On Employing a Constrained Nonlinear Optimizer to Constrained Economic Dispatch Problems

Abstract

Recently, different metaheuristic techniques, their variants, and hybrid forms have been extensively used to solve economic load dispatch (ELD) problems with and without valve point loading (VPL) effects. Due to the randomization involved in these metaheuristic techniques, one has to perform extensive runs for each experiment to get an optimal solution. The process may sometimes become laborious and time-consuming to converge to an optimal solution. On the other hand, advanced calculus-based techniques, being deterministic, perform iteration systematically and come up with the same solution on each run of the experiment. Since ELD problems are constrained optimization problems, we are proposing the constrained (deterministic) optimization algorithm for their solutions. Various 13-unit, 38-unit, and 40-unit thermal test systems are considered. Valve point loading (VPL) effects are also considered in some cases. Computer-based numerical results depict that the constrained optimization algorithm shows evidence of being almost as competitive in a total fuel cost as the metaheuristic optimization techniques, especially for the less-constrained ELD problems but with far reduced computation time. This finding validates the application of the constrained optimization technique to solve the economic dispatch problem.

1. Introduction

1.1. Background

One of the most critical and challenging issues that need to be solved in power systems is optimal economic dispatch [1]. The aim of economic dispatch problems (EDPs) is to determine the best power output combination across all generating units in order to reduce overall fuel costs while still meeting load demand and equality and inequality constraints [2]. It is important to note that the fuel consumption curve exhibits ripples due to the valve-point effects. The EDP is thus a large-scale, extremely nonlinear, constrained optimization problem. Significant cost savings can be realized by optimizing the unit output schedule. To reduce total fuel costs, as fuel prices are increasing daily, optimal output power from each generating unit needs to be achieved, which can be ensured by mathematical and metaheuristic optimization techniques. In this research, the performance of IPM is investigated when applied to small-scale to large-scale ELD problems and compared to metaheuristics.

1.2. Literature Review

Recently, researchers have extensively employed advanced calculus-based mathematical and metaheuristics-based optimization algorithms to figure out constrained ELD problems. Among the numerical optimization techniques, quadratic programming (QP) was employed to solve the economic load dispatch problem [3]. The best optimal dispatch is selected based on carrying out optimization independently for each period the generators operate. In [4], a gradient-based approach involving a primal-dual modified log-barrier function was adopted to solve a nonsmooth, nonconvex, and multimodal network-constrained economic dispatch problem in a reasonable amount of time. Other deterministic approaches, including linear [5] and nonlinear programming [6], base point and participation factors method [7], the gradient projection algorithm (GPA) [8], the dynamic programming (DP) method [9,10,11], the decomposition technique [12], the advantageous decision spaces approach [13], and others have also been used to resolve relatively less constrained ELD problems historically. However, with the introduction of more constraints, researchers have shown less interest in these techniques.

Deterministic approaches have solved not only single-objective but also multiobjective, nonconvex, and nondifferentiable environmental and economic dispatch problems with valve-point loading effects (EEDP-VP). For example, in [14], a deterministic approach encompassing a progressive bounded constraints (PBC) strategy, a modified logarithmic barrier function, and a smoothing technique to tackle the dispatch problem’s multiobjective nature, the subproblems originating due to the PBC strategy, and non-differentiability issues, respectively, were reported handling EEDP-VP having five generation systems. In [15], the optimal global point was successfully achieved using a highly effective Lagrange relaxation-based alternating iterative (AI) algorithm while solving a combined heat and power dispatch problem.

Nature-inspired metaheuristic techniques have recently been used in abundance to tackle ELD problems. References [16,17] presented all the variants and hybrid forms of particle swarm optimization used to solve ELD problems. The variations made to the structure of PSO to realize derived versions and its hybridization with other algorithms to avoid premature convergence for well-constructed constrained ELD problems have been summarized comprehensively. A robust learning mechanism-assisted grey wolf optimization algorithm [18] and a surrogate-assisted adaptive bat algorithm [19] successfully tackled large-scale economic load dispatch considering VPL effects. Similarly, in [20], to resolve the large-scale power dynamic economic dispatch of large-scale thermal power units while taking VPL effects and ramp-rate limits into consideration, a two-stage based multi-gradient PSO (MG-PSO) method was proposed.

Squirrel search algorithm [21], a novel swarm intelligence method, GA-SQP [22], modified HPSO-TVAC [23], ion motion optimization algorithm (IMA) [24], ameliorated grey wolf optimization [25], memory-based gravitational search algorithm [26], social optimization algorithm [27], novel social spider optimization algorithm [28], artificial cooperative search algorithm [29], granular computing method [30], hybrid algorithm integrating PSO and BA (PSO-BA) [31], and others have been applied to solve heavily constrained ELD problems. Reference [32] identified many renewable energy technologies contributing to economic growth.

1.3. Contribution

Researchers have extensively used metaheuristics algorithms to solve nonlinear economic dispatch problems. Although metaheuristic optimization techniques handle nonsmooth, nonlinear ELD problems well, they have to perform extensive computations to achieve optimal costs because their initialized populations and updating equations involve random numbers, thus giving different results on each trial. The algorithms have to be run multiple times to come up with minimum and average values of the cost function.

On the other hand, during the period, advanced calculus-based nonlinear constrained optimizers were ignored. In this research, a nonlinearly constrained optimizer named “the interior-point method” (IPM) has been revisited and applied to various nonlinear dispatch problems considering constraints like power balance equations and power generating limits. Different test cases containing thermal units ranging from three to 40 units are solved by IPM.

The reported results show that the interior point method handles less constrained, small/medium to large-scale ELD problems as efficiently as the metaheuristic without involving multiple runs on each experiment in a shorter time. This method has been investigated in this research.

1.4. Paper Organization

The mathematical formulation of the economic load dispatch problem involving fuel cost function subject to equality and inequality constraints is presented in Section 2. The primal-dual interior-point method is theoretically described in Section 3. Numerical results for various considered ELD problems are presented in Section 4. In the end, conclusions are drawn in Section 5.

2. ELD Problem Formulation

The ELD problem is a constrained nonlinear optimization problem that involves the minimization of the total fuel cost of a power plant’s online thermal units while calculating each unit’s power output within the lower and upper bounds, subject to equality and inequality constraints.

For a typical ELD problem, the fuel cost function $F_T$ that must be minimized is given by [33]:

$$\min F_T={\displaystyle \sum _{i=1}^n{F}_i(P_i)}$$

(1)

where $F_i(P_i)$ is the cost due to ith generating unit in $/h having $P_i$ output power in MW, $n$ signifies the total number generating units.

