Soft Computing

https://doi.org/10.1007/s00500-023-08584-0 (0123456789().,-volV)(0123456789().
,- volV)

OPTIMIZATION

Optimization of cost and emission for dynamic load dispatch problem

with hybrid renewable energy sources
Srinivasa Acharya1 • S. Ganesan2 • D. Vijaya Kumar1 • S. Subramanian3

Accepted: 14 May 2023

 The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023

Abstract
In the power system, the economic dispatch (ED) problem is the key issue, while fossil fuels cause environmental
pollution. The allocation of power generation is included in the actual economic load dispatch issue of power generation
for reducing the operating cost. This creates the economic load dispatch issue, a large-scale, highly nonlinear controlled
optimization issue. The major issue in the power systems is the loss, fuel cost, and emission. The existing algorithms can
optimize the parameters mentioned above, but it is not much better. Hence, this paper presents the multi-objective multi-
verse optimization (MOMVO) for the dynamic load dispatch problem. The dynamic load dispatch issue is evaluated by
cost and emission evaluation with hybrid renewable energy Sources (RES). Here, the proposed algorithm generates the
thermal, photovoltaic (PV) and wind power values, reducing the cost and emission values. The unit’s power generation is
the foremost aim of dynamic economic load dispatch (DELD) to meet the load demand while sustaining various opera-
tional constrictions; the generation’s total cost is reduced. The MVO algorithm is applied to nonlinear DELD issues and is
a reliable and robust optimization algorithm. The introduced scheme is implemented and tested over three test systems,
such as 6, 10, and 11 generating units. The implementation is performed on the MATLAB R2016a platform, and the
performance results are evaluated based on with and without valve-point loading (VPL). Finally, VPL produced better
solutions than the without VPL case.

Keywords Emission  Economic load dispatch  MOMVO algorithm  Fuel cost  Generation units  Fuel cost coefficients

1 Introduction The basic requirement of ELD for offering load demand

counting losses is to determine the best arrangement of a
The power plants emitting fossil fuel emissions cause high generator’s schedule. Designing the live ED system is the
operational costs and need great care to reduce pollution. key attribute of DLD. Permitting generators to load inside
Several techniques have been introduced for minimizing the framework in such a mode is crucial to confine the total
the emissions, which are classified into three groups, as cost of distributing the framework requirement at every
follows: (i) Power plant sites pollution removal device point (Qu et al. 2018). The lambda iteration method,
installation; (ii) Old devices replacement by new ones; and quadratic programming, nonlinear programming, interior
(iii) Environmental pollutants-based power plant operation point method, gradient method, and dynamic programming
(Ebrahimian et al. 2015). To solve the ED issue, emissions are the precise optimization approaches utilized for over-
from power plants based on several approaches have been coming the difficulties associated with ELD (Daniel et al.
proposed. Then, in related studies, this strategy was used to 2018).
control pollution. The pollution is reduced, and the total Several algorithms are introduced to increase and
cost is optimized by a molecular chemical reaction-based decrease the power generation and consider the ramp
optimization algorithm (Roy and Hazra 2014). To speed up characteristics in starting up and shutting down the gen-
the operational cost function’s convergence, the EMO erating units (Majd et al. 2020). Introducing ramp rate
algorithm was introduced in Ghorbani et al. (2014). control is a dynamic process. Dynamic programming (DP)
is a suitable procedure for treating the optimal control
problem for dynamic systems. The conventional
Extended author information available on the last page of the article

123
 S. Acharya et al.

Lagrangian relaxation (LR) approach was used in the Table 1 List of parameters
computationally more efficient approach, which was very Symbols Parameters
grateful to the power generation industry. Different
approaches are introduced to resolve the reserve-con- Vi or V~i Generator unity’s fuel cost
strained ED issue with a forbidden operating zone (Nari- mi ; ni ; and oi Cost coefficients
mani et al. 2018). An efficient Tabu Search (TS) (Joshi pi and qi Valve-point loading coefficients
2017) is developed to eliminate the SCED issue. To Xi Generating power
overcome the SCED issue, the exact procedure can be XD The total demand for the system
employed. Here, the problems are subjected to the equiv- XL Losses in total line
alence constraints of power balance, limits on the MVA Ximin and Ximax Generating the unit i’s maximum and minimum
line flow, and the active power creations as the disparity operation output
limits related to the contingency and base case states. TS is Pthermal ðtÞ The output power of thermal
a metaheuristic algorithm and is used to search the solution Ps ðtÞ Solar power unit
space without entrapping into a local optimum using some t Dispatch period
strategies for managing a local approach. The GWO bij ; b0i and b00 Loss coefficient
algorithm was utilized for the non-convex solution and Ftotal Minimized total operating cost
electric power system’s DELD issue (Kamboj et al. 2017). Ffuel ðPi Þ Generator’s fuel cost
In previous studies, the load dispatch problems are Eemission ðPi Þ Generator’s emission
addressed by many other solutions, such as the Newton Pi Generator’s output power
Raphson method (Chen et al. 2018), Analytical methods N Total generators
(Pandey et al. 2019), and a Lagrangian method (Tang et al. h A penalty factor of price
2019) are the initial schemes. The artificial immune system hi Climbing order
(Aragón et al. 2019), simulated annealing (Ziane et al. di ; gi Valve point effect emission coefficients
2017), frog algorithm (Anita and Raglend 2015), differ- Icost Investment costs
ential evolution (Pandit et al. 2015), genetic algorithm Mcost Maintenance costs
(Yeh et al. 2020) and particle swarm optimization (PSO) N Investment lifetime
(Al-Rubayi et al. 2020) are the evolutionary methods for R Interest rate
addressing the load dispatch issue. To get an economic
Ps Solar power
power flow with the optimum costs, the PSO-SIL algorithm
q Density of air
is introduced in Dong et al. (2023). There are many
Cp Wind power coefficient
advanced attempts to face the emission load dispatch
Sw Wind turbine blade’s swept area
(EELD) issue to reduce pollution control and generation
Vw Speed of the wind
costs simultaneously. The ELD problem is solved by
ristart ðtÞ Start-up ramp rate
employing many optimization strategies, namely linear
Ui ðtÞ Start-up state variable
programming (LP), classical, quadratic programming (QP),
Vi ðtÞ State variable of the shut-down process
and nonlinear programming (NLP) (Abbas et al. 2017). The
list of parameters used in this work is explained in Table 1. rishut ðtÞ Ramp rate shut-down
The foremost contribution of this research is given m Number of parameters
below: n Number of universes
Pm
n Element of the matrix G
• The optimal power creation of the units is the foremost r 2 ½0; 1 Random number in uniform
objective of DELD for getting load demand. Hence the WEP and TDR Coefficients
total generation cost is reduced while sustaining R2 ; R3 ; and R4 Random numbers
different operational constraints.
UBj jth variable’s upper bound
• Simultaneously, the DELD reduces emission dispersion
Pj jth parameter of the best universe
objectives and the total cost of thermal generating units.
LBj A lower bound of jth variable
Besides, wind power is combined with the grid to
min and max Constants
minimize the objectives and generate demand.
q Current iteration
• The increasing rate of the generation units makes the
Q Maximum iterations
cost high, making the system lose and emission. This
D Number of objects
can be optimized by the MOMVO algorithm, which is a
N Number of universes
reliable and robust optimization algorithm and is
illustrated in the nonlinear DELD issues. L High iterations

