Enhanced emperor penguin optimization algorithm for dynamic economic dispatch with renewable energy sources and microgrid

Abstract

In this paper, an enhanced version of the emperor penguin optimization algorithm is proposed for solving dynamic economic dispatch (DED) problem incorporating renewable energy sources and microgrid. Dynamic economic load dispatch optimally shares the power on an hourly basis for a day among the committed generating units to satisfy the feasible load demand. Emission of pollutants from the combustion fossil fuel and gradual depletion of fossil fuel encourages the usage of renewable energy sources. Implementation of renewable energy sources with the reinforcement of green energy transforms the fossil fuel-based plant into a hybrid generating plant. The increase in power production with the increase in electricity demand implicates challenges for economical operation. The proposed algorithm is applied to the DED problem for fossil fuel based and renewable energy system to find economic schedule of generated power among the committed generating units. The proposed optimization algorithm is inspired by the huddling behavior of the emperor penguin. The exploration strategy is enhanced by adapting oppositional based learning. Chaotic mapping is used to maintain a proper balance between exploration and exploitation in the entire search space, which minimizes the cost of generation in the power system.

Keywords

Dynamic economic dispatch (DED)emperor penguin optimization (EPO)chaotic oppositional learning-based emperor penguin optimization (COLEPO)constraints wind energy micro grid

1 Introduction

The electric power supply in the current scenario demands a cost-effective and reliable operation. Economic dispatch contributes significantly towards the optimum generation and operation in a power system. Economic dispatch optimally schedules and allocates the generated power among the committed generating units to meet the load demand satisfying all the operational and practical constraints. Dynamic economic dispatch is the enhanced version of static economic dispatch to schedule the power on an hourly basis for a specific day, as the load demand is varying stochastically in a real-time operation. The practical constraints like the valve point effect and ramp rate limit increase the complexity as the problem converted into a non-convex and non-linear problem [1]. The depletion of fossil fuel, reducing the environmental impact from the emission and availability of power from renewable sources tends to adapt the fossil fuel-based plant to the hybrid generating system [2]. The implementation of renewable energy in microgrid for power production reduces the losses, works as a replacement of large generators. Renewable energy also prevents system blackout with proper operation and placement. In the modern power system, the hybrid generating system performs a significant for optimal and low emission power supply. The microgrid is also considered as the hybrid generating system with the combination of distributed generators as renewable energy sources and fossil fuel-based system and a control system [3]. The dynamic economic dispatch problem considering the hybrid generating system and microgrid, schedules the generated power among the fossil fuel based generators, renewable energy sources and distributed generators to optimize the cost of generation [46].

Primarily the solution of Dynamic economic dispatch (DED) problem was obtained by some classical methods for fossil fuel based generators such as Lagrangian relaxation (LR) [4], dynamic programming (DP) [5], Linear Programming (LP) [6], Quadratic programming (QP) [7] and non-linear programming (NLP) [6] and many more. The solution of dynamic economic dispatch problem with the classical methods produces many challenges for the large power systems due to large dimensionality [47 –51]. The integration of valve point effect affects the linear programming method to solve the problem due to non-linear cost function. The objective function for solving precisely, the QP and NLP problem should be continuous and differentiable. Thus, the solution by QP and NLP is inaccurate in the final stage. The correction strategy of Lagrange multipliers in the LR method reduces efficiency by producing an oscillatory solution. Dynamic programming method used for solving DED problem with higher computational cost and time. Some classical methods did not able to find the solution for non-linear and non-convex cost of fossil fuel based generating units and several methods find the solution of the problem with larger convergence time. Therefore, to overcome such difficulties in the classical techniques for solving dynamic economic dispatch, many metaheuristics and swarm intelligence approach was introduced with their conventional version. Li et al. applied the conventional genetic algorithm (GA) [8] for the solution of the problem with the inclusion of transmission losses and ramp rate limit. The GA technique appropriately handled the constraints without any additional liability, and an improved strategy was demonstrated for a better optimal solution. The application of Differential Evolution (DE) techniques was applied by Balamurugan and Subramanian [9] with a simple structure and excellent convergence property. Many novel metaheuristics methods were applied such as Particle Search Optimization (PSO) [10], Symbiotic organisms search algorithm (SOSA) [11], Artificial Immune System (AIS) [12] for getting the solution of dynamic economic dispatch problem. The SOSA technique provides an optimized and robust result as compared to other technique for larger dimensionally system. These methods provide the optimum solution with an increased convergence time because of extensive power system used. The conventional methods attempt to find the optimum scheduling and cost. In the recent past, many conventional methods were further improvised and hybridized for enhancing the quality of solution regarding optimum scheduling for minimum cost with lesser convergence time. Lu et al. proposed a hybridized version of known as Chaotic differential bee colony optimization algorithm (CDBCO) [13] to increase the exploration capability by applying the chaotic sequence to the differential evolution method with the combination of chaotic local search method applied to bee colony optimization for better exploitation property and better convergence property. Elattar et al. [14] proposed a hybrid genetic algorithm and bacterial foraging (HGABF) to overcome the poor convergence property by conventional bacteria foraging algorithm. Hybridization of biogeography-based optimization and brainstorm optimization was proposed by Xiong et al. [15] for the application in DED problem with no linearities and non-convexity. Improvisation was prepared for better local search by the application of bio geography-based technique and improved global search by integrating brainstorm optimization.

Further, many approaches were used for the solution of dynamic economic dispatch problem incorporating the microgrid and renewable energy system. Basu et al. [3] used the cuckoo search algorithm for the solution of the economic dispatch problem with the wind and fuel cell. The proposed method shows superiority than the other applied method. Bi-population chaotic sequenced differential evolution was proposed for the DED problem integrating wind and thermal system.

These kinds of literature show the application of various conventional approaches to the dynamic economic dispatch with and without integrating renewable energy system [54 –61]. The implementation of renewable energy and distributed generators enables the power system to be more complex with the integration of uncertainty. The conventional techniques face a lot of challenges to optimally schedule the power among the generating units. The applied methods may not be able to find the optimal solution, according to no free lunch theorem [16]. No free lunch theorem justifies that, any optimization methods cannot claim as the best solution for the optimization problem. To overcome the challenges an enhanced version of an optimization technique is applied to manage the difficulties for large complexity and uncertainty.

Emperor penguin optimizer (EPO) is a novel optimization technique that acts with the inspiration of huddling behaviour of the emperor penguin. The conventional EPO method rarely converges prematurely considering all the non-linear constraints. Thus, an enhanced version of EPO is applied with the integration of chaotic mapping and oppositional based learning strategy. The proposed method used to get the solution of real time economic dispatch for hourly basis scheduling of power among the generating units. This chaotic based oppositional learning emperor penguin optimization (COEPO) maintains a perfect balance between exploration and exploitation for improved search quality and solution quality.

2 Modeling of dynamic economic dispatch problem with wind energy

The optimum scheduling of electric power among the generating units for a specified time to satisfy the load demand is known as Dynamic economic dispatch. The combination of thermal generating units and renewable energy introduces clean and green energy for the reduction of emission. The objective function of DED for thermal plants is to be minimized in accordance to cost function satisfying all the technical, practical and operational constraints. Generally, the overall objective function with the consideration of wind energy combines the objective function of the thermal system besides the underestimation and overestimation cost of wind power uncertainty. The overall cost function for hybrid generators (Fossil fuel based and renewable energy sources) is represented as C_f[17] as shown in Equation 1. $\begin{matrix} \min C_{f} (T, W_{i}) = \sum_{gu = 1}^{N_{g}} \sum_{t = 1}^{H} F_{tgu} (T_{gu}, t) \\ + \sum_{gu = 1}^{N_{w}} \sum_{t = 1}^{H} F_{wgu} (W_{gu}, t) \end{matrix}$ (1)

Here the C_f (T, W_i) shows the total cost function to be minimized. F_tgu and F_wgu represent the cost function of fuel of the thermal power plant and the cost function of wind power respectively. N_g and N_w show the number of thermal generating units and wind firm, respectively. T_gu represent the power generated from the thermal power plant from gu number of generators while W_gu represents the power generated from gu number of wind firm.

2.1 The cost function for thermal power plant

Each generator is used to produce power by the thermal generators were characterized according to their characteristics. Application of practical constraints to the turbine and the generators made the characteristics as nonlinear and non-convex. The objective function behaves as a piecewise linear characteristic is shown in Equation 2 for g_u number of generators [9]. $\begin{matrix} F_{tgu} (T_{gu . t}) = \sum_{gu = 1}^{N_{g}} \sum_{t = 1}^{H} a_{gu} T_{gu, t}^{2} + b_{gu} T_{gu, t} \\ + c_{gu} + | e_{gu} \times sin (f_{gu} (T_{gu}^{min} - T_{gu, t})) | \end{matrix}$ (2)

Here the cost coefficients with the effect of valve point are represented as a_gu, b_gu, c_gu, e_gu, f_gu in the objective function to be minimized T_gu represents the power generated from entire committed generating units. F_tgu (T_gu,t)shows the objective function for cost of gu^th at time t for the output power of T_gu. N_g shows the number of generating units and H shows the time for operation of generating units.