$F_i(P_i)$ is typically modeled by a quadratic function which may be superimposed by valve point loading (VPL) term originating due to the sequential opening of steam admission valves. This makes the cost curves even more nonlinear, nonconvex, and nonsmooth. $F_i(P_i)$ can also be modeled as a cubic cost function. As a result, $F_i(P_i)$ may be quadratic, quadratic with VPL effect, and cubic. It is given by the following:

$$F_i(P_i)=\{\begin{array}{ll}{a}_i{P}_i^2+b_i{P}_i+c_i& \mathrm{Quadratic}\\ a_i{P}_i^2+b_i{P}_i+c_i+\left|e_i\sin f_i(P_{i,\min }-P_i)\right|\hspace{1em}& \mathrm{with}\hspace{0.17em}\mathrm{VPL}\\ a_i{P}_i^3+b_i{P}_i^2+c_i{P}_i+d_i& \mathrm{Cubic}\end{array}$$

(2)

where $a_i$, $b_i$, and $c_i$ are the cost coefficients of the ith generating unit and $e_i$ and $f_i$ represent the ith generating unit’s cost coefficients reflecting the VPL effects.

The ELD problem is subjected to the following constraints:

Active Power Balance Equation:

An active power balance equation is essentially an equality constraint and is given by

$${\displaystyle \sum _{i=1}^n\left(P_i\right)}-\left(P_{Load}+P_{Loss}\right)=0$$

(3)

where $P_{Loss}$ and $P_{Load}$ are the transmission losses and total system demand, respectively. Typically, a (conventional) B matrix loss formula proposed by Kron [34] is used to account for the losses and is given by the equation,

$$P_{Loss}={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{P}_i{B}_{ij}{P}_j}}+{\displaystyle \sum _{i=1}^n{B}_{0i}{P}_i}+B_{00}$$

(4)

where $B_{ij}$, $B_{0i}$, and $B_{00}$ designate the loss coefficients or the B coefficients.

Generator Capacity Limits:

The ith generating unit is restricted to give its output power $P_i$ within its lower $P_{i,\min }$ and upper $P_{i,\max }$ bounds. According to the power bounds, the inequality constraint is specified by the following:

$$P_{i,\min }\le P_i\le P_{i,\max }$$

(5)

3. Interior Point Algorithm

Interior point methods (IPMs) and barrier methods, a particular class of algorithms for solving linear and nonlinear convex optimization problems, can deal with large-scale and sparse problems with many degrees of freedom, as well as small, dense problems [35,36]. Inequality constraints, as opposed to equality constraints, are prevented from being violated by supplementing the objective function with a barrier term, restricting the optimal unconstrained value within the feasible space. In addition, coding interior point approaches into a mathematical program is a fairly straightforward process. At each iteration, the algorithm satisfies the lower and upper bounds. IPMs use derivative information in the form of a Hessian function as an input to make the solution process faster, more accurate, and more robust for a constrained nonlinear minimization problem.

Realizing this fact, the IPM is used to solve small/medium-scale to large-scale constrained nonlinear ELD problems and is compared with other optimization techniques. IPM is theoretically described in the following way.

3.1. Original Problem

An original constrained nonlinear optimization problem involving the minimization of objective/cost function subject to nonlinear l equalities and m inequalities can be formulated as follows:

$$\begin{array}{lll}\min & f(x)& \\ \mathrm{s}.\mathrm{t}.& h_j(x)=0,\hspace{1em}& j=1,2,\dots ,l\\ & g_i(x)\le 0,\hspace{0.17em}& i=1,2,\dots ,m\end{array}$$

(6)

where $f:{ℝ}^n\to ℝ$, $h:{ℝ}^n\to {ℝ}^l$, and $g:{ℝ}^n\to {ℝ}^m$ are assumed to be twice continuously differentiable functions.

3.2. Approximate Problem

With the introduction of slack variables, all inequalities can be converted into non-negativities, and the problem takes the following approximate form:

$$\begin{array}{ll}\min & f(x)\\ \mathrm{s}.\mathrm{t}.& h(x)=0\\ \hspace{0.17em}& g(x)+s=0\end{array}$$

(7)

with $x\ge 0$ and $s\ge 0$.

For m inequalities, there exist m slack variables, i.e., $s={\left(s_i,\dots ,s_m\right)}^T\ge 0$. The approximate problem is converted into a sequence of equality-constrained problems, which are easier to handle than the original inequality-constrained problem.

3.3. Barrier Function

For the interior point strategy, with the addition of a logarithmic term called a barrier function $f_{\mu }$, the barrier problem in s and x is given by [37] as the following:

$$\begin{array}{ll}\min & f(x)-\mu {\displaystyle \sum _{i=1}^m\ln s_i}\\ \mathrm{s}.\mathrm{t}.& h(x)=0\\ \hspace{0.17em}& g(x)+s=0\end{array}$$

(8)

where $\mu >0$ represents a barrier parameter, which reduces to zero iteratively as the minimum of $f_{\mu }$ approaches the minimum of $f$.

3.4. Lagrangian Function

For the problem described in (8), the Lagrangian function is expressed by

$$L(x,s,{\lambda }_h,{\lambda }_g)=f(x)-\mu {\displaystyle \sum _{i=1}^m\ln s_i}+{\lambda }_h^{T}h(x)+{\lambda }_g^T(g(x)+s),$$

(9)

where ${\lambda }_h\in {ℝ}^l$ and ${\lambda }_g\in {ℝ}^m$ signify the Lagrange multipliers pertaining to the equality and inequality constraints, respectively.

3.5. KKT Conditions

The Karush-Khun-Tucker (KKT) optimality conditions, essentially the first-order derivative tests (necessary conditions), can be derived from the Lagrangian function and are given by

$$\begin{array}{l}{\nabla }_{x}L\left(x,s,{\lambda }_h,{\lambda }_g\right)=\nabla f(x)+J_h(x){\lambda }_h+J_g(x){\lambda }_g=0\\ {\nabla }_{s}L\left(x,s,{\lambda }_h,{\lambda }_g\right)=-\mu S^{-1}e+{\lambda }_g=0\\ {\nabla }_{{\lambda }_h}L\left(x,s,{\lambda }_h,{\lambda }_g\right)=h(x)=0\\ {\nabla }_{{\lambda }_g}L\left(x,s,{\lambda }_h,{\lambda }_g\right)=g(x)+s=0\end{array}$$

(10)

with

$$J_h(x)=\left(\nabla h_1(x),\dots ,\nabla h_l(x)\right),$$

$$J_g(x)=\left(\nabla g_1(x),\dots ,\nabla g_m(x)\right),$$

$$e={\left[1,\dots ,1\right]}^T, \mathrm{and}$$

$$S=\mathrm{diag}(s_1,\dots ,s_m);$$

Here $J_h(x)$ and $J_g(x)$ denote the Jacobian of the constraint function $h(x)$ and $g(x)$, respectively; e signifies the vector of those of the same size as $g(x)$; and $S$ represents a diagonal matrix with $s_i$.