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Optimization of cost and emission for dynamic load dispatch problem with...

Analysis
• With valve point effect
• Without valve point
effect
• Thermal generating
units
Considerations • PV, wind turbine and
• Fuel cost Constraints thermal generating unit
• Emission • Ramp rate limit
• Solar cost function • Shut-down and start-
• Turbine cost up constraint
function • Prohibited operating
• Start-up and shut- zones Parameters
down cost • Transmission loss
• Power generation
• Emission cost
• Fuel cost
• Total cost

Fig. 1 Parameters evolved in the methodology

Research organization: The remaining structure of this disturbance updating method, a moth-flame optimization
research paper is organized as follows: Sect. 2 discussed algorithm was developed, which aimed at high-dimen-
the recent related works review. Next, Sect. 3 gives the sional, nonlinear and non-convex characteristics of
proposed methodology, which includes the MOMVO HDEED issues. Second, the mathematical model of
algorithm, Sect. 4 gives the results and discussion, and HDEED was constructed based on the RES like solar, wind
Sect. 5 gives the overall conclusion. and thermal energy.
Ishraque et al. (2021) designed an islanded hybrid
microgrid by assessing different cost analyses, power sys-
2 Recent related works: a review tem responses and optimal component sizing for various
load dispatch strategies. Four islanded hybrid microgrids
Some of the recent techniques related to the economic load were designed using wind, solar, diesel generator and
dispatch problem are listed below battery for optimal resource planning and reliable opera-
To resolve the Combined Economic Emission Dispatch tion. The five dispatch methods were HOMER predictive
(CEED) with RES, Nagarajan et al. (2022) presented an dispatch method, load following, combined dispatch, gen-
improved mayfly optimization algorithm (IMA). Here, erator order and cycle charging. The microgrids were
wind and solar power were considered as cost functions. optimized to reduce the levelized cost of energy, CO2
The cost optimization method was examined to reduce the emission and Net present cost.
emission level and operational cost while satisfying the The greenhouse gas emission was reduced with the help
microgrid load demand. A novel IMA combined Levy of plug-in EVs and RESs. Ten units thermal system was
flight to solve CEED issues encountered in microgrids. In a rigorously analyzed along with plug-in EVs and RESs for
microgrid, the IMA was validated for its supremacy and dynamic CEED; moreover, the integrated system efficiency
efficiency under varying conditions. The minimization of was investigated. Behera et al. (2022) presented a fuzzy
emissions and the total cost were attained for different decision-making method for solving multi-objective CEED
scenarios. issues. The fuzzy decision-making method was tuned with
The mathematical model of RES-based hybrid dynamic the help of a constriction factor-based particle swarm
economic emission dispatch (HDEED) was constructed, optimization algorithm in ten-unit thermal generators with
and the enhanced moth-flame optimization algorithm was RESs and plug-in EVs. The optimal dynamic CEED
proposed by Liu et al. (2021). Based on the position

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 S. Acharya et al.

Fig. 2 Flowchart for proposed

MVO algorithm
Start

Universes Initialization and

inflation rate calculation

WEP and TDR Estimation

Inflation rate based population

sort

Search agents position update

Normalized inflation rate
calculation
Inflation rate determination for
Updated solution

No Is Maximum
iteration

Yes
Display the best optimal
solution and system
specification determination

End

problem outcomes were improved further using solar 3 Proposed methodology: formulation
power, wind power and plug-in EVs. of emission and DELD problems
Hazra et al. (2021) investigated the effectiveness of the
Grasshopper Optimization Algorithm (GOA) in the ELD Solar PV and wind are typically uncertain power sources
operation for minimizing the total cost of a 30-bus test on the source side, affected by weather conditions. There is
system. Minimizing the total cost was considered the main a large number of other uncertainties in modern renewable
objective of evaluating the system’s performance. The energy sources, like uncertainties of source through a
underestimation and overestimation cases were included transmission line, air conditioning load and leading to
for the uncertain nature of wind power availability. To power flow uncertainty. The electrical power market has
obtain an optimal solution, the GOA method requires fewer become more inexpensive since the last eras. The optimal
iterations. power generation that survives the environment minimizes

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Optimization of cost and emission for dynamic load dispatch problem with...

Table 2 Parameters setting of MOMVO algorithm generating units controlled the fluctuation to regain the
Parameters Values
balance constraint. The power dispatch issue further com-
plicated pollution reduction in environmental dispatch in
Maximum number of archive elements 15 addition to minimizing fuel costs and total power loss. The
z 6 optimized model minimized the total cost, fuel cost and
Min 0.2 pollutant emission. Several algorithms optimized the
Max 1 environmental/economic dispatch issue because of its
Maximum number of iterations 200 complexity.
Number of universes 50
3.1 Mathematical model of EELD/CEED Issues
the total cost. The main aim was to minimize the total cost Without violating any power system constraints, mini-
of generation while sustaining the operational constraints. mizing the total fuel cost at thermal power plants is the
In various existing workings, algorithms have been dis- foremost goal of EELD. Without VPE and with VPE are
cussed to solve the problem, even making the system the two types of EELD issues studied in this paper, which
expensive. But the proposed system can better eliminate are expressed below,
problems like emission and fuel costs. Figure 1 gives the
Xn X
n  
considerations and measures of the proposed method. min Vi ð Xi Þ ¼ mi Xi2 þ ni Xi þ oi
DELD simultaneously reduces the aim of emission i¼1 i¼1
dispersion and the total cost of thermal generating units. X X 
n n      
With the grid, wind power is integrated to aid this process min V~i ðXi Þ ¼ mi Xi2 þ ni Xi þ oi þ pi sin qi Ximin  P 
to minimize the objectives and meet the demand. The i¼1 i¼1

fluctuation occurs in a generation since the power of wind ð1Þ

output turns because of the stochastic nature. The demand where, the generator unity’s fuel cost is indicated as
balance constraint was not fulfilled in this case. Committed
Vi or V~i , cost coefficients are represented as mi ; ni ; and oi ,