2.2 The cost function for wind firm

The cost of power generation from the wind firm depends upon the overestimation cost and underestimation cost [53]. During overestimation, the power generated from the wind firm is below than the scheduled power, thus a penalty must be enforced for deficiency of the scheduled power. However, during underestimation, the actual generated power from the wind firm is excess than that of scheduled power. The extra power generated is used to charge the batteries. The cost function is represented in Equation 3 [17].

$\begin{matrix} F_{wgu} (W_{gu},_{t}) = & C_{oe, wgu} (W_{gu} - W_{av, wgu}) \\ + C_{ue, wgu} (W_{av, wgu} - W_{gu}) \end{matrix}$ (3)

Here C_oe,wgu and C_ue,wgu represents the coefficients due to the penalty for the overestimation and underestimation of the cost for wind power. W_av,wgu shows the availability of wind power from gu^th wind turbine.

2.3 Operational constraints

2.3.1 Power balance constraints

The power generated amongst the committed thermal generators must satisfy the projected demand of load with considering the transmission losses as shown in Equation 4 [17]. $\sum_{gu = 1}^{N_{g}} \sum_{t = 1}^{H} T_{gu, t} - P_{Dgu, t} - P_{Lgu, t} = 0$ (4)

The total power generated from all generating units at time t is T_gu, P_Dgu represents the power demand to be consumed; P_Lgu shows the losses during transmission of electric power. The transmission losses P_Lgu is represented with the B coefficient as shown in Equation 6 [9]. Balancing the power considering wind energy is represented in Equation 5 with W_gu,t is the power generated from wind firm [17]. $\sum_{gu = 1}^{N_{g}} \sum_{t = 1}^{H} T_{gu, t} + W_{gu, t} - P_{Dgu, t} - P_{Lgu, t} = 0$ (5)

$\begin{matrix} P_{L_{gu, t}} = & \sum_{gui = 1}^{N_{g}} \sum_{guj = 1}^{N_{g}} T_{gui, t} B_{guiguj, t} T_{guj, t} \\ + \sum_{gui = 1}^{N_{g}} B_{0 gui, t} T_{{gu}_{i, t}} + B_{00} \end{matrix}$ (6)

Here, B_guiguj, B_0gui and B₀₀ are transmission loss coefficient.

2.3.2 Capacity constraints

The output power generated from entire thermal and wind generators must fulfil their capacity constraints, i.e. should lie in between the upper and lower generation capacity as represented in Equations 7 and 8, respectively [17]. $T_{gumin} < T_{gu, t} < T_{gumax}$ (7) $0 < W_{gu, t} < W_{gumax}$ (8)

Where, T_gumin and T_gumax represents the lower capacity limitand upper capacity limit respectively for thermal generating unit and W_gumax is the maximum generating limit of the wind turbine.

2.3.3 Ramp rate constraints

During the real-time operation of generators, the effect of ramp rate constraints affects power production. The current generation of power impacts on the future scheduling to bound the generation, as shown in the Equation 9 [17]. ${\begin{matrix} T_{gu, t} - T_{gu, t - 1} ⩽ {UR}_{gu} & gu = 1, 2, \dots, N_{g} \\ T_{gu, t - 1} - T_{gu, t} ⩽ {DR}_{gu} & t = 1, 2, . . . ., H \end{matrix}$ (9)

Here, T_gu,t shows the generated active power during the previous operation of gu^th unit in MW. DR_gu, UR_gu are the respective down and up ramp rate limit.

2.3.4 Spinning reserve constraints

The sum of spinning reserve constraints of the generators can be used for sudden increase in load. A total of 5 % to 10% of the maximum load can be reserved from every generator for sudden change and source contingencies.

The positive and negative spinning reserve capacity constraint with wind power system is represented in Equations 10 and 11 respectively [17]. ${\begin{matrix} W_{av, gu} \times W_{u} % + P_{D, gu} \times L_{u} % < \sum_{gu = 1}^{Ng} U_{gu}, \\ U_{gu} = min (T_{gu, \max} - T_{gu} U_{Rgu} H_{10}) \end{matrix}$ (10)

${\begin{matrix} (W_{gu, \max} - W_{av}) \times W_{d} % + P_{D, gu} \times L_{d} % < \sum_{gu = 1}^{N_{g}} D_{gu} \\ D_{gu} = \min (T_{gu} - T_{gu, \min} D_{Rgu} H_{10}) \end{matrix}$ (11)

Here W_u% and W_d% shows the demand coefficients during the wind power prediction errors for positive and negative spinning reserve, respectively. L_u% and L_d% is the demand coefficients shows the system load prediction error. U_gu and D_gu are the spinning reserve capacity for gu^th plant. H₁₀ shows the response time for spinning reserve for 10 minutes.

3 Modelling of dynamic economic dispatch problem for microgrid

The considered microgrid comprises of multiple distributed generators assembling diesel generators, wind power plants and fuel-cell based generators. The dynamic economic dispatch of microgrid optimally schedules the electric power among the distributed generators to satisfy the load demand with consideration of constraints [52].

3.1 The cost function of diesel generators

The cost characteristics of diesel generators are identical as to the characteristics curve of the thermal plant without valve point effect. Thus, the objective function for cost considering the timeof the diesel generator in a microgrid in dynamic economic dispatch is presented in Equation 12 [18]. $F_{dg, t} = \sum_{dg = 1}^{N_{dg}} \sum_{t = 1}^{H} α_{di} T_{dg, t}^{2} + β_{di} T_{dg, t} + γ_{di}$ (12)

Where F_dg,t represents the cost of diesel generator for dg number of generators at t time. α_di, β_di, γ_di represents the coefficients for diesel generators. T_dg,t shows the power output at of dg^th generator at t time.

3.2 The cost function of wind generators

The objective function of the wind generator for the microgrid is calculated for power output as a function of wind speed [18]. $F_{wi, t} = \sum_{t = 1}^{H} \sum_{i = 1}^{N_{wi}} β_{wd} P_{wi, t}$ (13) $P_{wi} = P_{{wi}_{r} t, i} \times \frac{(V_{eit} - V_{ct, i})}{(V_{rt, i} - V_{ct, i})} KW,$ (14)

Here, V_ct,i ⩽ V_it ⩽ V_rt,i, t ∈ H

P_wi,it = P_{wi
_{rt
_,i}}KW, V_rt,i ⩽ V_it ⩽ V_ctot,i, t ∈ H

P_wi,it = 0, V_it ⩽ V_ct,i and V_it > V_ctot,i, t ∈ H

where, the cost function of a wind turbine is represented by F_wi,t for N_wd number of the wind turbine at a time t. β_wd shows the maintenance and operating cost in $/KW. The output power depends upon the wind velocity V_it at time (t) with the rated velocity of V_rt, V_ct,i and V_ctot,i represents the cut-in velocity and cut-out velocity, respectively. The generated power is P_wi while the rated power of wind is P_{wi_rt,i}.

3.3 The cost function of fuel-cell plant

The cost of operation of the fuel-cell system considers both costs of fuel and efficiency of fuel for electricpower generation as represented in Equation 15 [18]. $F_{fc, t} = \sum_{t = 1}^{H} (β_{nat} \sum_{i = 1}^{N_{ft}} \frac{P_{ft, it}}{η_{ft, i}})$ (15)

Here, F_fc,t shows the generation cost of fuel-cell, β_nat is the cost of natural gas in $/KG, P_ft,it is output power from the fuel-cell from i_th units at t^th time. The efficiency and the number of fuel-cell plants are denoted as η_ft and N_ft respectively.

3.4 Overall objective function for a microgrid with the combination of distributed generation

The overall cost function of the microgrid is the summation of all the cost function of distributed generators as shown in Equation 16 [18]. $F_{oa, t} {= F}_{dg, t} {+ F}_{wi, t} {+ F}_{fc, t} {+ F}_{by}$ (16)

Here, the F_by represents the cost of purchasing the power from a grid and calculated as per Equation 17. $F_{by} = \sum_{t = 1}^{H} (K_{p} P_{S, by, t})$ (17)

Here, K_p represents the cost of buying power while P_S,by,t shows the purchasing power from the transmission network at t time.

3.5 Constraints for economic dispatch of microgrid

3.5.1 Constraints for power balance

The total power generated from distributed generators and the purchasing power should meet the load demand and the power losses as shown in Equation 18 [18].

$\begin{matrix} \sum_{dg = 1}^{N_{dg}} \sum_{t = 1}^{H} T_{dg, t} + \sum_{i = 1}^{N_{wi}} \sum_{t = 1}^{H} P_{wi, t} + \sum_{i = 1}^{N_{ft}} \sum_{t = 1}^{H} P_{ft, it} \\ + P_{s, by, t} = \sum_{t = 1}^{H} P_{Dl, t} + P_{ls, t}, t \in H \end{matrix}$ (18)

Here, P_Dl,t is the load demand and P_ls,t is the power loss at time t.