Alternatively, the KKT optimality conditions can be written as the following:

$$\left(\begin{array}{c}\nabla f(x)+J_h(x){\lambda }_h+J_g(x){\lambda }_g\\ -\mu S^{-1}e+{\lambda }_g\\ h(x)\\ g(x)+s\end{array}\right)=0$$

(11)

By multiplying the second equation of (11) with $S$, the primal-dual system in the standard form can be obtained as

$$\left(\begin{array}{c}\nabla f(x)+J_h(x){\lambda }_h+J_g(x){\lambda }_g\\ S{\lambda }_g-\mu e\\ h(x)\\ g(x)+s\end{array}\right)=0$$

(12)

This also establishes a correspondence between primal-dual interior-point methods and the SQP approach.

3.6. Solution to KKT System

At each iteration, the algorithm chooses between two primary steps to solve the approximate problem.

3.6.1. Newton or Direct Step

The algorithm first attempts to take Newton step, also called a direct step in (x, s), to solve the KKT equations for the approximate problem via linear approximation at each iteration. The solution of the standard primal-dual system can be found in Newton’s method, which gives the iteration as

$$\left(\begin{array}{cccc}{\nabla }_{xx}^{2}L& 0& J_h(x)& J_g(x)\\ 0& \Lambda & 0& S\\ J_h^T(x)& 0& 0& 0\\ J_g^T(x)& I& 0& 0\end{array}\right)\left(\begin{array}{c}\Delta x\\ \Delta s\\ \Delta {\lambda }_h\\ \Delta {\lambda }_g\end{array}\right)=-\left(\begin{array}{c}\nabla f(x)+J_h(x){\lambda }_h+J_g(x){\lambda }_g\\ S{\lambda }_g-\mu e\\ h(x)\\ g(x)+s\end{array}\right)$$

(13)

with

$$\Lambda =\mathrm{diag}({\lambda }_{g,1},\dots ,{\lambda }_{g,m})$$

$${\nabla }_{xx}^{2}L=H={\nabla }^{2}f(x)+{\displaystyle \sum _{j=1}^l{\lambda }_{h,j}{\nabla }^2{h}_j(x)}+{\displaystyle \sum _{i=1}^m{\lambda }_{g,i}{\nabla }^2{g}_i(x)}$$

Here H denotes the Hessian of the Lagrangian function. The equation can be symmetrized by premultiplying the second variable $\Delta s$ by $S^{-1}$.

$$\left(\begin{array}{cccc}{\nabla }_{xx}^{2}L& 0& J_h(x)& J_g(x)\\ 0& S\Lambda & 0& S\\ J_h^T(x)& 0& 0& 0\\ J_g^T(x)& S& 0& 0\end{array}\right)\left(\begin{array}{c}\Delta x\\ S^{-1}\Delta s\\ \Delta {\lambda }_h\\ \Delta {\lambda }_g\end{array}\right)=-\left(\begin{array}{c}\nabla f(x)+J_h(x){\lambda }_h+J_g(x){\lambda }_g\\ S{\lambda }_g-\mu e\\ h(x)\\ g(x)+s\end{array}\right)$$

(14)

3.6.2. Conjugate Gradient (CG) Step

For a locally nonconvex approximate problem near the current iterate, the algorithm does not take a direct step. In situations where the algorithm cannot take a direct step, it takes a conjugate gradient (CG) step using a trust region. The step involves minimizing a quadratic approximation to the approximate problem in a trust region with radius R, subject to linearized constraints. That is to say,

$$\min \hspace{0.17em}\nabla f{(x)}^T{d}_x+\frac{1}{2}{d}_x^T{\nabla }_{xx}^{2}L(x,s,{\lambda }_h,{\lambda }_g)d_x-\mu e^T{S}^{-1}{d}_s+\frac{1}{2}{d}_s^T{S}^{-1}Z{d}_s$$

(15)

subject to the linearized constraints

$$\begin{array}{l}h(x)+J_h\Delta x=0\\ g(x)+J_g\Delta x+\Delta s=0\end{array}$$

(16)

3.7. Next Iteration and Step Size

After the determination of the step $\Delta =(\Delta x,\Delta s,\Delta {\lambda }_h,\Delta {\lambda }_g)$, the new iterate is computed by

$$\begin{array}{l}{x}_{k+1}=x_k+{({\alpha }_s^{\max })}_k{(\Delta x)}_k\\ s_{k+1}=s_k+{({\alpha }_s^{\max })}_k{(\Delta s)}_k\\ {({\lambda }_h)}_{k+1}={({\lambda }_h)}_k+{({\alpha }_{{\lambda }_g}^{\max })}_k{(\Delta {\lambda }_h)}_k\\ {({\lambda }_g)}_{k+1}={({\lambda }_g)}_k+{({\alpha }_{{\lambda }_g}^{\max })}_k{(\Delta {\lambda }_g)}_k\end{array}$$

(17)

with

$${\alpha }_s^{\max }=\max \{\alpha \in (0,1]:s+\alpha \Delta s\ge (1-\tau )s\}$$

$${\alpha }_{{\lambda }_g}^{\max }=\max \{\alpha \in (0,1]:{\lambda }_g+\alpha \Delta {\lambda }_g\ge (1-\tau ){\lambda }_g\}$$

where $\tau \in (0,1)$ takes a typical value of 0.995 (close to one). The condition named the fraction to the boundary rule prevents the variables s and λ_g from approaching their lower bounds of 0 too quickly. The ${\alpha }_s^{\max }$ and ${\alpha }_{{\lambda }_g}^{\max }$ are the most significant step sizes without violating the nonnegativity constraints and are selecta ed using line search procedure.

3.8. Merit Function

While applying Newton’s method to the KKT condition, not only do the variables s and λ_g remain positive but also the nondifferentiable merit function reduces sufficiently at each iteration, such as,

$${\phi }_{\upsilon }(x,s)=f(x)-\mu {\displaystyle \sum _{i=1}^m\ln s_i}+\upsilon {\Vert h(x)\Vert }_2+\upsilon {\Vert (g(x)+s)\Vert }_2$$

(18)

The penalty parameter may be increased with the number of iterations to force the solution towards feasibility. The merit function controls the quality of the steps. If ${\phi }_{\upsilon }$ does not decrease sufficiently, the algorithm rejects the attempted step and attempts a new step.