Table 3 Pseudo code of MVO

Initialize TDR, WER and best universe
algorithm
NI=Normalize the inflation rate of the universes and SU=Sorted universes
while the end criterion is not satisfied
Calculate the fitness function for all universes
for every universe indexed by i
Update TDR and WEP
Black_hole_index=I;
for every object indexed by j
r1=random ([0,1])
if r1<NI(Ui)
U (Black_hole_index,j)=SU(White_hole_index,j);
White_hole_index= Roulette Wheel Selection (-NI);
end if
r2=random([0,1]);
if r2<Wormhole_existance_probability
r3=random([0,1]);
r4=random([0,1]);
if r3<0.5
U(i,j)=Best_universe(j) + Travelling_distance_rate*
*
((ub(j)-lb(j)) r4+lb(j));
Else
U(i,j)=Best_universe(j) - Travelling_distance_rate*
((ub(j)-lb(j))*r4+lb(j));
end if
end if
end for
end for
end while

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 Optimization of cost and emission for dynamic load dispatch problem with...

b Fig. 3 The total generated power of thermal units (a), (c), (e) 6, 10, Here Ftotal represents the minimized total operating cost,
and 11 generation units without VPL, (b), (d), (f) 6, 10, and 11 Ffuel ðPi Þ represents the generator’s fuel cost in ($/h),
generation units with VPL for case 1
Eemission ðPi Þ denotes the generator’s emission in ($/h), Pi
represents the generator’s output power, the total genera-
tors are represented as N and h indicates the penalty factor
valve-point loading coefficients are indicated as pi and qi ,
of price, $/lb ($/kg), which is evaluated by Eq. (6),
the generating power is represented as Xi (in MW) and n   max
represents the overall generators quantity. Besides satis- Ffuel Pmaxi Pi
hi ¼ ; i ¼ 1; 2; 3; . . .; 24 ð6Þ
fying the two types of necessities, such as inequality and Eemission ðPi Þ=Pmax
max
i
equality constraints given by,
The values hi are organized in climbing order. Each
X
n
generating unit’s maximum capacity is added one at a time,
Xi  X L  XD ¼ 0
initiating from the unit taking the minimum hi until
i¼1 P max
Pi  Pdemand . hi represents the price penalty factor
Ximin  Xi  Ximax ð2Þ associated with the last unit for the given load.
The total demand of the system is represented as XD , the The total emission can be formulated by inserting the
losses in total line are represented as XL (in MW); the VPL effect can be expressed as Eq. (7),
generating unit i’s maximum and minimum operation M X
X N  
Eemission ðPi Þ ¼ ai P2i þ bi Pi þ ci þ g exp dPi ðlb=hrÞ
output is indicated as Ximin and Ximax respectively. Then the
m¼1 i¼1
output power of thermal Pthermal ðtÞ and solar power units
ð7Þ
Ps ðtÞ is determined as Pi ðtÞ, which is described in Eq. (3),
X
N Here, Eemission ðPi Þ is the total emission with the valve

Pi ðtÞ ¼ Ps ðtÞ þ Pthermal ðtÞ ð3Þ point effect ai ðkg=hÞ; bi ðkg=MWhÞ and ci ðkg MW2 hÞ are
i¼1 the coefficients of emission for the generator i and di ; gi are
Here, Pdemand ðtÞ and Ploss ðtÞ represents the system load described as the valve point effect emission coefficients of
demand and loss of transmission network for the dispatch jth the generating unit.
period t, respectively. A power flow computation is utilized
to evaluate the transmission power losses. A mutual prac- 3.1.1 Solar power sources cost function
tice is to imprecise the overall transmission losses through
a simplified linear formula or as a power output’s quadratic Based on the DELD solution, the rest of the load demand is
function of the generating units called Kron’s loss for distributed over the conventional generators. The following
abruption and simplification of the issue. The b-coefficients Eq. (8) describes the solar power cost function as,
!
are utilized to express the transmission loss as Eq. (4), r
N X
X N X
N Scost ðPs Þ ¼ Icost þ Mcost Ps ð8Þ
½1  ð1 þ r ÞN 
Ploss ðP; tÞ ¼ Pi ðtÞ:bij :Pj ðtÞ þ b0i :Pi ðtÞþb00
i¼1 j¼1 i¼1
Here, Icost and Mcost indicates the investment and
ð4Þ maintenance costs per unit installed power ($/kW),
respectively, N indicates the investment lifetime, r repre-
Here, bij ; b0i and b00 are the elements of the loss
sents the interest rate, and the solar power is represented as
coefficient. During the dispatch procedure, b matrix coef-
Ps .
ficients are constant. When the original operating settings
Uncertainty modeling of solar
adjacent to coefficients were computed, reasonable accu-
The Weibull probability distribution function is used for
racy was evaluated. The power flow program must be run
modeling the solar PV system, which is expressed as
in advance to regulate the b-coefficients.
follows,
The main objective of the EELD/CEED issue is to "   #
   k1 1
reduce the total emission and generation cost of the power k1 G G k1
system inside a defined time duration; it can be expressed fG ðGÞ ¼ x exp 
c1 c1 c1
as Eq. (5),   k2 1 "   #
k2 G G k2
X
N þ ð1  xÞ exp  0
c2 c2 c2
Ftotal ¼ Ffuel ðPi Þ þ ðh  Eemission ðPi ÞÞ ð5Þ
i¼1 G1
ð9Þ

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Table 4 The analysis of with VPL effect presence in 6 generating units
Time Generation schedule Total Generated power Load demand Power loss With VPL
(h) (MW) (MW) (MW)
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Fuel cost Emission cost Total cost
($/h) (lb/h) ($/h)