3.5.2 Constraints for power generation capacity of distributed generators

The generated power from each distributed generator should be within the upper and lower limit as given in Equation 19 [18]. $\begin{matrix} T_{dg, t}^{min} ⩽ T_{dg, it} ⩽ T_{dg, t}^{max}, dg \in N_{dg}, t \in H \\ P_{wi, it}^{min} ⩽ P_{wi, it} ⩽ P_{wi, it}^{max}, i \in N_{wi}, t \in H \\ P_{ft, it}^{min} ⩽ P_{ft, it} ⩽ P_{ft, it}^{max}, i \in N_{ft}, t \in H \end{matrix}$ (19)

4 Emperor penguin optimizer

The emperor penguin optimizer is encouraged from the huddling attitude of the emperor penguins which were originated in the Antarctic. Emperor penguins usually process in colonies for foraging [19]. This huddling behaviour is the unique characteristic as observed in these social animals during foraging. Hence, in the mathematical model, the prime objective will be to identify an effective mover from the swarm. To achieve this, the distances between Emperor Penguins (EPs) (Z_ep) are calculated followed by their temperature profile (T_mp). From this, the effective mover is identified, and the locations of other EPs are updated to get the optimum value [19].

The temperature profile of the Emperor Penguins is calculated by using Equation 20 [19]. $T_{mp}^{'} = (T_{mp} - \frac{{Itr}_{\max}}{k - {Itr}_{\max}})$ (20) $T_{mp} = {\begin{matrix} 0 if R_{nd} > 0.5 \\ 1 if R_{nd} < 0.5 \end{matrix}$ (21) where, k represents the recent iteration, Itr_max presents the maximum iteration and R_nd shows the random number between [0, 1].

The generated huddle boundary signifies the distance of EPs to the best optimal solution. The optimum solution is determined by considering the fitness value nearer to the optimal solution. The other emperor penguins updated their position depending upon the optimum solutions, as shown in Equation 22 [19]. $\vec{D_{eps}} = | Xs (\vec{A}) . \vec{Z_{b} (k)} - \vec{C} . \vec{Z_{ep} (k)} |$ (22)

Here, $\vec{D_{eps}}$ show the distance from EPs to the best solution. $\vec{A}$ and $\vec{C}$ are two vectors responsible for escaping from collision from other EPs. $\vec{Z_{b}}$ represent the optimum solutions, $\vec{Z_{ep}}$ shows the EP’s position vector. Xs () indicate the social forces of EPs.

Since EPs generally huddle together to maintain temperature. Thus, special care is to be taken to make them safe from collisions among the neighbours. For this reason, two vectors ( $\vec{A}$ ) and ( $\vec{C}$ ) are calculated in Equations 23 and 24 and the social forces are represented in Equation 26. $\vec{A} = {M \times (T_{mp}^{'} + X_{grid} (acrcy)) \times Rand ()} - T_{mp}^{'}$ (23) $\vec{C} = Rand ()$ (24) $X_{grid} (acrcy) = | {\vec{Z}}_{b} - {\vec{Z}}_{ep} |$ (25)

Here, M indicates the movement parameter and is set as 2, Rand is the random value in the range [0,1], X_grid (acrcy) shows the absolute difference between EPs and the optimal solution. $X_{s} (\vec{A}) = (\sqrt{f . e^{- \frac{k}{v}} - e^{- k}})$ (26)

Here, e represent the expression function, f and v represent the control parameters for a better exploration and exploitation lie within the range [2, 3] and [1.5, 2] respectively. The positions of the EPs are updated as per the optimal agent obtained using Equation 27. ${\vec{Z}}_{eps} (k + 1) = {\vec{Z}}_{b} (k) - \vec{A} \cdot {\vec{D}}_{eps}$ (27)

5 Oppositional based learning and β chaotic mapping

Most of the meta-heuristic techniques use a random parameter for initialization with the identical distribution. In recent times, implementation of chaotic mapping to the meta-heuristic approaches retains analogous characteristics and features to that of randomness with better dynamic and statistical behaviour.

Thus, the introduction of chaotic mapping to the novel swarm intelligence method is used as an alternative of random values. This mapping provides proper initialization, arbitrariness, and ergodicity as the additional features to the optimization approach [20]. Chaotic mapping improves the performance as compared to random values by the generation of non-repeated number with periodicity. These features, due to chaotic mapping, deliver the techniques to outflow local optima for maintaining the distinction and improvising the ability of global search.

Proper initialization refers to insignificant adaptation to the preliminary points originates a substantial result to the concluding outcomes presuming the result depends prudently to the initial point. Random variables are replaced with chaotic mapping in the process of arbitrariness. Ergodicity refers to the ability of chaotic variables to discover non-repeated variables within a certain range.

Various chaotic functions have been implemented in different meta-heuristics techniques. Amongst all proposed technique, a new technique is announced to the chaotic function family by Zahmoul et.al. in the year 2017 [21]. This mapping is mathematically represented in Equation 28. $\begin{matrix} β (x; u, v, x_{1}, x_{2}) \\ = {\begin{matrix} {(\frac{(x - x_{1})}{(x_{c} - x_{1})})}^{u} {(\frac{(x_{2} - x)}{(x_{2} - x_{c})})}^{v} if x \in (x_{1}, x_{2}) \\ 0 otherwise \end{matrix} \end{matrix}$ (28)

Here x₁, x₂, u and v ∈ R_x, x₁ > x₂ and calculated as follows $xc = \frac{({ux}_{1} + {vx}_{2})}{u + v},$ (29) $u = a_{1} + b_{1} \times c$ (30) $v = a_{2} + b_{2} \times c$ (31) $X_{t + 1} = k β (x_{t}; u, v, x_{1}, x_{2})$ (32)

Where a₁, a₂, b₁, b₂ are the constants and c represent the parameter for bifurcation.

The β-chaotic map shows sensitiveness towards initialization and variation. The Fig. 1 shows the generation of chaotic sequence for a respective iterative process and Fig. 2 represents a bifurcation diagram of the β-chaotic map.

Fig. 1

Chaotic sequence output of the β-chaotic function.

Fig. 2

Bifurcation diagram of a β-chaotic map.

Application of chaotic mapping to the conventional method of the emperor penguin maintains the proper balance between exploration and exploitation. The random function used in conventional optimization technique is replaced by β chaotic mapping. Then the Equations 21, 23 and 24 will be updated with chaotic mapping. $T_{mp} = {\begin{matrix} 0 if x_{t} > 0.5 \\ 1 if x_{t} < 0.5 \end{matrix}$ (33)

$\vec{A} = {M \times (T_{mp}^{'} + X_{grid} (acrcy)) \times x_{t}} - T_{mp}^{'}$ (34)

$\vec{C} = x_{t}$ (35)

Here, x_t shows the chaotic mapping for t_th iteration.

Opposition based learning is a novel and operative theory for enhancement of performance in optimization perspective for various meta-heuristic approach. The concept of OBL is to consider the associated opposite candidate as another solution for achieving a closer solution to the global optimum rather finding a random solution candidate. The main theory of OBL in the optimization approach is to enhance the quality of the solution by estimating the candidate solution for the corresponding pair. The best solution candidate is considered for the next individual solution [22].

If O_i is considered as candidate solution, then the corresponding opposite solution $O_{i}^{'}$ can be obtained by the Equation 33. $O_{i}^{'} = a + b - O_{i}$ (36)

Here a defined as the upper bound and b denoted as the lower bound of the search space.

Applying oppositional based learning improves the global search capability and by applying this theory, the solution is represented in Equation 37 with the application of chaotic mapping. ${\vec{Z}}_{eps}^{'} = {\vec{Z}}_{max} + {\vec{Z}}_{min} - {\vec{Z}}_{best} + x_{t} ({\vec{Z}}_{best} - {\vec{Z}}_{eps})$ (37)

Here, $Z_{eps}^{'}$ shows the opposite candidate penguin, Z_min and Z_max shows the lower bound and upper bound, respectively. Z_best shows the best penguin’s solution and Z_eps shows the position vector of the penguin. x_t is the chaotic mapping. The flow chart of the proposed technique is given in Fig. 3.

Fig. 3

Flow chart of the Enhanced EPO.

6 Result and analysis

6.1 Time complexity

The time complexity of the proposed algorithm may be computed as:

Time taken for population initialization.

Time for computing the fitness function of each particle.

Thus, the algorithm takes $O (lb + ub \times n \times \dim \times MaxIterations)$ where lb represents the lower bound, ub represents the upper bound, dim is the dimension of the problem, and MaxIterations represent the maximum number of iterations.

6.2 Validation of the optimization algorithm

The effectiveness and efficacy of the proposed technique are validated by using five different benchmark functions. After 100 trials, the enhanced version of conventional EPO showed more optimal performance. The proposed improved version provides superior accuracy, robustness with lesser computational time. The optimum solution of the proposed technique exhibits better efficiency for the search process and a perfect balance between exploration and exploitation towards handling of benchmark functions. The comparison table with the conventional EPO is depicted in Table 1. Table 1 shows the optimum value of the proposed method as compared to the original EPO. The convergence graph is presented in Fig. 4 –8 for different benchmark function for comparison purpose. The convergence graphs show the optimization technique provides more optimum value as compared to conventional EPO [19] technique.