3.9. Updating Barrier Parameter

The barrier parameter must get reduced as the iterations go on for the approximate problem to get closer to the original problem [38]. The algorithm updates the barrier parameter monotonically or by predictor–corrector.

3.9.1. Monotonically

The parameter μ decreases monotonically by a factor of 1/100 or 1/5 when the approximate problem has been satisfactorily solved in the preceding iteration. When the algorithm only needs one or two iterations to attain adequate precision, the option applies a factor of 1/100; otherwise, it applies a factor of 1/5.

The following test, which assesses if the magnitude of all terms on the right side of Equation (14) is smaller than μ, serves as a measure of accuracy:

$$\max \left(\Vert \nabla f(x)+J_h^T(x){\lambda }_h+J_g^T(x){\lambda }_g\Vert ,\Vert S{\lambda }_g-\mu e\Vert ,\Vert h(x)\Vert ,\Vert g(x)+s\Vert \right)<\mu$$

(19)

3.9.2. By Predictor–Corrector

The “predictor–corrector” algorithm for updating the barrier parameter μ resembles the “predictor–corrector’” approach used in linear programming. The predictor step employs a linearized step without a barrier function, i.e., µ = 0. The predictor–corrector technique has two effects: it frequently improves step directions and, at the same time, adaptively adjusts the barrier parameter and the centering parameter σ to encourage iterates to follow the central path.

In summary, the pseudocode of the primal–dual interior–point algorithm is outlined in Algorithm 1.

Algorithm 1. Primal–dual interior–point algorithm
1:	Initialize $x_0,s_0>0$ and Lagrangian multipliers ${\lambda }_h,{\lambda }_g>0$.
2:	Choose threshold parameter $0<{\alpha }_{\min }<1$, parameter $\eta >0$, trust-region radius $R>0$, barrier parameter $\mu >0$ and $\sigma ,\tau \in (0,1)$.
3:	Set $k=0$.
4:	Define $z=(x_k,s_k)$ and $\Delta z=(\Delta x,\Delta s)$.
5:	Repeat until an original nonlinear stopping test is met:
6:	Repeat until KKT conditions are met:
7:	Factor the primal-dual system and count its coefficient matrix’s negative eigenvalues nEig.
8:	Set LineSearch = False,
9:	If nEig $\le m+l$,
10:	Obtain the search direction $\Delta =(\Delta x,\Delta s,\Delta {\lambda }_h,\Delta {\lambda }_g)$.
11:	Compute ${\alpha }_s^{\max }$, ${\alpha }_{\lambda }^{\max }$.
12:	If $\min \left({\alpha }_s^{\max },{\alpha }_{\lambda }^{\max }\right)>{\alpha }_{\min }$,
13:	Update the penalty parameter ${\upsilon }_k$.
14:	Compute a steplength ${\alpha }_s=\overline{\alpha }{\alpha }_s^{\max },\alpha \in (0,1)$ such that
15:	${\varphi }_{\upsilon }(z+{\alpha }_s\Delta z)\le {\varphi }_{\upsilon }(z)+\eta {\alpha }_{s}D{\varphi }_{\upsilon }(z;\Delta z)$
16:	If ${\alpha }_s>{\alpha }_{\min }$
17:	Set ${\alpha }_s=\overline{\alpha }{\alpha }_s^{\max }$
18:	Set $(x_{k+1},s_{k+1},{\lambda }_{h,k+1},{\lambda }_{g,k+1})$.
19:	Set LineSearch = True.
20:	Endif
21:	Endif
22:	Endif
23:	If LineSearch = False,
24:	Compute $(x_{k+1},s_{k+1},{\lambda }_{h,k+1},{\lambda }_{g,k+1})$
25:	Endif
26:	Compute $\Delta (k+1)$.
27:	Set ${\mu }_{k+1}\leftarrow {\mu }_k$.
28:	Set $k\leftarrow k+1$.
29:	End
30:	Reduce barrier parameter µ monotonically or by “predictor–corrector.”
31:	End

In the context of ELD problems, the decision vector $x=(x_1,\dots ,x_n)$ refers to the power contribution of each generating unit to meet the load demand. The algorithm starts with an initial guess of each unit’s power contribution, which is taken as the average of the minimum and maximum power.

4. Simulation Results

To validate the performance of the interior point method (IPM), different power systems containing small/medium to large-scale thermal units with and without VPL effects are considered. Generating units’ lower and upper bounds as inequalities are considered. The system/software used for carrying out the simulations for all the cases has the following specifications:

Processor: Intel(R) Core(TM) i7-4600U CPU @ 2.10 GHz 2.70 GHz

RAM: 8.00 GB (7.88 GB usable)

Software: MATLAB R2022b

4.1. Case Study 1: 3-Unit System

A 3-unit system with and without VPL effects is considered for this case. Fuel cost coefficients with and without VPL effects are presented in

Table 1. Fuel cost coefficients and operating generator limits for a 3-unit system.

Case	Unit	Pmin (MW)	Pmax (MW)	a	b	c	e	f
Without VPL	1	100	600	0.001562	7.92	561	-	-
	2	50	200	0.004820	7.97	078	-	-
	3	100	400	0.001940	7.85	310	-	-
With VPL	1	100	600	0.001562	7.92	561	300	0.0315
	2	50	200	0.004820	7.97	078	150	0.0630
	3	100	400	0.001940	7.85	310	200	0.0420

. The data for the VPL-neglected and VPL-considered cases are taken from [39,40], respectively.

4.1.1. Without VPL Effects

The total power demand (PD) considered for this case is 800 MW. The power contribution of each thermal unit and total cost (TC) for a 3-unit system is shown in

Table 2. Power contribution of each thermal unit for a 3-unit system.

Unit	PSO [39]	Interior Point
1	369.9323	369.6871
2	315.5234	315.6969
3	114.5465	114.6160
PD (MW)	800	800
TC ($/h)	7738.797	7738.77

. The total fuel cost offered by IPM is 7738.77 $/h.

From

Table 3. Total fuel cost offered by various methods (3-unit system).

Method	Best Cost ($/h)
Conventional Method [39]	7738.5189
GA [39]	7756.80
PSO [39]	7738.797
Interior Point	7738.77

, it can be observed that IPM shows almost the same total fuel cost as metaheuristics (GA and PSO) and conventional methods without performing multiple runs to obtain the optimal cost. Since the ELD problem is of low dimension, the difference in total fuel cost offered by various methods is insignificant.