1 28.62047 57.24095 85.86142 114.48190 143.10238 171.72285 603.1703 598 5.17028 3785.212 1890.541 5675.752
2 28.71476 57.42952 86.14428 114.85904 143.5738 172.28857 605.2425 600 5.24254 3787.898 1896.379 5684.277
3 29.14595 58.29190 87.43785 116.58381 145.72976 174.87571 614.3273 609 5.32728 3790.537 1980.769 5771.306
4 31.68285 63.36571 95.04857 126.73142 158.41428 190.09714 667.6853 662 5.68528 4238.482 2075.53 6314.012
5 32.21023 64.42047 96.63071 128.84095 161.0511 193.26142 678.7455 673 5.74548 4298.168 2106.959 6405.127
6 32.54333 65.08666 97.63 130.17333 162.71666 195.26 685.7865 680 5.78654 4349.336 2122.893 6472.228
7 33.83547 67.67095 101.5064 135.34190 169.17738 203.01285 712.9599 707 5.95988 4528.695 2221.193 6749.888
8 34.79309 69.58619 104.3792 139.17238 173.96547 208.75857 733.115 727 6.11498 4689.372 2267.036 6956.409
9 35.17690 70.35380 105.5307 140.70761 175.88452 211.06142 741.1881 735 6.1881 4740.592 2297.253 7037.845
10 35.56023 71.12047 106.6807 142.24095 177.80119 213.36142 749.2962 743 6.29617 4792.755 2305.499 7098.254
11 35.84690 71.69380 107.5407 143.38761 179.23452 215.08142 755.3211 749 6.32114 4836.336 2325.806 7162.142
12 35.94214 71.88428 107.8264 143.76857 179.71071 215.65285 757.3587 751 6.35874 4836.74 2360.476 7197.217
13 36.08619 72.17238 108.2585 144.34476 180.43095 216.51714 760.3971 754 6.39713 4842.951 2360.784 7203.734
14 36.37238 72.74476 109.1171 145.48952 181.86190 218.23428 766.4278 760 6.42776 4903.709 2361.959 7265.668
15 37.23523 74.47047 111.7057 148.94095 186.17619 223.41142 784.6141 778 6.61406 5046.415 2434.913 7481.329
16 37.85595 75.71190 113.5678 151.42381 189.27976 227.13571 797.6586 791 6.65864 5173.387 2456.766 7630.153
17 39.62809 79.25619 118.8842 158.51238 198.14047 237.76857 834.9333 828 6.9333 5356.825 2594.641 7951.466
18 40.77523 81.55047 122.3257 163.10095 203.87619 244.65142 859.1256 852 7.12564 5530.275 2655.627 8185.902
19 44.17357 88.34714 132.5207 176.69428 220.86785 265.04142 930.4891 923 7.48912 5998.392 2879.839 8878.231
20 44.31761 88.63523 132.9528 177.27047 221.58809 265.90571 933.567 926 7.56697 6022.407 2908.991 8931.398
21 45.13166 90.26333 135.395 180.52666 225.65833 270.79 950.655 943 7.65501 6133.274 2952.275 9085.55
22 45.65619 91.31238 136.9685 182.62476 228.28095 273.93714 961.7093 954 7.70929 6195.564 3002.106 9197.67
23 45.89785 91.79571 137.6935 183.59142 229.48928 275.38714 966.8022 959 7.80225 6235.975 3002.983 9238.958
24 49.77380 99.54761 149.3214 199.09523 248.86904 298.64285 1048.243 1040 8.24318 6766.352 3267.492 10,033.84
S. Acharya et al.
 Table 5 The analysis of without VPL effect presence in 6 generating units
Time Generation schedule Total Generated power Load demand Power loss Without VPL
(h) (MW) (MW) (MW)
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Fuel cost Emission cost Total cost
($/h) (lb/h) ($/h)

1 28.618762 57.2375 85.85628 114.47505 143.0938 171.7125 601.6243 598 3.624253 3784.319 1868.113 5652.432
2 28.714321 57.42864 86.14296 114.85728 143.5716 172.2859 603.7014 600 3.701371 3784.882 1869.472 5654.354
3 29.145114 58.29022 87.43534 116.58045 145.7255 174.8706 612.7452 609 3.745245 3784.968 1869.985 5654.953
4 31.681487 63.36297 95.04446 126.72594 158.4074 190.0889 666.0156 662 4.015565 4211.015 2048.604 6259.619
5 32.207944 64.41588 96.62383 128.83177 161.0397 193.2476 677.1472 673 4.147173 4298.054 2104.535 6402.589
6 32.5429714 65.08594 97.62891 130.17188 162.7148 195.2578 684.1921 680 4.192058 4322.214 2110.489 6432.703
7 33.8351987 67.67039 101.5055 135.34079 169.1759 203.0111 711.3365 707 4.336514 4526.918 2220.886 6747.804
8 34.7921665 69.58433 104.3764 139.16866 173.9608 208.7529 731.4632 727 4.463237 4643.714 2267.023 6910.737
9 35.1754298 70.35085 105.5262 140.70171 175.8771 211.0525 739.5347 735 4.534711 4729.432 2295.635 7025.067
10 35.5583070 71.11661 106.6749 142.23322 177.7915 213.3498 747.5769 743 4.576866 4755.208 2305.293 7060.501
11 35.8452337 71.69046 107.5357 143.38093 179.2261 215.0714 753.6163 749 4.616299 4815.259 2323.496 7138.754
Optimization of cost and emission for dynamic load dispatch problem with...

12 35.9409472 71.88189 107.8228 143.76378 179.7047 215.6456 755.6593 751 4.659308 4816.613 2360.373 7176.986
13 36.0844465 72.16889 108.2533 144.33778 180.4222 216.5066 758.6785 754 4.67853 4817.761 2360.716 7178.477
14 36.3718571 72.74371 109.1155 145.48742 181.8592 218.2311 764.7605 760 4.760497 4833.055 2361.532 7194.587
15 37.2330329 74.46606 111.6990 148.93213 186.1651 223.3981 782.8563 778 4.856267 5009.998 2432.459 7442.456
16 37.8550529 75.71010 113.5651 151.42021 189.2752 227.1303 795.9429 791 4.942851 5132.039 2456.266 7588.305
17 39.6260858 79.25217 118.8782 158.50434 198.1304 237.7565 833.1317 828 5.131747 5333.02 2592.256 7925.276
18 40.7744713 81.54894 122.3234 163.09788 203.8723 244.6468 857.274 852 5.274037 5509.402 2655.411 8164.813
19 44.1722579 88.34451 132.5167 176.68903 220.8612 265.0335 928.6333 923 5.633265 5945.762 2878.957 8824.719
20 44.3159066 88.63181 132.9477 177.26362 221.5795 265.8954 931.6612 926 5.661219 5971.105 2908.425 8879.53
21 45.1293316 90.25866 135.3879 180.51732 225.6466 270.7759 948.8109 943 5.810896 6110.893 2950.29 9061.183
22 45.6557771 91.31155 136.9673 182.62310 228.2788 273.9346 959.8766 954 5.876563 6176.609 2979.723 9156.332
23 45.8954486 91.79089 137.6863 183.58179 229.4772 275.3726 964.9102 959 5.91021 6178.401 2984.472 9162.873
24 49.7718838 99.54376 149.3156 199.08753 248.8594 298.6313 1046.373 1040 6.372944 6759.506 3267.203 10,026.71