Table 1
Comparison table for validation by testing with benchmark functions

Function Name Equation Range Optimization technique Best Worst Mean Std. Dev. Time (S)

Ackely $\begin{matrix} F_{1} = - 20 exp (- 0.2 \sqrt{n^{- 1} \sum_{i = 1}^{n} x_{i}^{2}} \\ - exp n^{- 1} \sum_{i = 1}^{n} cos 2 π x) \\ + 20 + e \end{matrix}$ [–32, 32]ⁿ EPO 1.0447×10^-12 1.6233 3.1834 2.5939 47.81

CEPO 3.2163×10^-12 0.4079 1.5902 2.9240 46.11

OLEPO 1.7343×10^-12 0.9693 0.9861 2.5884 43.29

COLEPO 3.6608×10^-13 0.3972 1.6479 1.8019 37.00

Levi $\begin{matrix} F_{2} = \sum_{i = 1}^{n} {(xi - 1)}^{2} \\ [1 + 10 {sin}^{2} (3 π x_{i} + 1)] \\ + {sin}^{2} (3 π x_{i}) + \\ | x_{n} - 1 | [1 + {sin}^{2} (3 π x_{n})] \end{matrix}$ [–10, 10]ⁿ EPO 0.6910×10^-10 0.7430 0.3935 2.1858 52.36

CEPO 1.0024×10^-10 0.9694 0.5385 1.1841 44.89

OLEPO 0.5531×10^-10 1.6858 3.8436 2.9758 40.01

COLEPO 0.5171×10^-16 0.2006 3.9010 0.7550 33.47

Rastrigin $F_{3} = \sum_{i = 1}^{n} [x_{i}^{2} - 10 cos (2 π x_{i}) + 10]$ [–5.1, 5.1]ⁿ EPO 3.4772×10^-13 0.6170 2.3468 1.7454 49.50

CEPO 2.2029×10^-13 3.8169 0.2439 0.7333 42.97

OLEPO 2.0894×10^-13 1.9831 0.9579 1.4724 41.94

COLEPO 0.5508×10^-15 1.9766 1.4409 2.5000 35.39

Scfwel $F_{4} = \sum_{i = 1}^{n} (\sum_{j = 1}^{i} x_{j})$ [–100, 100]ⁿ EPO 3.2415×10^-10 1.3644 3.3505 3.1178 41.37

CEPO 2.3638×10^-10 3.6362 0.0628 0.3242 39.19

OLEPO 1.3336×10^-10 1.4918 0.1755 3.7138 36.34

COLEPO 1.9504×10^-13 0.4492 0.6895 3.0998 33.11

Sphere $F_{5} = \sum_{i = 1}^{n} (x_{i}^{2})$ [–100, 100]ⁿ EPO 0.6918×10^-10 0.5620 2.1167 1.6073 47.94

CEPO 1.0035×10^-12 3.4768 0.2200 0.6753 41.61

OLEPO 0.5537×10^-15 1.8064 0.8640 1.3560 40.62

COLEPO 0.5176×10^-17 1.8004 1.2996 2.3023 34.27

Function Name	Equation	Range	Optimization technique	Best	Worst	Mean	Std. Dev.	Time (S)
Ackely	$\begin{matrix} F_{1} = - 20 exp (- 0.2 \sqrt{n^{- 1} \sum_{i = 1}^{n} x_{i}^{2}} \\ - exp n^{- 1} \sum_{i = 1}^{n} cos 2 π x) \\ + 20 + e \end{matrix}$	[–32, 32]ⁿ	EPO	1.0447×10^-12	1.6233	3.1834	2.5939	47.81
			CEPO	3.2163×10^-12	0.4079	1.5902	2.9240	46.11
			OLEPO	1.7343×10^-12	0.9693	0.9861	2.5884	43.29
			COLEPO	3.6608×10^-13	0.3972	1.6479	1.8019	37.00
Levi	$\begin{matrix} F_{2} = \sum_{i = 1}^{n} {(xi - 1)}^{2} \\ [1 + 10 {sin}^{2} (3 π x_{i} + 1)] \\ + {sin}^{2} (3 π x_{i}) + \\ \| x_{n} - 1 \| [1 + {sin}^{2} (3 π x_{n})] \end{matrix}$	[–10, 10]ⁿ	EPO	0.6910×10^-10	0.7430	0.3935	2.1858	52.36
			CEPO	1.0024×10^-10	0.9694	0.5385	1.1841	44.89
			OLEPO	0.5531×10^-10	1.6858	3.8436	2.9758	40.01
			COLEPO	0.5171×10^-16	0.2006	3.9010	0.7550	33.47
Rastrigin	$F_{3} = \sum_{i = 1}^{n} [x_{i}^{2} - 10 cos (2 π x_{i}) + 10]$	[–5.1, 5.1]ⁿ	EPO	3.4772×10^-13	0.6170	2.3468	1.7454	49.50
			CEPO	2.2029×10^-13	3.8169	0.2439	0.7333	42.97
			OLEPO	2.0894×10^-13	1.9831	0.9579	1.4724	41.94
			COLEPO	0.5508×10^-15	1.9766	1.4409	2.5000	35.39
Scfwel	$F_{4} = \sum_{i = 1}^{n} (\sum_{j = 1}^{i} x_{j})$	[–100, 100]ⁿ	EPO	3.2415×10^-10	1.3644	3.3505	3.1178	41.37
			CEPO	2.3638×10^-10	3.6362	0.0628	0.3242	39.19
			OLEPO	1.3336×10^-10	1.4918	0.1755	3.7138	36.34
			COLEPO	1.9504×10^-13	0.4492	0.6895	3.0998	33.11
Sphere	$F_{5} = \sum_{i = 1}^{n} (x_{i}^{2})$	[–100, 100]ⁿ	EPO	0.6918×10^-10	0.5620	2.1167	1.6073	47.94
			CEPO	1.0035×10^-12	3.4768	0.2200	0.6753	41.61
			OLEPO	0.5537×10^-15	1.8064	0.8640	1.3560	40.62
			COLEPO	0.5176×10^-17	1.8004	1.2996	2.3023	34.27

Fig. 4

Convergence graph of the Ackley function.

Fig. 5

Convergence curve of the Levi function.

Fig. 6

Convergence curve of the Rastrigin function.

Fig. 7

Convergence curve of the Scfwel function.

Fig. 8

Convergence curve of the Sphere function.

6.3 Application of the optimization algorithm to the dynamic economic dispatch

The proposed technique with an enhancement to the conventional EPO is applied for five different test system for thermal generators, hybrid generators and distributed generators. The application of the proposed technique was also verified for the dynamic dispatch of microgrid considering three types of distributed generators. All the test system was optimized by considering the practical and operational constraints with non-linearities and non-convexity. The optimum generation schedule for economical cost of generation is intended for hourly load variation for a day. The comparison of cost with the conventional technique and other referred techniques is depicted for every case. The application of proposed enhanced technique is conducted through MATLAB R2018a software.

Five generating units with transmission losses and without wind energy

Ten generating units with loss and without wind energy

Ten generating units with loss and predicted wind energy

Ten generating units with the consideration of uncertainty wind power

Dynamic economic dispatch problem for a microgrid with generators

6.3.1 Test system-1: Five generating units without wind energy

The hourly demand of a specific day is to be satisfied by the five thermal generating units. Integrating the effect of the steam valve to the thermal generating units creates non-convexity in the cost characteristics. Effect of ramp rate constraints, transmission losses and spinning reserve constraints converts the linear curve to the nonlinear curve. The system data for input coefficients and transmission losses are referred from [9]. The generated power from the committed generating units and transmission losses needs to be satisfied for hourly load demand to find the optimum cost is depicted in Table 3. The optimum cost obtained with the application of the proposed technique is calculated as 41,565.32 $. Table 2 presents the cost of comparison with other referred techniques with variation of cost during the iteration process. Table 2 explores the effectiveness of the proposed technique to the application of the dynamic economic dispatch problem compared to recently applied techniques. The graphical comparison of the proposed technique for the optimum cost is depicted in Fig. 9. Figure 10 represents the convergence graph of the enhanced version of the EPO technique. The convergence graph represents the optimum cost with the comparison with original EPO technique.

Table 2
Comparison of cost with other applied techniques with the proposed method for case 1

Techniques Best cost($) Mean cost($) Maximum cost($)

CMA-ES [23] 49,703.14 – –

ABC [24] 44,045.83 44,064.73 44,218.64

BBOSB [25] 43,017.95 43,066.40 43,197.01

BBPSO [25] 43,233 43,732 44,252

BBPSO-DCS [25] 43,725 – –

DE [9] 43,213 43,813 44,247

DE/BBO [15] 43,047.46 43,292.72 43,683.79

E-DE [26] 42,571.2 – –

E-GA [26] 42,580.6 – –

GA–DE [27] 42,547.6 – –

HIGA [28] 43,125.37 43,162.24 43,259.35

LDISS [29] 43,213 – –

MBF-SSO [30] 43,048 43,068 43,093

ODO-ABC [31] 42,528 42,528 42,528

ODO-PSO [31] 42,528 42,572 42,537

SLPSO [32] 43,125.09 43,681.86 44,237.20

SOS [11] 43,090.59 43,103.08 43,162.21

TVAC-IPSO [33] 43,136.56 43,185.66 43,302.23

EPO 41,970.36 42,021.44 42,104.98

CEPO 41,895.24 41,997.34 42,012.37

OLEPO 41,605.74 41,696.37 41,758.81

COLEPO 41,565.32 41,595.74 41,617.25

Techniques	Best cost($)	Mean cost($)	Maximum cost($)
CMA-ES [23]	49,703.14	–	–
ABC [24]	44,045.83	44,064.73	44,218.64
BBOSB [25]	43,017.95	43,066.40	43,197.01
BBPSO [25]	43,233	43,732	44,252
BBPSO-DCS [25]	43,725	–	–
DE [9]	43,213	43,813	44,247
DE/BBO [15]	43,047.46	43,292.72	43,683.79
E-DE [26]	42,571.2	–	–
E-GA [26]	42,580.6	–	–
GA–DE [27]	42,547.6	–	–
HIGA [28]	43,125.37	43,162.24	43,259.35
LDISS [29]	43,213	–	–
MBF-SSO [30]	43,048	43,068	43,093
ODO-ABC [31]	42,528	42,528	42,528
ODO-PSO [31]	42,528	42,572	42,537
SLPSO [32]	43,125.09	43,681.86	44,237.20
SOS [11]	43,090.59	43,103.08	43,162.21
TVAC-IPSO [33]	43,136.56	43,185.66	43,302.23
EPO	41,970.36	42,021.44	42,104.98
CEPO	41,895.24	41,997.34	42,012.37
OLEPO	41,605.74	41,696.37	41,758.81
COLEPO	41,565.32	41,595.74	41,617.25