4.1.2. With VPL Effects

For the VPL considered case, the total power demand is 850 MW. The dispatching problem becomes more nonlinear with the inclusion of VPL effects. The power contribution of each thermal unit and total fuel cost for a 3-unit system with VPL effects is shown in

Table 4. Power contribution of each thermal unit for a 3-unit system with VPL effects.

Unit	GA [40]	EP [40]	EP–SQP [40]	PSO [40]	PSO–SQP [40]	GSA [41]	Interior Point
1	398.700	300.264	300.267	300.268	300.267	300.2102	300.2631
2	50.100	149.736	149.733	149.732	149.733	149.7953	149.7369
3	399.600	400.000	400.000	400.000	400.000	399.9958	400.0000
PD (MW)	848.400	850	850	850	850	850	850
TC ($/h)	8222.07	8234.07	8234.07	8234.07	8234.07	8234.10	8234.07

. The total fuel cost offered by IPM is 8234.07 $/h.

Table 5. Total fuel cost offered by various methods (3-unit system with VPL effects).

Method	Best Cost ($/h)
GA [40]	8222.07
EP [40]	8234.07
EP–SQP [40]	8234.07
PSO [40]	8234.07
PSO–SQP [40]	8234.07
GAB [42]	8234.08
GAF [42]	8234.07
CEP [42]	8234.07
FEP [42]	8234.07
MFEP [42]	8234.08
IFEP [42]	8234.07
GSA [41]	8234.10
Interior Point	8234.07

shows that IPM yields almost the same total fuel cost (8234.07 $/h) as metaheuristics (GA, EP, EP–SQP, CEP, etc.) without performing multiple runs to obtain the optimal cost, thus justifying its applicability for small-scale ELD problems.

4.2. Case Study 2: 10-Unit System

Table 6. Generator unit capacity and coefficients for a 10-unit system.

Unit	Pmin (MW)	Pmax (MW)	a	b	c
1	23.00	92	0.2162	42.5118	4088.5375
2	23.00	92	0.4108	20.5021	4547.8075
3	47.25	189	0.0562	32.9483	4601.9649
4	47.25	189	0.1266	22.2655	4316.1074
5	10.25	41	0.6210	50.6244	3707.7500
6	10.25	41	0.1255	69.7050	3459.6950
7	23.00	95	3.6454	370.6642	9045.7750
8	23.00	95	0.3981	31.9013	1124.9075
9	23.00	95	2.3185	484.7006	8549.5500
10	41.25	165	0.1142	31.8112	4486.6174

shows the input data for ten generating units of the East Java 150 kV power system located in East Java, Indonesia. The total load demand for this practical case is 616 MW. The cost function for this realistic ELD problem is smooth. Transmission losses and VPL effects are neglected. This ELD optimization problem is, thus, less-constrained. The power contribution of each thermal unit and the total fuel cost offered by different algorithms for a 10-unit system is shown in

Table 7. Power contribution of each thermal unit for a 10-unit system.

Unit	PSO [43]	MIW-PSO [43]	dBA [44]	BA [44]	GA [44]	Interior Point
1	38.63	36.34	35.84	23	42.469	34.1381
2	38.94	46.58	44.47	52.51	54.964	44.7553
3	178.00	189.00	189	185.97	69.765	189.0000
4	142.20	139.16	138.40	150.56	73.755	138.2612
5	13.43	11.06	10.25	10.25	32.788	10.2500
6	13.42	10.25	10.25	10.25	37.772	10.2500
7	29.00	23.00	23	23	23.009	23.0000
8	26.84	29.90	31.62	23	93.591	31.8658
9	29.00	23.00	23	23	23.032	23.0000
10	106.54	107.71	110.17	114.47	164.854	111.4796
PD (MW)	616	616	616	616	616	616
TC ($/h)	95,840.57	95,835.53	95,633.00	95,745.54	100,207.15	95,632.12

.

From

Table 8. Comparative analysis of the total minimum cost of algorithms (10-unit system).

Method	Best Cost ($/h)
PSO [43]	95,840.57
MIW-PSO [43]	95,835.53
dBA [44]	95,633.00
BA [44]	95,745.54
GA [44]	100,207.15
HPSOBA [45]	96,062.547
MHPSO-BAAC [45]	95,768.798
MHPSO-BAAC-χ [45]	95,759.119
Interior Point	95,632.12

, it is surprising to note that IPM outshines the well-established metaheuristic techniques such as PSO, MIW-PSO, dBA, BA, GA, HPSOBA, MHPSO-BAAC, and MHPSO-BAAC-χ for this relatively “less-constrained” ELD problem, justifying its workability for ELD problems. The total fuel cost offered by IPM is 95,632.12 $/h, whereas GA offers the worst cost of 100,207.15 $/h. The IPM offers 4.56557% reduced fuel cost compared to GA.

Throughout the IPM’s execution, numerous progress indicators are plotted in Figure 1. The first (top) curve of Figure 1 represents the contribution of each generating unit, clearly indicating that the third unit contributes heavily, 189 MW, compared to all other generating units. The second (middle) curve represents the progression of function/cost value, which settles to its optimal value of 95,632.12 $/h at the end. First-order optimality is performed in the third curve (bottom) of Figure 1. How near to optimal a point x is is determined by its first-order optimality. IPM exhibits better convergence characteristics.

4.3. Case Study 3: 13-Unit System

The 13-unit case is also used as a benchmark in the literature for various techniques to satisfy the total demand of 1800 MW and 2520 MW. Fuel cost coefficients with and without VPL effects are presented in

Table 9. Fuel cost coefficients and operating generator limits for a 13-unit system.

Unit	Pmin (MW)	Pmin (MW)	a	b	c	e	f
1	0	680	0.00028	8.10	550	300	0.035
2	0	360	0.00056	8.10	309	200	0.042
3	0	360	0.00056	8.10	307	200	0.042
4	60	180	0.00324	7.74	240	150	0.063
5	60	180	0.00324	7.74	240	150	0.063
6	60	180	0.00324	7.74	240	150	0.063
7	60	180	0.00324	7.74	240	150	0.063
8	60	180	0.00324	7.74	240	150	0.063
9	60	180	0.00324	7.74	240	150	0.063
10	40	120	0.00284	8.6	126	100	0.084
11	40	120	0.00284	8.6	126	100	0.084
12	55	120	0.00284	8.6	126	100	0.084
13	55	120	0.00284	8.6	126	100	0.084

. We presume that no transmission losses exist (P_L = 0).