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Table 6 The analysis of with VPL effect presence in 10 generating units
Time Generation schedule Total Load Power With VPL
(h) Generated demand loss
P1 P2 P3 P4 (MW) P5 P6 P7 P8 P9 P10 power (MW) (MW) Fuel Emission Total
(MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) cost ($/ cost (lb/h) cost ($/
h) h)

1 16.40751 32.81503 49.22254 65.63006 82.03757 98.44509 114.8526 131.2601 147.6676 164.0751 902.47 900 2.4700 5822.29 2879.31 8701.617
2 20.24436 40.48873 60.73310 80.977472 101.22184 121.46620 141.71057 161.95494 182.19931 202.44368 1113.434 1110 3.4337 7218.49 3602.44 10,820.94
3 21.59686 43.19373 64.79059 86.387465 107.98433 129.58119 151.17806 172.77493 194.37179 215.96866 1187.814 1184 3.8143 7717.70 3850.95 11,568.65
4 22.94770 45.89541 68.84311 91.790821 114.73852 137.68623 160.63393 183.58164 206.52934 229.47705 1262.173 1258 4.1732 8201.48 4072.24 12,273.73
5 24.29756 48.59512 72.89268 97.190243 121.48780 145.78536 170.08292 194.38048 218.67804 242.97560 1336.526 1332 4.5261 8729.69 4346.75 13,076.45
6 25.65190 51.30381 76.95572 102.60763 128.25954 153.91145 179.56336 205.21527 230.86718 256.51908 1410.77 1406 4.7701 9205.96 4568.54 13,774.5
7 27.00262 54.00525 81.00788 108.01051 135.01314 162.01577 189.01840 216.02103 243.02366 270.02629 1485.122 1480 5.1221 9632.32 4788.76 14,421.09
8 26.99965 53.99930 80.99896 107.99861 134.99826 161.99792 188.99757 215.99722 242.99688 269.99653 1485.146 1480 5.1456 9645.73 4789.98 14,435.72
9 27.36790 54.73580 82.10370 109.47161 136.83951 164.20741 191.57532 218.94322 246.31112 273.67903 1505.226 1500 5.2261 9846.08 4835.28 14,681.37
10 28.34981 56.69963 85.04945 113.39926 141.74908 170.09890 198.44872 226.79853 255.14835 283.49817 1559.33 1554 5.3302 10,182.2 5036.58 15,218.82
11 29.70203 59.40407 89.10611 118.80815 148.51019 178.21223 207.91427 237.61631 267.31835 297.02039 1633.64 1628 5.6395 10,672.3 5249.28 15,921.6
12 29.70160 59.40320 89.10481 118.80641 148.50801 178.20962 207.91122 237.61283 267.31443 297.01603 1633.695 1628 5.6953 10,673.8 5266.85 15,940.72
13 31.12822 62.25645 93.38468 124.51291 155.64113 186.76936 217.89759 249.02581 280.15404 311.28227 1712.121 1706 6.1212 11,218.2 5544.86 16,763.06
14 32.40586 64.81173 97.21760 129.62346 162.02933 194.43520 226.84107 259.24693 291.65280 324.05867 1782.149 1776 6.1489 11,649.0 5751.74 17,400.78
15 32.40271 64.80543 97.20814 129.61086 162.01358 194.4163 226.81901 259.22173 291.62445 324.02716 1782.248 1776 6.2480 11,674.9 5767.26 17,442.19
16 32.40399 64.80798 97.21197 129.61596 162.01996 194.42395 226.82794 259.23193 291.63592 324.03992 1782.291 1776 6.2912 11,679.6 5768.95 17,448.62
17 35.10852 70.21704 105.3255 140.43408 175.54260 210.65112 245.75964 280.86816 315.97668 351.08520 1930.78 1924 6.7802 12,613 6240.37 18,853.38
18 35.10724 70.21448 105.3217 140.42896 175.53620 210.64344 245.75068 280.85792 315.96516 351.07240 1930.938 1924 6.9382 12,621.6 6243.15 18,864.78
19 35.10617 70.21235 105.3185 140.4247 175.5308 210.63705 245.74323 280.84941 315.95558 351.06176 1930.94 1924 6.9401 12,642.6 6245.46 18,888.08
20 37.80775 75.61550 113.4232 151.2310 189.0387 226.8465 264.6542 302.4620 340.2697 378.0775 2079.441 2072 7.4409 13,590.7 6750.82 20,341.54
21 37.80811 75.61622 113.4243 151.2324 189.0405 226.8486 264.6568 302.4649 340.2730 378.0811 2079.448 2072 7.4483 13,593.9 6755.89 20,349.81
22 37.80868 75.61736 113.4260 151.2347 189.0434 226.8520 264.6607 302.4694 340.2781 378.0868 2079.646 2072 7.6455 13,659.4 6793.20 20,452.6
23 39.16082 78.32165 117.4824 156.6433 195.8041 234.9649 274.1258 313.2866 352.4474 391.6082 2153.816 2146 7.8159 14,113.5 6980.23 21,093.78
24 40.51096 81.02193 121.5328 162.0438 202.5548 243.0657 283.5767 324.0877 364.5986 405.1096 2228.103 2220 8.1025 14,637.9 7205.08 21,843.06
S. Acharya et al.
 Table 7 The analysis of without VPL effect presence in 10 generating units
Time Generation schedule Total Load Power With Out VPL
(h) Generated demand loss
P1 P2 P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 P8 P9 P10 power (MW) (MW) Fuel cost Emission Total cost
(MW) (MW) (MW) (MW) (MW) (MW) (MW) ($/h) cost (lb/ ($/h)
h)