Table 3

Scheduled generation from five generating units considering transmission losses for case 1

Hours	Load Demand (MW)	P₁ (MW)	P₂ (MW)	P₃ (MW)	P₄ (MW)	P₅ (MW)	Total Output (MW)	Loss (MW)
1	410	20.522	98.651	29.996	125.040	139.620	413.828	3.828
2	435	10.018	98.207	66.286	125.041	139.621	439.173	4.173
3	475	10.138	98.648	106.285	125.035	139.746	479.851	4.851
4	530	10.033	98.650	112.570	175.047	139.651	535.950	5.950
5	558	10.014	92.593	112.689	209.861	139.652	564.809	6.809
6	608	10.014	98.605	112.689	209.861	185.008	616.177	8.177
7	626	10.014	72.528	112.690	209.861	229.493	634.586	8.586
8	654	12.710	98.648	112.657	209.861	229.493	663.369	9.369
9	690	42.736	105.595	112.657	209.862	229.493	700.342	10.342
10	704	64.056	98.650	112.656	209.862	229.493	714.717	10.717
11	720	75.079	104.148	112.656	209.862	229.493	731.239	11.239
12	740	75.079	124.834	112.656	209.862	229.493	751.925	11.925
13	704	64.067	98.651	112.656	209.862	228.493	713.729	9.729
14	690	43.240	105.128	112.656	209.862	228.493	699.378	9.378
15	654	13.216	98.606	112.656	209.862	228.493	662.833	8.833
16	580	10.013	75.771	112.656	159.371	229.493	587.304	7.304
17	558	10.013	87.676	112.656	125.035	229.493	564.872	6.872
18	608	10.013	98.605	112.656	165.248	229.493	616.014	8.014
19	654	12.711	98.605	112.656	209.861	229.494	663.327	9.327
20	704	42.736	120.144	112.656	209.861	229.494	714.891	10.891
21	680	39.332	98.605	112.656	209.861	229.494	689.948	9.948
22	605	10.013	98.605	112.656	162.143	229.494	612.911	7.911
23	527	10.013	98.605	112.656	124.933	186.747	532.954	5.954
24	463	10.013	80.240	112.656	124.933	139.751	467.594	4.594
						Total Cost	41,565.32 ($)

Fig. 9

Convergence characteristics for five generators considering transmission losses.

Fig. 10

Graphical comparison of proposed techniques with other techniques for five generating units.

6.3.2 Test system-2: Ten generating units with loss and without wind energy

This test system comprises of ten fossil fuel-based generators to meet hourly basis load demand considering the practical constraints like of transmission losses, valve point effect and ramp rate limit. The transmission losses, generator’s cost coefficients and hourly basis load profile for a day are referred from [44]. The optimum cost obtained with 200 iterations is 1,036,074.21 $, as shown in Table 4 with the scheduled generations among the ten committed generating units for 24 hours. From Table 5, it has been observed that the proposed techniques provides an optimum cost of generation as compared to ODO-ABC [31], MTLA [41], MIQP [39], MACO [38], LETLBO [26], LDISS [29], EBSO [36], CSAPSO [34], CDBCO [33], TSMILP [43] and SOS [11] techniques. The convergence characteristic is shown in Fig. 12 with conventional EPO for the optimum cost. The graphical representation for the optimal with comparison to other techniques is represented in Fig. 11.

Table 4
Scheduled generation from ten generating units considering transmission losses without wind energy

Hour Load Demand (MW) P₁ (MW) P₂ (MW) P₃ (MW) P₄ (MW) P₅ (MW) P₆ (MW) P₇ (MW) P₈ P₉ (MW) P₁₀ (MW) Generated Power (MW) P_L

1 1036 150.108 222.068 172.491 60.022 73.012 121.387 129.613 47.001 20.010 55.000 1050.711 14.711

2 1110 150.102 222.517 188.278 60.624 122.847 129.058 129.658 48.412 20.019 55.000 1126.515 16.515

3 1258 226.743 222.404 186.413 110.524 122.896 123.486 129.983 78.417 20.021 55.000 1275.888 17.888

4 1406 303.351 222.404 257.442 60.664 122.896 141.229 129.600 85.421 50.031 55.000 1428.038 22.038

5 1480 303.448 222.297 310.523 73.895 172.740 125.618 129.815 89.654 20.021 55.000 1503.011 23.011

6 1628 379.954 229.390 303.907 120.310 173.014 122.431 129.880 119.796 20.014 55.000 1653.695 25.695

7 1702 303.307 309.378 303.396 134.868 222.633 135.569 129.647 119.995 20.052 55.000 1733.845 31.845

8 1776 379.838 311.620 298.461 181.000 222.894 122.846 129.640 90.132 20.061 55.000 1811.492 35.492

9 1924 380.081 391.550 339.930 191.527 222.591 122.846 129.640 85.296 50.087 55.000 1968.548 44.548

10 2072 458.220 403.154 340.035 241.550 223.454 159.968 129.657 90.796 20.396 55.000 2122.230 50.230

11 2146 456.385 396.654 329.900 291.561 238.364 159.968 129.657 120.002 20.001 55.000 2197.493 51.493

12 2220 456.623 459.949 325.644 299.191 223.293 159.534 129.657 120.002 50.001 55.000 2278.895 58.895

13 2072 456.579 396.707 297.355 249.195 222.600 150.100 129.629 115.247 50.052 55.000 2122.465 50.465

14 1924 456.372 396.014 284.424 199.205 222.600 122.431 129.629 85.421 20.021 55.000 1971.116 47.116

15 1776 379.694 396.477 293.664 169.626 172.627 122.431 129.528 76.870 20.012 55.000 1815.928 39.928

16 1554 303.247 316.529 248.912 120.291 222.630 122.431 129.629 47.032 20.012 55.000 1585.713 31.713

17 1480 302.945 309.378 184.019 118.913 221.030 122.488 129.613 47.032 20.012 55.000 1510.430 30.430

18 1628 303.289 389.298 262.659 118.686 172.386 158.335 129.613 55.231 20.012 55.000 1664.509 36.509

19 1776 379.954 396.754 318.653 124.656 172.553 132.857 129.987 85.165 20.001 55.000 1815.580 39.580

20 2072 456.536 459.845 332.244 174.138 222.685 159.920 129.613 115.247 20.291 55.000 2125.520 53.520

21 1924 456.572 396.766 300.042 180.832 222.872 123.470 129.624 85.321 20.001 55.000 1970.500 46.500

22 1628 380.044 316.823 298.433 131.925 173.188 127.209 101.985 55.302 20.054 55.000 1659.963 31.963

23 1332 303.311 236.851 219.125 83.681 172.657 122.584 93.239 47.014 20.147 55.000 1353.609 21.609

24 1184 226.500 221.569 183.673 60.014 172.653 122.584 93.035 47.014 20.001 55.000 1202.043 18.043