For a load requirement of 1800 MW,

Table 10. Dispatch results for the different methods for 1800 MW demand.

Unit	NN-EPSO [46]	SGA [47]	Interior Point
1	490	359.04	359.0391
2	189	154.18	223.8693
3	214	225.18	223.8693
4	160	159.74	109.8665
5	90	109.87	109.8665
6	120	109.91	109.8665
7	103	159.74	109.8665
8	88	109.87	109.8665
9	104	109.87	109.8665
10	13	77.45	76.6115
11	58	77.40	76.6115
12	66	92.40	90.4000
13	55	55.01	90.4000
PD (MW)	1750	1800	1800
TC ($/h)	18,442.59	18,083.29	18,081.91829

lists the dispatch results for the different methods, including NN-EPSO, SGA, and the proposed interior point method.

Table 11. Final fuel costs calculated using different approaches for 1800 MW demand.

Method	Best Cost ($/h)
ACOR [48]	18,438.73
SGA [47]	18,083.29
PSO [49]	18,132.33
GA [49]	18,138.67
NN-EPSO [46]	18,442.59
Classical PSO [50]	18,239.7537
QPSO [50]	18,321.4745
HQPSO(1) [50]	18,146.7234
HQPSO(2) [50]	18,083.6341
HQPSO(3) [50]	18,134.1893
HQPSO(4) [50]	18,092.7130
Interior Point	18,081.91829

summarizes the final fuel costs calculated using different approaches described in the literature to meet an 1800 MW power demand.

The IPM offers a reduced cost of 18,081.91829 $/h compared to other metaheuristics like ACO_R, SGA, PSO, GA, and NN-EPSO, and this has been achieved without performing multiple runs on each experiment as in metaheuristics cases.

It should not be concluded that IPM always offers an optimal solution. Variants and the hybrid versions of recently developed metaheuristics may give better fuel cost but at the expense of performing various trial runs. The idea is to realize that IPM may handle medium-scale dispatch problems efficiently.

Similarly, for a load requirement of 2520 MW,

Table 12. Dispatch results for the different methods for 2520 MW demand.

Unit	SA [40]	Interior Point
1	668.40	628.3185
2	359.78	359.9766
3	358.20	359.9766
4	104.28	159.7331
5	60.36	159.7331
6	110.64	159.7331
7	162.12	159.7331
8	163.03	159.7331
9	161.52	159.7331
10	117.09	63.3036
11	75.00	40.0087
12	60.00	55.0087
13	119.58	55.0087
PD (MW)	2520	2520
TC ($/h)	24,970.91	24,383.462

lists the dispatch results for the different methods, including SA and the proposed Interior point method.

Table 13. Final fuel costs calculated using different approaches for 2520 MW demand.

Method	Best Cost ($/h)
SA [40]	24,970.91
Interior Point	24,383.462

summarizes the final fuel costs calculated using different approaches described in the literature to meet a 2520 MW power demand.

The IPM offers a reduced cost of 24,573.04 $/h compared to SA, and this has been achieved without performing multiple runs on each experiment as in metaheuristics cases. This justifies the applicability of IPM for solving medium-scale dispatch problems efficiently.

The contribution of each unit, cost convergence curve, and first-order optimality curve for a 10-unit system are shown in Figure 2.

4.4. Case Study 4: 38-Unit System

A large-scale 38-unit system without VPL effects is considered for this case. Fuel cost coefficients with minimum and maximum generation capacity are presented in

Table 14. Fuel cost coefficients and operating generator limits for a 38-unit system.

Unit	Pmin (MW)	Pmin (MW)	a	b	c	Unit	Pmin (MW)	Pmin (MW)	a	b	c
1	220	550	0.3133	796.9	64,782	20	120	272	0.4921	696.1	39,197
2	220	550	0.3133	796.9	64,782	21	120	272	0.5728	660.2	45,576
3	200	500	0.3127	795.5	64,670	22	110	260	0.3572	803.2	28,770
4	200	500	0.3127	795.5	64,670	23	80	190	0.9415	818.2	36,902
5	200	500	0.3127	795.5	64,670	24	10	150	52.123	33.5	105,510
6	200	500	0.3127	795.5	64,670	25	60	125	1.1421	805.4	22,233
7	200	500	0.3127	795.5	64,670	26	55	110	2.0275	707.1	30,953
8	200	500	0.3127	795.5	64,670	27	35	75	3.0744	833.6	17,044
9	114	500	0.7075	915.7	172,832	28	20	70	16.765	2188.7	81,079
10	114	500	0.7075	915.7	172,832	29	20	70	26.355	1024.4	124,767
11	114	500	0.7515	884.2	176,003	30	20	70	30.575	837.1	121,915
12	114	500	0.7083	884.2	173,028	31	20	70	25.098	1305.2	120,780
13	110	500	0.4211	1250.1	91,340	32	20	60	33.722	716.6	104,441
14	90	365	0.5145	1298.6	63,440	33	25	60	23.915	1633.9	83,224
15	82	365	0.5691	1298.6	65,468	34	18	60	32.562	969.6	111,281
16	120	325	0.5691	1290.8	77,282	35	8	60	18.362	2625.8	64,142
17	65	315	2.5881	238.1	190,928	36	25	60	23.915	1633.9	103,519
18	65	315	3.8734	1149.5	285,372	37	20	38	8.482	694.7	13,547
19	65	315	3.6842	1269.1	271,676	38	20	38	9.693	655.9	13,518

. The data for the VPL-neglected case is taken from [46]. The total load demand the 38 generating units have to meet is 6000 MW. For a load requirement of 6000 MW,

Table 15. Power contribution of each thermal unit for a 38-unit system by various methods.