1 16.3721 32.7443 49.1165 65.4887 81.8609 98.2331 114.605 130.977 147.349 163.721 900.4913 900 0.49127 5801.485 2876.047 8677.532
2 20.2082 40.4164 60.6246 80.8328 101.041 121.249 141.457 161.665 181.873 202.082 1111.503 1110 1.50286 7197.677 3555.777 10,753.45
3 21.5610 43.1221 64.6832 86.2443 107.805 129.366 150.927 172.488 194.049 215.610 1185.859 1184 1.85927 7696.886 3798.436 11,495.32
4 22.9124 45.8248 68.7372 91.6497 114.562 137.474 160.387 183.299 206.211 229.124 1260.187 1258 2.18748 8180.671 4045.931 12,226.6
5 24.2629 48.5258 72.7888 97.0517 121.314 145.577 169.840 194.103 218.366 242.629 1334.365 1332 2.36476 8708.884 4304.73 13,013.61
6 25.6146 51.2292 76.8438 102.458 128.073 153.687 179.302 204.916 230.531 256.146 1408.887 1406 2.88654 9185.147 4527.868 13,713.01
7 26.9661 53.9323 80.8985 107.864 134.830 161.797 188.763 215.729 242.695 269.661 1483.091 1480 3.09101 9620.511 4788.641 14,409.15
8 26.9649 53.9299 80.8949 107.859 134.824 161.789 188.754 215.719 242.684 269.649 1483.109 1480 3.10878 9644.917 4789.268 14,434.19
9 27.3293 54.6586 81.9879 109.317 136.646 163.975 191.305 218.634 245.963 273.293 1503.176 1500 3.1764 9827.068 4835.182 14,662.25
10 28.3172 56.6344 84.9516 113.268 141.586 169.903 198.220 226.537 254.855 283.172 1557.417 1554 3.41733 10,161.42 5029.372 15,190.79
Optimization of cost and emission for dynamic load dispatch problem with...

11 29.6646 59.3293 88.9940 118.658 148.323 177.988 207.652 237.317 266.982 296.646 1631.489 1628 3.48914 10,352.01 5238.067 15,590.07
12 29.6653 59.3307 88.9961 118.661 148.326 177.992 207.657 237.323 266.988 296.653 1631.738 1628 3.73796 10,668.05 5250.99 15,919.04
13 31.0917 62.1834 93.2751 124.366 155.458 186.550 217.642 248.733 279.825 310.917 1710.158 1706 4.15846 11,197.38 5517.411 16,714.79
14 32.3674 64.7348 97.1022 129.469 161.837 194.204 226.572 258.939 291.306 323.674 1780.208 1776 4.2083 11,628.22 5739.561 17,367.79
15 32.3699 64.7398 97.1097 129.479 161.849 194.219 226.589 258.959 291.329 323.699 1780.335 1776 4.33507 11,638.11 5740.149 17,378.26
16 32.3648 64.7296 97.0944 129.459 161.824 194.188 226.553 258.918 291.283 323.648 1780.427 1776 4.42678 11,651.85 5746.988 17,398.84
17 35.0719 70.1439 105.215 140.287 175.359 210.431 245.503 280.575 315.647 350.719 1928.795 1924 4.79546 12,612.19 6220.032 18,832.22
18 35.0737 70.1474 105.221 140.294 175.368 210.442 245.516 280.589 315.663 350.737 1928.841 1924 4.84131 12,615.81 6220.399 18,836.21
19 35.0683 70.1367 105.205 140.273 175.341 210.410 245.478 280.547 315.615 350.683 1928.917 1924 4.91669 12,640.8 6222.545 18,863.35
20 37.7730 75.5461 113.319 151.092 188.865 226.638 264.411 302.184 339.957 377.730 2077.437 2072 5.43664 13,443.9 6695.54 20,139.44
21 37.7748 75.5496 113.324 151.099 188.874 226.648 264.423 302.198 339.973 377.748 2077.461 2072 5.461 13,546.49 6703.276 20,249.77
22 37.7720 75.5441 113.316 151.088 188.860 226.632 264.404 302.176 339.948 377.720 2077.493 2072 5.49269 13,656.28 6713.556 20,369.84
23 39.1263 78.2527 117.379 156.505 195.631 234.758 273.884 313.010 352.137 391.263 2151.821 2146 5.82141 14,092.73 6953.217 21,045.95
24 40.4750 80.9501 121.425 161.900 202.375 242.850 283.325 323.800 364.275 404.750 2226.152 2220 6.15221 14,617.15 7161.282 21,778.43

123
123
Table 8 The analysis of with VPL effect presence in 11 generating units
Time Generation schedule Total Load Power With VPL
(h) Generated demand loss
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 power (MW) (MW) (MW) Fuel cost Emission Total
(MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) ($/h) cost (lb/ cost ($/
h) h)