Total Cost 1,036,074.21 $

Hour	Load Demand (MW)	P₁ (MW)	P₂ (MW)	P₃ (MW)	P₄ (MW)	P₅ (MW)	P₆ (MW)	P₇ (MW)	P₈	P₉ (MW)	P₁₀ (MW)	Generated Power (MW)	P_L
1	1036	150.108	222.068	172.491	60.022	73.012	121.387	129.613	47.001	20.010	55.000	1050.711	14.711
2	1110	150.102	222.517	188.278	60.624	122.847	129.058	129.658	48.412	20.019	55.000	1126.515	16.515
3	1258	226.743	222.404	186.413	110.524	122.896	123.486	129.983	78.417	20.021	55.000	1275.888	17.888
4	1406	303.351	222.404	257.442	60.664	122.896	141.229	129.600	85.421	50.031	55.000	1428.038	22.038
5	1480	303.448	222.297	310.523	73.895	172.740	125.618	129.815	89.654	20.021	55.000	1503.011	23.011
6	1628	379.954	229.390	303.907	120.310	173.014	122.431	129.880	119.796	20.014	55.000	1653.695	25.695
7	1702	303.307	309.378	303.396	134.868	222.633	135.569	129.647	119.995	20.052	55.000	1733.845	31.845
8	1776	379.838	311.620	298.461	181.000	222.894	122.846	129.640	90.132	20.061	55.000	1811.492	35.492
9	1924	380.081	391.550	339.930	191.527	222.591	122.846	129.640	85.296	50.087	55.000	1968.548	44.548
10	2072	458.220	403.154	340.035	241.550	223.454	159.968	129.657	90.796	20.396	55.000	2122.230	50.230
11	2146	456.385	396.654	329.900	291.561	238.364	159.968	129.657	120.002	20.001	55.000	2197.493	51.493
12	2220	456.623	459.949	325.644	299.191	223.293	159.534	129.657	120.002	50.001	55.000	2278.895	58.895
13	2072	456.579	396.707	297.355	249.195	222.600	150.100	129.629	115.247	50.052	55.000	2122.465	50.465
14	1924	456.372	396.014	284.424	199.205	222.600	122.431	129.629	85.421	20.021	55.000	1971.116	47.116
15	1776	379.694	396.477	293.664	169.626	172.627	122.431	129.528	76.870	20.012	55.000	1815.928	39.928
16	1554	303.247	316.529	248.912	120.291	222.630	122.431	129.629	47.032	20.012	55.000	1585.713	31.713
17	1480	302.945	309.378	184.019	118.913	221.030	122.488	129.613	47.032	20.012	55.000	1510.430	30.430
18	1628	303.289	389.298	262.659	118.686	172.386	158.335	129.613	55.231	20.012	55.000	1664.509	36.509
19	1776	379.954	396.754	318.653	124.656	172.553	132.857	129.987	85.165	20.001	55.000	1815.580	39.580
20	2072	456.536	459.845	332.244	174.138	222.685	159.920	129.613	115.247	20.291	55.000	2125.520	53.520
21	1924	456.572	396.766	300.042	180.832	222.872	123.470	129.624	85.321	20.001	55.000	1970.500	46.500
22	1628	380.044	316.823	298.433	131.925	173.188	127.209	101.985	55.302	20.054	55.000	1659.963	31.963
23	1332	303.311	236.851	219.125	83.681	172.657	122.584	93.239	47.014	20.147	55.000	1353.609	21.609
24	1184	226.500	221.569	183.673	60.014	172.653	122.584	93.035	47.014	20.001	55.000	1202.043	18.043
											Total Cost	1,036,074.21 $

Table 5

Comparison of cost with other applied techniques for test system 2

Techniques	Best Cost ($)	Mean Cost($)	Maximum Cost($)
ABC [24]	1,043,381	1,044,963	1,046,805
BBOSB [25]	1,038,362.01	1,039,968.8	1,041,538.96
CDBCO [33]	1,042,900	1,044,700	–
CSAPSO [34]	1,038,251	1,039,543	–
CSO [15]	1,038,320	1,039,374	1,042,518
DE/BBO [15]	1,039,983.199	1,041,074.566	1,042,431.806
EAPSO [35]	1,037,685	1,038,109	1,038,238
EBSO [36]	1,038,915	1,039,188	1,039,272
EFA [26]	1,037,673	1,038,037	1,038,090
E-GA [26]	1,037,020	1,036,430	1,036,460
GA-DE [27]	1,036,240	1,036,280	1,036,360
IPSO [37]	1,046,275	1,048,145	–
LDISS [29]	1,039,083	1,041,091	1,042,630
LETLBO [26]	1,049,007.6	1,051,724.855	1,056,496.537
LWOA [26]	1,057,177.92	1,060,359.4	1,065,293.43
MACO [38]	1,042,274.44	1,042,790.5	1,044,117.15
MIQP [39]	1,038,376	–	–
MPSO-JAC [40]	1,038,211	1,038,995	1,040,618
MTLA [41]	1,037,489	1,037,712	1,038,090
ODO-ABC [31]	1,036,289	1,036,876	1,037,310
SAMFA [42]	1,037,698	1,037,938	1,039,199
SOS [11]	1,042,868.789	1,042,900.383	1,042,945.28
TSMILP [43]	1,037,487	1,037,487	1,037,487
EPO	1,037,657.57	1,037,714.22	1,037,784.34
CEPO	1,036,898.33	1,036,914.21	1,036,987.32
OLEPO	1,036,421.28	1,036,445.22	1,036,461.87
COLEPO	1,036,074.21	1,036,085.33	1,036,095.37

Fig. 11

Graphical comparison of proposed techniques with other techniques for test system 2.

Fig. 12

Convergence characteristics of the proposed technique for test system 2.

6.3.3 Test system-3: Ten generating units with predicted wind energy

This test system comprises of ten thermal generating units with connection to a wind firm. The wind firm consists of 50 wind turbines for an installed capacity of 100 MW. The input parameter for thermal power plants and the wind turbines are referred from [45]. The output of wind firm is considered as the predicted value as referred to in [17]. The optimal cost for the DED problem is found as 2,35,2357.38 $, which is optimum as compared to IMOEA [45], NSGA-II [45], MAPSO [17], EPO and many more. As the novel approach finds a better global search process, the efficiency for finding the optimum cost is more. Table 6 presents the scheduled output power from the thermal plants and wind firms considering the losses to fulfill the load demand. Figure 13 graphically represents the power production at each hour for a day from each generating units, including wind firm and transmission losses. The comparison and efficacy of the proposed technique is depicted in Table 7, with the other recently applied technique. The convergence graph for the optimal cost is presented in Fig. 14.

Table 6
Generated output from ten generating units with transmission losses and predicted wind energy

Hour Load Demand (MW) P₁ (MW) P₂ (MW) P₃ (MW) P₄ (MW) P₅ (MW) P₆ (MW) P₇ (MW) P₈ (MW) P₉ (MW) P₁₀ (MW) Predicted Wind (MW) Generated Power (MW) P_L (MW)

1 1036 150.015 135.014 73.000 60.000 159.294 159.997 111.168 119.988 20.000 10.111 55 1053.586 17.586

2 1110 150.015 135.014 115.948 60.000 209.289 159.997 129.997 89.991 20.000 10.001 50 1130.252 20.252

3 1258 151.005 135.014 176.502 109.989 171.353 147.167 129.997 119.988 50.005 27.423 65 1283.443 25.443

4 1406 150.015 135.014 256.494 159.984 221.348 159.997 128.687 109.919 20.272 49.285 48 1439.015 33.015

5 1480 150.015 135.014 277.142 209.979 242.976 159.997 129.997 111.259 28.333 34.643 38 1517.355 37.355

6 1628 151.355 138.974 339.966 259.974 242.976 159.997 106.438 119.988 58.336 46.985 48 1672.988 44.988

7 1702 150.015 137.534 339.966 287.541 241.836 159.997 129.997 119.988 80.008 49.255 55 1751.137 49.137

8 1776 170.327 201.570 339.966 299.970 233.797 159.997 129.997 119.988 80.008 46.995 48 1830.615 54.615

9 1924 250.185 281.578 339.966 299.970 242.976 159.997 129.997 119.988 80.008 55.000 32 1991.665 67.665

10 2072 299.530 361.586 339.966 299.970 242.976 159.997 129.997 119.988 80.008 45.965 20 2099.983 27.983

11 2146 301.960 440.494 339.966 299.970 242.976 159.997 129.997 119.988 60.516 55.000 40 2190.864 44.864

12 2220 374.967 387.929 339.966 299.970 242.976 159.997 129.997 119.988 80.008 51.285 50 2237.083 17.083

13 2072 313.631 345.925 339.966 298.750 242.976 159.997 129.997 118.998 80.008 55.000 65 2150.248 78.248

14 1924 237.794 265.917 339.966 297.520 242.976 159.997 129.997 119.988 79.988 41.944 72 1988.087 64.087

15 1776 157.786 185.909 339.966 295.030 225.187 159.997 125.297 119.988 80.008 49.055 90 1828.224 52.224

16 1554 150.015 135.014 259.974 245.575 224.138 145.597 129.997 102.960 55.586 43.544 100 1592.400 38.400

17 1480 150.225 135.014 201.560 195.580 242.976 159.997 129.987 119.988 51.715 43.234 85 1515.276 35.276

18 1628 150.015 156.056 281.552 232.227 242.976 159.997 125.767 119.988 80.008 55.000 68 1671.585 43.585

19 1776 150.015 236.064 339.966 282.222 242.976 159.997 129.997 118.968 63.646 46.415 60 1830.265 54.265

20 1972 230.023 316.072 339.966 299.970 242.976 159.997 129.997 119.988 80.008 52.065 70 2041.062 69.062

21 1924 211.161 287.799 339.966 299.970 235.286 159.997 124.398 119.988 80.008 55.000 75 1988.573 64.573

22 1628 150.015 207.791 259.974 249.975 221.778 155.227 129.997 119.988 50.455 35.824 90 1671.024 43.024

23 1332 150.015 135.014 181.322 199.980 179.122 136.867 128.087 116.258 20.452 33.113 80 1360.231 28.231

24 1184 150.015 135.014 141.066 158.774 134.207 130.997 129.997 87.411 50.455 12.971 75 1205.907 21.907