Unit	(BBO) [46]	λ-Logic-Based Method [51]	PS [46]	GWO [46]	Interior Point
1	550	426.6061	258.3397	429.7056	426.6062
2	550	426.6061	258.3397	416.2439	426.6063
3	500	429.6633	238.3397	408.4052	429.6631
4	500	429.6633	238.3397	412.4527	429.6632
5	375.6216	429.6633	238.3397	433.6422	429.6632
6	200	429.6633	238.3397	425.6522	429.6630
7	200	429.6633	238.3397	435.6207	429.6631
8	200	429.6633	238.3397	437.6536	429.6632
9	114	114	196.2345	115.2751	114.0000
10	114.6486	114	196.2345	116.883	114.0000
11	162.1622	119.7681	196.2345	130.7939	119.7681
12	114	127.0729	196.2345	153.2393	127.0732
13	129.2432	110	196.2345	110	110.0000
14	90	90	196.2345	90.028	90.0000
15	153.2432	82	196.2345	82.0111	82.0000
16	120	120	196.2345	120	120.0000
17	204.3243	159.5981	196.2345	157.1682	159.5982
18	65	65	196.2345	65	65.0000
19	65	65	196.2345	65.0326	65.0000
20	120	272	196.2345	271.9524	272.0000
21	182.4324	272	196.2345	271.959	272.0000
22	110	160	196.2345	259.81	260.0000
23	187.2973	130.6487	190	120.8832	130.6483
24	27.027	10	150	12.3567	10.0000
25	125	113.3051	125	107.634	113.3050
26	110	88.0669	110	92.4117	88.0670
27	75	37.5051	75	39.6668	37.5049
28	70	20	70	20.005	20.0000
29	70	20	70	20.0014	20.0000
30	70	20	70	20.0302	20.0000
31	70	20	70	20.013	20.0000
32	60	20	60	20.007	20.0000
33	60	35	60	25.0032	25.0000
34	60	18	60	18.008	18.0000
35	60	8	60	8.006	8.0000
36	60	25	60	25.002	25.0000
37	38	21	38	22.4379	21.7820
38	38	21	38	20.0048	21.0622
PD (MW)	6000	6000	6000	6000	6000
TC ($/h)	10,630,807.3057	-	12,055,832.4091	9,419,270.188	9,417,235.7866

lists the dispatch results for the different methods, including biogeography-based optimization (BBO), λ-logic-based method, pattern search (PS), grey wolf optimizer (GWO), and the proposed interior point method.

Table 16. Total fuel cost offered by various methods (38-unit system).

Method	Best Cost ($/h)
MBDE [52]	9,417,235.786392
SADE [52]	9,417,241.934475
MDE [52]	9,417,235.786397
IDE [52]	9,417,235.786392
λ-logic [51]	9,447,031.7754
SPSO [53]	9,543,984.777
PSO_Crazy [53]	9,520,024.601
New PSO [53]	9,516,448.312
PSO_TVAC [53]	9,500,448.307
BBO [54]	9,417,633.6376
DE/BBO [54]	9,417,235.7864
MsEBBO [55]	9,417,235.7757
Interior Point	9,417,235.7866

summarizes the final fuel costs calculated using different approaches described in the literature to meet a 6000 MW power demand.

The IPM offers a reduced cost of 9,417,235.7866 $/h compared to other metaheuristics tabulated in Table 16, which has been achieved without performing multiple runs on each experiment, as in metaheuristics cases. Among them, SPSO (9,543,984.777 $/h), PSO_Crazy (9,520,024.601 $/h), New PSO (9,516,448.312 $/h), and PSO_TVAC (9,500,448.307 $/h) offered the worst cost. The contribution of each unit, the cost convergence curve, and the first-order optimality curve for a 38-unit system are shown in Figure 3. The results depict that IPM finds its workability even for large-scale ELD problems.

4.5. Case Study 5: 40-Unit System

To further confirm the effectiveness of the interior point method, this test system considers 40 thermal generators with valve–point effects. The total demand considered for this system is 10,500 MW.

Table 17. Fuel cost coefficients and operating generator limits for a 40-unit system.

Unit	Pmin (MW)	Pmax (MW)	a	b	c	e	f	Unit	Pmin (MW)	Pmax (MW)	a	b	c	e	f
1	36	114	0.0069	6.73	94.705	100	0.084	21	254	550	0.00298	6.63	785.96	300	0.035
2	36	114	0.0069	6.73	94.705	100	0.084	22	254	550	0.00298	6.63	785.96	300	0.035
3	60	120	0.02028	7.07	309.54	100	0.084	23	254	550	0.00284	6.66	794.53	300	0.035
4	80	190	0.00942	8.18	369.03	150	0.063	24	254	550	0.00284	6.66	794.53	300	0.035
5	47	97	0.0114	5.35	148.89	120	0.077	25	254	550	0.00277	7.1	801.32	300	0.035
6	68	140	0.01142	8.05	222.33	100	0.084	26	254	550	0.00277	7.1	801.32	300	0.035
7	110	300	0.00357	8.03	278.71	200	0.042	27	10	150	0.52124	3.33	1055.1	120	0.077
8	135	300	0.00492	6.99	391.98	200	0.042	28	10	150	0.52124	3.33	1055.1	120	0.077
9	135	300	0.00573	6.6	455.76	200	0.042	29	10	150	0.52124	3.33	1055.1	120	0.077
10	130	300	0.00605	12.9	722.82	200	0.042	30	47	97	0.0114	5.35	148.89	120	0.077
11	94	375	0.00515	12.9	635.2	200	0.042	31	60	190	0.0016	6.43	222.92	150	0.063
12	94	375	0.00569	12.8	654.69	200	0.042	32	60	190	0.0016	6.43	222.92	150	0.063
13	125	500	0.00421	12.5	913.4	300	0.035	33	60	190	0.0016	6.43	222.92	150	0.063
14	125	500	0.00752	8.84	1760.4	300	0.035	34	90	200	0.0001	8.95	107.87	200	0.042
15	125	500	0.00708	9.15	1728.3	300	0.035	35	90	200	0.0001	8.62	116.58	200	0.042
16	125	500	0.00708	9.15	1728.3	300	0.035	36	90	200	0.0001	8.62	116.58	200	0.042
17	220	500	0.00313	7.97	647.85	300	0.035	37	25	110	0.0161	5.88	307.45	80	0.098
18	220	500	0.00313	7.95	649.69	300	0.035	38	25	110	0.0161	5.88	307.45	80	0.098
19	242	550	0.00313	7.97	647.83	300	0.035	39	25	110	0.0161	5.88	307.45	80	0.098
20	242	550	0.00313	7.97	647.81	300	0.035	40	242	550	0.00313	7.97	647.83	300	0.035

displays the generation limits and generator cost coefficients. Transmission losses are ignored for this test system. The data for the VPL-considered case is taken from [56].

For a load requirement of 10,500 MW,

Table 18. Power contribution of each thermal unit for a 40-unit system by various methods.