1 19.6288 39.257 58.88645 78.51527 98.14409 117.7729 137.4017 157.0305 176.6593 196.2881 215.917 1079.449 1077 2.4493 6990.073 3485.676 10,475.75
2 20.9992 41.998 62.99781 83.99709 104.9963 125.9956 146.9949 167.9941 188.9934 209.9927 230.992 1155.448 1152 3.4482 7517.401 3746.346 11,263.75
3 22.0225 44.045 66.06763 88.09018 110.1127 132.1352 154.1578 176.1803 198.2029 220.2254 242.248 1211.812 1208 3.8115 7890.09 3929.67 11,819.76
4 22.5707 45.141 67.71218 90.28290 112.8536 135.4243 157.9950 180.5658 203.1365 225.7072 248.278 1242.155 1238 4.1548 8068.779 4003.213 12,071.99
5 23.3199 46.639 69.95972 93.27963 116.5995 139.9194 163.2393 186.5592 209.8791 233.1990 256.519 1283.576 1279 4.5756 8329.979 4126.453 12,456.43
6 23.7219 47.443 71.16572 94.88763 118.6095 142.3314 166.0533 189.7752 213.4971 237.2190 260.941 1305.882 1301 4.8823 8522.995 4181.41 12,704.4
7 23.7401 47.480 71.22054 94.96072 118.7009 142.4410 166.1812 189.9214 213.6616 237.4018 261.142 1307.044 1302 5.0443 8525.184 4182.147 12,707.33
8 23.8680 47.736 71.60427 95.47236 119.3404 143.2085 167.0766 190.9447 214.8128 238.6809 262.549 1314.146 1309 5.1463 8561.034 4232.072 12,793.11
9 24.5259 49.051 73.57772 98.10363 122.6295 147.1554 171.6813 196.2072 220.7331 245.2590 269.785 1350.238 1345 5.2378 8801.684 4322.42 13,124.1
10 24.7269 49.453 74.18072 98.90763 123.6345 148.3614 173.0883 197.8152 222.5421 247.2690 271.996 1361.306 1356 5.3059 8840.655 4362.027 13,202.68
11 24.8548 49.709 74.56445 99.41927 124.2740 149.1289 173.9837 198.8385 223.6933 248.5481 273.403 1368.651 1363 5.6506 8927.377 4402.9 13,330.28
12 33.041 66.082 99.123 132.164 165.205 198.246 231.287 264.328 297.369 330.41 363.451 1816.698 1811 5.6983 11,905.72 5855.275 17,761
13 33.4795 66.959 100.4386 133.9181 167.3977 200.8772 234.3568 267.8363 301.3159 334.7954 368.275 1840.979 1835 5.9792 12,060.87 5927.4 17,988.27
14 34.0642 68.128 102.1928 136.2570 170.3213 204.3856 238.4499 272.5141 306.5784 340.6427 374.707 1873.351 1867 6.3512 12,273.42 6038.204 18,311.62
15 34.8317 69.663 104.4951 139.3269 174.1586 208.9903 243.8220 278.6538 313.4855 348.3172 383.149 1915.37 1909 6.3697 12,534.32 6170.154 18,704.47
16 37.1340 74.268 111.4022 148.5363 185.6704 222.8045 259.9386 297.0727 334.2068 371.3409 408.475 2041.376 2035 6.3760 13,382.6 6609.849 19,992.45
17 38.4131 76.826 115.2395 153.6527 192.0659 230.4790 268.8922 307.3054 345.7186 384.1318 422.545 2111.853 2105 6.8526 13,878.12 6807.38 20,685.5
18 39.0161 78.032 117.0485 156.0647 195.0809 234.0970 273.1132 312.1294 351.1456 390.1618 429.178 2144.98 2138 6.9795 14,055.1 6938.323 20,993.42
19 39.6374 79.274 118.9123 158.5498 198.1872 237.8247 277.4621 317.0996 356.7370 396.3745 436.012 2179.017 2172 7.0171 14,311.75 7030.725 21,342.47
20 40.7338 81.467 122.2014 162.9352 203.6690 244.4029 285.1367 325.8705 366.6043 407.3381 448.072 2239.642 2232 7.6419 14,723.93 7224.302 21,948.23
21 42.7620 85.524 128.2862 171.0483 213.8104 256.5725 299.3346 342.0967 384.8588 427.6209 470.383 2350.653 2343 7.6534 15,447.07 7578.856 23,025.93
22 43.0179 86.035 129.0537 172.0716 215.0895 258.1074 301.1253 344.1432 387.1611 430.1790 473.197 2364.739 2357 7.7388 15,558.43 7636.202 23,194.63
23 43.0361 86.072 129.1085 172.1447 215.1809 258.2170 301.2532 344.2894 387.3256 430.3618 473.398 2365.827 2358 7.8274 16,583.66 7667.274 24,250.93
24 45.6309 91.261 136.8927 182.5236 228.1545 273.7854 319.4163 365.0472 410.6781 456.3090 501.94 2508.117 2500 8.1171 16,688.68 8124.152 24,812.84
S. Acharya et al.
 Table 9 The analysis of without VPL effect presence in 11 generating units
Time Generation schedule Total Load Power Without VPL
(h) generated demand loss
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 power (MW) (MW) (MW) Fuel cost Emission Total cost
(MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) (MW) ($/h) cost (lb/ ($/h)
h)

1 19.625 39.250 58.875 78.500 98.125 117.75 137.37 157.00 176.62 196.25 215.87 1079.376 1077 2.37625 6969.492 3474.477 10,443.97
2 20.995 41.991 62.986 83.982 104.97 125.97 146.96 167.96 188.96 209.95 230.95 1154.753 1152 2.75321 7496.82 3720.351 11,217.17
3 22.018 44.037 66.056 88.075 110.09 132.11 154.13 176.15 198.16 220.18 242.20 1211.031 1208 3.03102 7869.509 3904.806 11,774.31
4 22.566 45.133 67.700 90.267 112.83 135.40 157.96 180.53 203.10 225.66 248.23 1241.186 1238 3.18641 8048.198 3980.82 12,029.02
5 23.316 46.632 69.948 93.264 116.58 139.89 163.21 186.52 209.84 233.16 256.47 1282.394 1279 3.39413 8309.397 4105.588 12,414.98
6 23.718 47.436 71.154 94.872 118.59 142.30 166.02 189.74 213.46 237.18 260.89 1304.503 1301 3.50348 8507.414 4181.022 12,688.44
7 23.736 47.472 71.209 94.945 118.68 142.41 166.15 189.89 213.62 237.36 261.10 1305.505 1302 3.5047 8516.603 4181.966 12,698.57
8 23.864 47.728 71.593 95.457 119.32 143.18 167.05 190.91 214.77 238.64 262.50 1312.54 1309 3.53967 8540.452 4215.842 12,756.29
9 24.522 49.044 73.566 98.088 122.61 147.13 171.65 196.17 220.69 245.22 269.74 1348.718 1345 3.71777 8784.103 4321.117 13,105.22
10 24.723 49.446 74.169 98.892 123.61 148.33 173.06 197.78 222.50 247.23 271.95 1359.777 1356 3.77656 8820.074 4350.381 13,170.45
Optimization of cost and emission for dynamic load dispatch problem with...