Total Cost 2,35,2357.38 $

Hour	Load Demand (MW)	P₁ (MW)	P₂ (MW)	P₃ (MW)	P₄ (MW)	P₅ (MW)	P₆ (MW)	P₇ (MW)	P₈ (MW)	P₉ (MW)	P₁₀ (MW)	Predicted Wind (MW)	Generated Power (MW)	P_L (MW)
1	1036	150.015	135.014	73.000	60.000	159.294	159.997	111.168	119.988	20.000	10.111	55	1053.586	17.586
2	1110	150.015	135.014	115.948	60.000	209.289	159.997	129.997	89.991	20.000	10.001	50	1130.252	20.252
3	1258	151.005	135.014	176.502	109.989	171.353	147.167	129.997	119.988	50.005	27.423	65	1283.443	25.443
4	1406	150.015	135.014	256.494	159.984	221.348	159.997	128.687	109.919	20.272	49.285	48	1439.015	33.015
5	1480	150.015	135.014	277.142	209.979	242.976	159.997	129.997	111.259	28.333	34.643	38	1517.355	37.355
6	1628	151.355	138.974	339.966	259.974	242.976	159.997	106.438	119.988	58.336	46.985	48	1672.988	44.988
7	1702	150.015	137.534	339.966	287.541	241.836	159.997	129.997	119.988	80.008	49.255	55	1751.137	49.137
8	1776	170.327	201.570	339.966	299.970	233.797	159.997	129.997	119.988	80.008	46.995	48	1830.615	54.615
9	1924	250.185	281.578	339.966	299.970	242.976	159.997	129.997	119.988	80.008	55.000	32	1991.665	67.665
10	2072	299.530	361.586	339.966	299.970	242.976	159.997	129.997	119.988	80.008	45.965	20	2099.983	27.983
11	2146	301.960	440.494	339.966	299.970	242.976	159.997	129.997	119.988	60.516	55.000	40	2190.864	44.864
12	2220	374.967	387.929	339.966	299.970	242.976	159.997	129.997	119.988	80.008	51.285	50	2237.083	17.083
13	2072	313.631	345.925	339.966	298.750	242.976	159.997	129.997	118.998	80.008	55.000	65	2150.248	78.248
14	1924	237.794	265.917	339.966	297.520	242.976	159.997	129.997	119.988	79.988	41.944	72	1988.087	64.087
15	1776	157.786	185.909	339.966	295.030	225.187	159.997	125.297	119.988	80.008	49.055	90	1828.224	52.224
16	1554	150.015	135.014	259.974	245.575	224.138	145.597	129.997	102.960	55.586	43.544	100	1592.400	38.400
17	1480	150.225	135.014	201.560	195.580	242.976	159.997	129.987	119.988	51.715	43.234	85	1515.276	35.276
18	1628	150.015	156.056	281.552	232.227	242.976	159.997	125.767	119.988	80.008	55.000	68	1671.585	43.585
19	1776	150.015	236.064	339.966	282.222	242.976	159.997	129.997	118.968	63.646	46.415	60	1830.265	54.265
20	1972	230.023	316.072	339.966	299.970	242.976	159.997	129.997	119.988	80.008	52.065	70	2041.062	69.062
21	1924	211.161	287.799	339.966	299.970	235.286	159.997	124.398	119.988	80.008	55.000	75	1988.573	64.573
22	1628	150.015	207.791	259.974	249.975	221.778	155.227	129.997	119.988	50.455	35.824	90	1671.024	43.024
23	1332	150.015	135.014	181.322	199.980	179.122	136.867	128.087	116.258	20.452	33.113	80	1360.231	28.231
24	1184	150.015	135.014	141.066	158.774	134.207	130.997	129.997	87.411	50.455	12.971	75	1205.907	21.907
												Total Cost	2,35,2357.38 $

Fig. 13

Output power from each generating units with predicted wind energy and losses.

Table 7

Comparison of cost with other applied techniques for case 3 considering predicted wind energy

Techniques	Best Cost ($)	Mean Cost($)	Maximum Cost($)
IMOEA [45]	2359800	–	–
NSGA-II [45]	2377700	–	–
MAPSO [17]	2355671	–	–
EPO	2362476.58	2362657.67	2362814.53
CEPO	2360549.74	2360780.45	2360842.37
OLEPO	2357584.35	2357641.24	2357712.28
COLEPO	2352357.38	2352389.74	2352417.56

Fig. 14

Convergence characteristics of the proposed technique for test system 3.

6.3.4 Test system-4: Ten generating units with the consideration of uncertainty wind power

The randomness of wind power is considered in this test system instead of predicted wind power in test system 3. The cost function of wind power is calculated using the overestimation and underestimation of penalty cost as represented in Equation 3. All the input and system data for all generating units is considered from [17]. The power generated from each generating units considering the uncertainty wind power and transmission losses for 24 hours in a day is provided in Table 8. The considered wind firm is same as that of in test system 3. The cost of generation for 24 hours is found optimum as shown in Table 10. The output power form each thermal generating units and the wind firm with consideration of transmission losses is presented graphically in Fig. 15 and the convergence graph is presented in Fig. 16.

Table 8
Generated output from ten generating units with transmission losses and uncertainty wind energy

Hour Load Demand (MW) P₁ (MW) P₂ (MW) P₃ (MW) P₄ (MW) P₅ (MW) P₆ (MW) P₇ (MW) P₈ (MW) P₉ (MW) P₁₀ (MW) Wind Energy (MW) Generated Power (MW) P_L

1 1036 150.015 135.001 73.000 60.010 189.398 57.000 129.974 79.099 71.293 55.000 54.588 1054.379 18.379

2 1110 160.616 135.001 135.499 60.010 159.898 106.998 112.278 109.099 41.296 55.000 54.79452 1130.489 20.489

3 1258 150.015 135.001 215.498 106.389 182.898 122.498 82.284 119.999 53.395 51.099 65.29347 1284.369 26.369

4 1406 150.015 135.001 240.998 156.384 223.898 142.997 112.278 115.599 64.594 38.801 58.29417 1438.858 32.858

5 1480 150.015 135.001 320.997 204.280 238.598 127.497 95.281 112.999 79.992 10.000 43.29567 1517.955 37.955

6 1628 221.022 135.001 333.897 243.076 188.598 159.997 125.275 119.999 79.992 10.000 56.39436 1673.251 45.251

7 1702 150.015 177.902 316.197 293.071 238.598 159.997 127.974 109.299 76.092 40.000 61.19388 1750.338 48.338

8 1776 150.015 257.903 339.997 299.970 231.998 159.997 129.974 101.499 59.394 51.802 48.9951 1831.543 55.543

9 1924 230.023 334.903 339.997 299.970 242.998 159.997 129.974 101.899 79.992 35.299 36.9963 1992.048 68.048

10 2072 281.128 370.304 339.997 295.770 242.998 159.997 126.075 119.999 71.393 55.000 35.89641 2098.556 26.556

11 2146 361.136 372.204 336.997 299.970 239.698 155.497 122.176 117.299 67.793 55.000 60.79392 2188.562 42.562

12 2220 319.632 401.904 339.997 299.970 240.598 159.997 121.376 119.999 74.593 55.000 100 2233.064 13.064

13 2072 300.030 354.804 339.997 299.970 242.098 159.997 129.974 119.499 79.992 24.997 36.598 2087.954 15.954

14 1924 241.024 274.803 339.997 299.970 232.598 159.997 108.878 107.699 79.992 55.000 87.651 1987.608 63.608

15 1776 161.016 208.402 339.997 299.970 242.998 129.297 129.974 119.999 49.995 55.000 91.228 1827.876 51.876

16 1554 150.015 135.001 310.397 249.975 229.698 147.397 120.676 100.799 19.998 28.641 100 1592.597 38.597

17 1480 150.015 135.001 339.997 199.980 179.698 138.697 129.974 99.199 19.998 34.287 88.199 1515.045 35.045

18 1628 150.015 183.802 330.097 249.975 229.698 159.997 99.980 101.599 42.196 55.000 70.007 1672.365 44.365

19 1776 184.918 225.502 339.997 299.970 206.098 159.997 129.974 119.999 52.095 50.321 61.321 1830.192 54.192

20 1972 227.923 301.003 339.997 293.971 242.998 159.997 129.974 119.999 79.992 41.998 100 2037.850 65.850

21 1924 213.021 307.203 339.997 299.970 220.898 129.597 129.974 119.999 76.292 49.654 100 1986.605 62.605

22 1628 150.015 227.202 275.597 256.274 215.998 126.397 99.980 119.999 79.992 19.612 100 1671.067 43.067

23 1332 150.015 147.201 195.598 247.775 165.998 76.398 86.383 119.999 77.792 10.000 82.954 1360.114 28.114

24 1184 150.015 140.901 115.599 197.780 174.998 82.298 86.383 106.499 65.793 10.020 75.998 1206.285 22.285