Unit	Classical PSO [56]	PSO_TV AC [56]	DE3 [56]	BFO [56]	BA [57]	BA-Penalty [57]	Interior Point
1	78.1003	79.8086	79.8090	79.8090	113.1233	111.9952	113.6341
2	113.3127	113.3186	113.3222	113.3222	111.4569	110.9453	113.6337
3	119.7509	120.0000	120.0000	120.0000	120	97.39597	119.9828
4	129.8665	129.8666	129.8666	129.8666	179.9948	179.7417	180.0212
5	88.0047	87.9899	87.9902	87.9902	97	88.92837	89.2895
6	140.0000	140.0000	140.0000	140.0000	139.9736	105.4038	139.9917
7	274.6463	274.6963	274.6946	274.6946	300	259.6279	299.9830
8	299.8646	299.8627	299.8632	299.8632	296.7893	284.6572	284.6222
9	284.6040	284.5998	284.5997	284.5997	292.5603	284.6307	288.2776
10	200.0000	200.0000	200.0000	200.0000	130.0603	131.9808	204.7540
11	94.0000	94.0000	94.0000	94.0000	94	168.7988	168.8652
12	94.0000	94.0000	94.0000	94.0000	94.16944	318.3965	94.0185
13	394.2794	394.2794	394.2794	394.2794	484.0661	375.8561	304.5780
14	300.0000	300.0000	300.0000	300.0000	125.0045	394.2805	394.3340
15	484.0392	484.0392	484.0392	484.0392	125.0941	125.0027	304.7904
16	214.7990	214.7598	214.7598	214.7598	304.6026	394.2744	304.5801
17	489.2795	489.2794	489.2794	489.2794	489.5124	489.2821	399.8920
18	489.3031	489.2794	489.2794	489.2794	489.3235	489.3007	399.5876
19	528.0891	527.1423	527.1309	527.1309	547.7208	511.2816	511.3170
20	511.2794	511.2794	511.2794	511.2794	549.9241	511.2772	511.3000
21	523.2794	523.2794	523.2794	523.2794	548.6068	523.2853	523.4055
22	523.2973	523.2935	523.2834	523.2834	545.562	523.2868	523.4275
23	523.2817	523.3203	523.2794	523.2994	545.9307	523.2973	523.8080
24	523.2817	523.3203	523.2991	523.2991	543.7959	514.5068	524.1301
25	528.9245	527.0591	526.8115	526.8115	549.7956	523.2821	523.4521
26	523.2796	523.2794	523.2807	532.2807	543.9368	523.8991	523.5345
27	10.0000	10.0000	10.0000	10.0000	10	10.00444	10.0143
28	10.0000	10.0000	10.0000	10.0000	10.04373	9.999218	10.0143
29	10.0000	10.0000	10.0000	10.0000	10.00774	9.999577	10.0143
30	90.5193	90.2770	90.2777	90.2777	96.83174	89.70938	89.3020
31	190.0000	190.0000	190.0000	190.0000	189.9952	110.7659	189.9906
32	190.0000	190.0000	190.0000	190.0000	189.8675	191.6123	189.9906
33	190.0000	190.0000	190.0000	190.0000	190	191.5734	189.9906
34	166.7258	166.6847	166.9471	166.9471	199.9782	164.8092	199.9813
35	199.3297	199.9938	200.0000	200.0000	199.9634	165.5802	199.9808
36	171.3223	171.7952	171.7950	171.7950	200	164.9268	199.9808
37	89.1152	89.1182	89.1183	89.1183	110	90.73679	109.9885
38	110.0000	110.0000	110.0000	110.0000	110	111.304	109.9885
39	89.1462	89.1368	89.1460	89.1460	110	111.1426	109.9885
40	511.2509	511.2789	511.2789	511.2794	511.3088	511.3018	511.5650
PD (MW)	10,500	10,500	10,500	10,500	10,500	10,500	10,500
TC ($/h)	122,729.6552	122,710.4302	122,709.5039	122,709.5039	123,757.39	122,936.74	122,264.8799

lists the dispatch results for the different methods, including classical PSO, PSO_TV AC, DE3, BFO, BA, BA-Penalty, and the proposed interior point method.

Table 19. Total fuel cost offered by various methods (40-unit system).

Method	Fuel Cost ($/h)
	Minimum	Maximum	Average
ACOR [48]	127,734.93	129,695.74	128,840.14
MBDE [42]	124,135.413297	126,953.92874	125,547.293196
CEP [42]	123,488.29	126,902.89	124,793.48
FEP [42]	122,679.71	127,245.59	124,119.37
MFEP [42]	122,647.57	124,356.47	123,489.74
IFEP [42]	122,624.35	125,740.63	123,382.00
PSO [58]	122,323.97	123,690.62	125,103.28
BA [57]	123,757.39	125,979.26	128,510.43
BA-Penalty [57]	122,936.74	126,093.09	129,218.58
Classical PSO [56]	122,729.6552	-	-
PSO_TVAC [56]	122,710.4302	-	-
DE3 [56]	122,709.5039	-	-
BFO [56]	122,709.5039	-	-
Interior Point	122,264.8799	-	-

provides a summary of the final fuel costs calculated using different approaches described in the literature to meet a 10,500 MW power demand.

The IPM offers a reduced cost of 122,264.8799 $/h compared to other metaheuristics tabulated in Table 19, and this has been achieved without performing multiple runs on each experiment as in metaheuristics cases. ACO_R (127,734.93 $/h) offered the worst cost among them. Alternatively, each unit’s contribution, total fuel cost curve, and first-order optimality curve are shown in Figure 4. The results show that IPM finds its workability even for large-scale ELD problems with 40 generating units.

5. Conclusions

This research uses the interior point method to handle small-scale to large-scale ELD problems rather than abundantly available metaheuristic optimization techniques. Although metaheuristics find optimal solutions for ELD problems, they usually require multiple runs for each experiment to reach the optimal solution, making finding the optimal solution more complex. Metaheuristics involve randomization in population initialization and updating equations, giving different results on each run. Minimum, maximum, and average values have to be recorded to ensure optimal fuel cost. Numerical results revealed that the IPM offered reduced or competitive total fuel costs compared to many metaheuristics for relatively less constrained ELD problems. This result justifies once scorned, now respectable, IPM as a good choice for figuring out ELD problems. Various power systems consisting of 3-unit, 10-unit, 13-unit, 38-unit, and 40-unit were tested. It is concluded that if the ELD problems do not include heavy constraints, such as ramp rate limits, prohibited operating zones (POZ), line flow constraints, multi-fuels options, etc., the constrained IPM can be employed to handle ELD problems without performing multiple runs for each experiment. In addition, these constrained optimizers can be hybridized with metaheuristics to solve nondifferentiable, discontinuous, nonsmooth fuel cost functions. In future work, the constrained nonlinear IPM method can be applied to ELD problems involving more constraints, and their performance can be investigated compared to metaheuristics to validate their applicability. Also, IPM may be applied to multi-objective combined economic emission dispatch (CEED) and renewables integrated dispatch problems to check their performance compared to metaheuristics.