 (MVAr)	− 140	200	45.57596	18.17372	49.64064	25.94357
QTG2 (MVAr)	− 10	60	36.31086	29.81988	35.71654	34.45734
QTG3 (MVAr)	− 140	200	43.5552	85.87377	34.83814	43.29191
Qws (MVAr)	− 150	155	46.29675	59.18807	47.6041	81.64951
Qss (MVAr)	− 17	50	49.99998	49.97691	49.94311	49.41857
Qssh (MVAr)	− 8	25	− 1.50499	− 1.96302	1.580005	4.335169
QTG1 (MVAr)	− 3	9	8.994968	8.999653	8.966335	− 0.36129
Total generation cost ($/h)			20,390.6424	20,403.7133	20,393.6679	20,414.4213
Ploss (MW)			10.6183	11.3273	10.9847	11.7644
VD (p.u.)			1.702914	1.406348	1.484137	1.454489
Computation time (sec.)			815.8	851.59	879.48	825.57

Fig. 20.

Voltages of the load buses of the modified IEEE -57 bus power system

Fig. 21.

Convergence characteristics (Case 8)

Conclusion

In this work, an OPF model including both conventional and RESs is provided by modifying IEEE- 30 power system. The used RESs are solar, wind, and small-hydro units that have been modeled by lognormal, Weibull, and Gumbel PDFs, respectively. The overestimation and underestimation of output power of RESs are added to the total power cost in the form of reserve and penalty costs, respectively. A recent meta-heuristic algorithm is used for solving the proposed OPF models. The objective functions are to minimize total cost with and without tax emissions, as well as total emissions. The impacts of valve point, ramp rate limits, and prohibited operating zones (POZs) of thermal power generators on total cost are also studied. The effect of varying load demand is also investigated using four different loading scenarios. The impact of varying the reserve and penalty cost coefficients on the costs of different power sources and the total cost is also observed and obtained. Minimizing the total generation cost in the adapted IEEE-57 bus power system is also considered to verify the validity of INFO in solving the OPF problem in another standard test system. The simulation results of the case studies that have been performed in the two modified power systems prove the effectiveness of INFO in solving the OPF problem as compared to other optimization algorithms in terms of minimizing the total generation cost with the lowest computational time and the fastest convergence to optimal solutions.

List of symbols

$C_{\text{FF}}$

The cost of fossil fuel

a_i, b_i, and c_i

ith TPG’s cost coefficients

P_TG,i

ith TPG’s output power

N_TG

Number of TPGs

d_i and $e_i$

Valve-point loading coefficients

$P_{\text{TG},i}^{\text{min}}$

ith TPG’s minimum power

$C_{\text{wd}}$

The direct cost of wind power

$P_{\text{ws}}$

WPG’s scheduled output power

$g_w$

WPG’s direct cost coefficient

$C_{\text{sd}}$

SPG’s direct cost

$P_{\text{ss}}$

SPG’s scheduled output power

$h_s$

SPG’s direct cost coefficient

C_shd

SHPG’s direct cost

$P_{\text{ssh},s}$

Scheduled output power from the SPG in the combined SHPG

$P_{\text{ssh},h}$

Scheduled output power from the small-hydro unit in the combined SHPG

$m_h$

Direct cost coefficient from small-hydro unit

$C_{\text{rw}}$

WPG’s reserve cost

$P_{\text{wav}}$

WPG’s available power

$K_{\text{rw}}$

WPG’s reserve cost coefficient

$f_w\left(P_w\right)$

The Weibull PDF of the WPG output power

$C_{\text{pw}}$

WPG’s penalty cost

$K_{\text{pw}}$

WPG’s Penalty cost coefficient

$P_{\text{wr}}$

WPG’s rated output power

$K_{\text{rs}}$

SPG’s reserve cost

$P_{\text{sav}}$

SPG’s available power

$f_s(P_{\text{sav}}<P_{\text{ss}})$

The probability of shortage occurrence of solar power

$\text{Ex}(P_{\text{sav}}<P_{\text{ss}})$

The expectation of solar PV power below $P_{\text{ss}}$

$K_{\text{ps}}$

Penalty cost coefficient of the SPG

$f_s(P_{\text{sav}}>P_{\text{ss}})$

The probability of solar power being excess of the P_ss

$\text{Ex}(P_{\text{sav}}>P_{\text{ss}})$

The expectation of solar PV power above the $P_{\text{ss}}$

$K_{\text{rsh}}$

Reserve cost coefficient of SHPG

$f_{\text{sh}}(P_{\text{shav}}<P_{\text{ssh}})$

The probability of shortage occurrence of solar-hydro power

$\text{Ex}(P_{\text{shav}}<P_{\text{ssh}})$

The expectation of solar-hydro power below $P_{\text{shs}}$

φ_i, ψ_i, ω_i, τ_i ,and ζ_i

The coefficients of emissions from the ith TPG

$C_E$

Cost of emission

$C_{\text{tax}}$

Carbon tax

$E$

Emissions (ton/h)

NB

The total number of grid buses

${\delta }_{\text{ij}}=({\delta }_{i-}{\delta }_j)$

The voltage difference between angles of buses i and j

$P_{\text{Gi}}\text{and}{Q}_{\text{Gi}}$

The produced active and reactive power components at bus i, respectively

$P_{\text{Di}}\text{and}{Q}_{\text{Di}}$

The consumed active and reactive components at buses i and j, respectively

$B_{\text{ij}}\text{and}{G}_{\text{ij}}$

The susceptance and conductance between buses i and j, respectively

$P_{\text{TGi}}^{\text{minpoz},j}$

Lower limit (in MW) of the jth POZ of the ith TPG

$P_{\text{TGi}}^{\text{maxpoz},j}$

Upper limit (in MW) of the jth POZ of the ith TPG

$P_{\text{TGi}}^0$

Output power from the ith TPG at previous hour

${\text{Dr}}_i,{\text{Ur}}_i$

Limits of down and up ramp-rate for the ith TPG

Author’s contributions

MF and SK contributed to the conceptualization, methodology, and software. AMA was involved in the conceptualization, data curation, and writing—original draft preparation. AYA assisted in the visualization and investigation. MT-V was involved in the visualization, writing—original draft preparation, and investigation.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this manuscript.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed Consent

Not Applicable.