Total Cost 23,64,127.45 $

Hour	Load Demand (MW)	P₁ (MW)	P₂ (MW)	P₃ (MW)	P₄ (MW)	P₅ (MW)	P₆ (MW)	P₇ (MW)	P₈ (MW)	P₉ (MW)	P₁₀ (MW)	Wind Energy (MW)	Generated Power (MW)	P_L
1	1036	150.015	135.001	73.000	60.010	189.398	57.000	129.974	79.099	71.293	55.000	54.588	1054.379	18.379
2	1110	160.616	135.001	135.499	60.010	159.898	106.998	112.278	109.099	41.296	55.000	54.79452	1130.489	20.489
3	1258	150.015	135.001	215.498	106.389	182.898	122.498	82.284	119.999	53.395	51.099	65.29347	1284.369	26.369
4	1406	150.015	135.001	240.998	156.384	223.898	142.997	112.278	115.599	64.594	38.801	58.29417	1438.858	32.858
5	1480	150.015	135.001	320.997	204.280	238.598	127.497	95.281	112.999	79.992	10.000	43.29567	1517.955	37.955
6	1628	221.022	135.001	333.897	243.076	188.598	159.997	125.275	119.999	79.992	10.000	56.39436	1673.251	45.251
7	1702	150.015	177.902	316.197	293.071	238.598	159.997	127.974	109.299	76.092	40.000	61.19388	1750.338	48.338
8	1776	150.015	257.903	339.997	299.970	231.998	159.997	129.974	101.499	59.394	51.802	48.9951	1831.543	55.543
9	1924	230.023	334.903	339.997	299.970	242.998	159.997	129.974	101.899	79.992	35.299	36.9963	1992.048	68.048
10	2072	281.128	370.304	339.997	295.770	242.998	159.997	126.075	119.999	71.393	55.000	35.89641	2098.556	26.556
11	2146	361.136	372.204	336.997	299.970	239.698	155.497	122.176	117.299	67.793	55.000	60.79392	2188.562	42.562
12	2220	319.632	401.904	339.997	299.970	240.598	159.997	121.376	119.999	74.593	55.000	100	2233.064	13.064
13	2072	300.030	354.804	339.997	299.970	242.098	159.997	129.974	119.499	79.992	24.997	36.598	2087.954	15.954
14	1924	241.024	274.803	339.997	299.970	232.598	159.997	108.878	107.699	79.992	55.000	87.651	1987.608	63.608
15	1776	161.016	208.402	339.997	299.970	242.998	129.297	129.974	119.999	49.995	55.000	91.228	1827.876	51.876
16	1554	150.015	135.001	310.397	249.975	229.698	147.397	120.676	100.799	19.998	28.641	100	1592.597	38.597
17	1480	150.015	135.001	339.997	199.980	179.698	138.697	129.974	99.199	19.998	34.287	88.199	1515.045	35.045
18	1628	150.015	183.802	330.097	249.975	229.698	159.997	99.980	101.599	42.196	55.000	70.007	1672.365	44.365
19	1776	184.918	225.502	339.997	299.970	206.098	159.997	129.974	119.999	52.095	50.321	61.321	1830.192	54.192
20	1972	227.923	301.003	339.997	293.971	242.998	159.997	129.974	119.999	79.992	41.998	100	2037.850	65.850
21	1924	213.021	307.203	339.997	299.970	220.898	129.597	129.974	119.999	76.292	49.654	100	1986.605	62.605
22	1628	150.015	227.202	275.597	256.274	215.998	126.397	99.980	119.999	79.992	19.612	100	1671.067	43.067
23	1332	150.015	147.201	195.598	247.775	165.998	76.398	86.383	119.999	77.792	10.000	82.954	1360.114	28.114
24	1184	150.015	140.901	115.599	197.780	174.998	82.298	86.383	106.499	65.793	10.020	75.998	1206.285	22.285
												Total Cost	23,64,127.45 $

Table 9

Generated output from distributed generators for a microgrid

Hour	T_dg1	T_dg2	P_w1	P_w2	P_fc1	P_fc2	P_fc3
1	0	298.3197	171.2699	130.377	53.6639	0	0
2	0	139.3368	300	43.3846	67.553	0	0.0145
3	199.6581	176.0755	0	84.1035	4.4504	100.001	80.7001
4	16.2046	171.6078	300	200.196	0	0	0
5	0	356.2198	0	299.994	150.015	36.616	0
6	399.92	451.9558	0	147.89	100.01	18.197	0
7	306.4896	515.3622	0	257.148	81.2025	64.191	100.001
8	399.92	422.2871	300	20.9547	150.015	100	0
9	399.92	707.5804	0	125.187	98.4187	96.493	0
10	0	643.1776	277.5463	299.994	72.5477	100	0
11	206.9003	151.3759	300	299.994	150.015	30.127	99.9716
12	0	323.326	288.1575	299.994	25.4849	62.58	84.0877
13	0	682.0682	0	149.997	100.01	100	0
14	292.4543	298.438	300	99.2505	7.4272	0	0
15	228.9373	450.0919	4.6599	299.904	0	0	100.001
16	0	436.7258	300	40.2288	149.553	6.838	98.7132
17	349.3365	218.6155	300	149.997	100.01	0	0
18	156.4387	469.577	300	299.994	150.015	0	0
19	305.0689	609.9586	300	278.344	0	100	75.0238
20	399.92	645.671	300	144.163	100.01	61.428	0
21	399.92	337.7812	300	299.994	138.553	100	57.7144
22	321.8738	374.8098	300	299.994	65.296	0	100.001
23	286.4408	466.4734	300	65.0639	23.7122	99.901	100.001
24	0	466.4466	300	299.994	0	0	0
				Total Cost ($)		33699.33 $

Table 10

Comparison of cost for case 4 considering uncertainty wind energy

Techniques	Best Cost ($)	Mean Cost($)	Maximum Cost($)
MAPSO [17]	2368764	–	–
EPO	2376571.91	2376845.25	2377101.74
CEPO	2374546.54	2374632.33	2374721.35
OLEPO	2366327.53	2366418.36	2366527.14
COLEPO	2364127.45	2364184.27	2364221.57

Fig. 15

Output power from each generating units with uncertainty wind energy and losses.

Fig. 16

Convergence characteristics of the proposed technique for test system 4 with uncertainty wind energy.

6.3.5 Test system-5: Dynamic economic dispatch problem for microgrid with consideration of distributed generators

This test system represents the microgrid with diesel-based generators, wind firm and fuel cell-based plant considered as the distributed generators. The proposed improved technique is applied to the microgrid to satisfy the load demand. The input data for all the distributed generators are referred from [3]. The hourly scheduled generation from two diesel generators, two wind turbines and three fuel cell-based distributed generators are depicted in Table 9. The total generation cost is obtained as 33699.33 $, through optimum scheduling among the considered distributed generators which is 106.20 $ less than the obtained cost by MOSHEPO [18] technique as presented in Table 11. The scheduled generation from the distributed generators to meet the load demand and convergence characteristics is presented graphically in Figs. 17 and 18 respectively.

Table 11
Comparison of cost for DED for microgrid

Techniques Best Cost ($)

CSA [3] 33824.10

DE [3] 33930.94

PSO [3] 38189.31

MOSHEPO [18] 33804.53

EPO 34154.21

CEPO 33965.35

OLEPO 33865.47

COLEPO 33699.33

Techniques	Best Cost ($)
CSA [3]	33824.10
DE [3]	33930.94
PSO [3]	38189.31
MOSHEPO [18]	33804.53
EPO	34154.21
CEPO	33965.35
OLEPO	33865.47
COLEPO	33699.33

Fig. 17

Output power from each distributed generators for microgrid.

Fig. 18

Convergence characteristics of the proposed technique for microgrid.

7 Conclusion

An enhanced version of novel EPO is proposed and applied dynamic economic dispatch problem with renewable energy sources and incorporating microgrid. The scheduled generation from all the power generating sources to achieve the optimum cost for five different test systems was presented. The operated generators satisfy the practical and operational constraints including the uncertainty of wind power for solving the problem. The β-chaotic mapping is used replacing the random function for perfect balance, and oppositional based learning was adapted for reducing the searching time. The effectiveness and efficacy of the proposed technique were validated for different benchmark function to find the optimum value with lesser time. The improved version of EPO is tested for five test system for DED problem for thermal generators, renewable energy sources and distributed generators for microgrid. The applied technique provides the optimum cost as compared to different optimization techniques with a lesser variation. The convergence graph and comparison graph is presented in the result and analysis section. The tabular and graphical comparison shows the superiority of the proposed technique with a better balance between exploration and exploitation. This technique can be used in the dynamic economic dispatch considering the emission effect.

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Enhanced emperor penguin optimization algorithm for dynamic economic dispatch with renewable energy sources and microgrid

Abstract

Keywords

1 Introduction

2 Modeling of dynamic economic dispatch problem with wind energy

2.3.1 Power balance constraints

3.1 The cost function of diesel generators

3.5.1 Constraints for power balance

6.1 Time complexity

6.2 Validation of the optimization algorithm

6.3.1 Test system-1: Five generating units without wind energy

Table 11 Comparison of cost for DED for microgrid Techniques Best Cost ($) CSA [3] 33824.10 DE [3] 33930.94 PSO [3] 38189.31 MOSHEPO [18] 33804.53 EPO 34154.21 CEPO 33965.35 OLEPO 33865.47 COLEPO 33699.33

References

Table 11
Comparison of cost for DED for microgrid

Techniques Best Cost ($)

CSA [3] 33824.10

DE [3] 33930.94

PSO [3] 38189.31

MOSHEPO [18] 33804.53

EPO 34154.21

CEPO 33965.35

OLEPO 33865.47

COLEPO 33699.33