11 24.851 49.702 74.553 99.404 124.25 149.10 173.95 198.80 223.65 248.51 273.36 1366.812 1363 3.81242 8886.796 4354.359 13,241.15
12 33.037 66.074 99.111 132.14 165.18 198.22 231.26 264.29 297.33 330.37 363.40 1816.055 1811 5.05489 11,885.14 5810.432 17,695.58
13 33.475 66.951 100.42 133.90 167.37 200.85 234.33 267.80 301.28 334.75 368.23 1840.167 1835 5.16669 12,040.29 5875.564 17,915.85
14 34.060 68.121 102.18 136.24 170.30 204.36 238.42 272.48 306.54 340.60 374.66 1873.228 1867 6.22793 12,252.84 6001.713 18,254.55
15 34.828 69.656 104.48 139.31 174.14 208.96 243.79 278.62 313.45 348.28 383.10 1915.239 1909 6.2393 12,513.74 6143.508 18,657.25
16 37.130 74.260 111.39 148.52 185.65 222.78 259.91 297.04 334.17 371.30 408.43 2041.373 2035 6.37274 13,362.02 6543.969 19,905.99
17 38.409 76.818 115.22 153.63 192.04 230.45 268.86 307.27 345.68 384.09 422.50 2111.522 2105 6.52166 13,857.54 6757.754 20,615.29
18 39.012 78.024 117.03 156.04 195.06 234.07 273.08 312.09 351.11 390.12 429.13 2144.687 2138 6.68665 14,034.51 6845.97 20,880.48
19 39.633 79.267 118.90 158.53 198.16 237.80 277.43 317.06 356.70 396.33 435.97 2179.002 2172 7.00231 14,291.16 6979.197 21,270.36
20 40.730 81.460 122.19 162.92 203.65 244.38 285.11 325.84 366.57 407.30 448.03 2239.155 2232 7.15529 14,703.34 7173.763 21,877.11
21 42.758 85.516 128.27 171.03 213.79 256.55 299.30 342.06 384.82 427.58 470.34 2350.606 2343 7.60564 15,426.49 7544.34 22,970.83
22 43.014 86.028 129.04 172.05 215.07 258.08 301.09 344.11 387.12 430.14 473.15 2364.713 2357 7.71285 15,537.85 7593.727 23,131.58
23 43.032 86.064 129.09 172.12 215.16 258.19 301.22 344.25 387.29 430.32 473.35 2365.785 2358 7.78492 15,540.08 7600.14 23,140.22
24 45.627 91.254 136.88 182.50 228.13 273.76 319.38 365.01 410.64 456.27 501.89 2508.098 2500 8.09762 16,468.1 8042.265 24,510.37

123
 S. Acharya et al.

350 350

300 300

250 250

200 200
Power(MW)

Power(MW)
150 150

100 100

50 50

0 0
2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24
Time (Hour) Time (Hour)

(a) (b)
500 500

450 450

400 400

350 350

300 300

Power(MW)
Power(MW)

250 250

200
200

150
150

100
100

50
50

0
0 2 4 6 8 10 12 14 16 18 20 22 24
2 4 6 8 10 12 14 16 18 20 22 24
Time (Hour)
Time (Hour)

(d)
450 450

400
400

350
350

300
300

250
Power(MW)

250
Power(MW)

200
200

150
150
100

100
50

50
0
2 4 6 8 10 12 14 16 18 20 22 24
0
Time (Hour)
2 4 6 8 10 12 14 16 18 20 22 24

(e) Time (Hour)

Fig. 4 The total generated power of thermal units (a), (c), (e) 6, 10, and 11 generation units without VPL, (b), (d), (f) 6, 10, and 11 generation
units with VPL for case 2

123
 Table 10 The analysis of without VPL effect presence in 6 generating units
Time Thermal power PV power Wind Total power Load Power Without VPL
(h) power P (MW) demand loss (MW)
p1 p2 p3 P4 P5 P6 MW) Fuel Thermal Thermal PV cost Wind cost Total
cost ($/ emission cost (lb/ cost ($/h) ($/kW) ($/kW) cost ($/
h) h) h)

1 66.777 133.55 200.33 44.520 89.041 66.781 601.00 598 2.99 995.61 899.00 1894.62 282.52 282.52 2459.6
2 67.001 134.00 201.00 44.669 89.339 67.004 603.02 600 3.00 996.23 903.70 1899.94 283.75 283.75 2467.4
3 68.005 136.01 204.01 45.339 90.678 68.005 612.05 609 3.04 1021.8 908.65 1930.47 289.26 289.26 2508.9
4 73.924 147.84 221.77 49.284 98.569 73.926 665.32 662 3.31 1102.3 991.19 2093.57 321.67 321.70 2736.9
5 75.152 150.30 225.45 50.101 100.20 75.160 676.38 673 3.37 1126.4 999.24 2125.73 328.40 328.42 2782.5
6 75.933 151.86 227.8 50.623 101.24 75.935 683.40 680 3.40 1128.5 1025.3 2153.95 332.66 332.70 2819.3
7 78.949 157.89 236.84 52.633 105.26 78.957 710.55 707 3.53 1189.8 1062.1 2252.04 349.18 349.19 2950.4
8 81.181 162.36 243.54 54.123 108.24 81.188 730.64 727 3.63 1229.6 1094.4 2324.08 361.42 361.46 3046.9
9 82.076 164.15 246.23 54.717 109.43 82.077 738.68 735 3.67 1252.4 1104.6 2357.14 366.32 366.31 3089.7
10 82.968 165.93 248.90 55.315 110.63 82.976 746.73 743 3.72 1252.9 1115.4 2368.37 371.22 371.23 3110.8
Optimization of cost and emission for dynamic load dispatch problem with...

11 83.639 167.27 250.91 55.759 111.52 83.647 752.76 749 3.75 1266.4 1121.9 2388.31 374.89 374.91 3138.1
12 83.862 167.72 251.58 55.908 111.81 83.862 754.75 751 3.75 1275.6 1126.9 2402.58 376.11 376.10 3154.7
13 84.197 168.39 252.59 56.131 112.26 84.205 757.78 754 3.77 1275.7 1132.9 2408.71 377.95 377.96 3164.6
14 84.867 169.73 254.60 56.579 113.15 84.873 763.81 760 3.80 1287.7 1136.3 2424.07 381.61 381.63 3187.3
15 86.876 173.75 260.63 57.920 115.84 86.884 781.90 778 3.89 1322.5 1162.9 2485.5 392.60 392.65 3270.7
16 88.329 176.65 264.98 58.888 117.77 88.333 794.97 791 3.95 1349.0 1196.8 2545.94 400.58 400.57 3347.1
17 92.461 184.92 277.38 61.641 123.28 92.464 832.15 828 4.14 1411.7 1256.9 2668.66 423.20 423.23 3515.1
18 95.141 190.28 285.42 63.428 126.85 95.148 856.28 852 4.26 1452.4 1288.2 2740.69 437.87 437.88 3616.4
19 103.06 206.13 309.20 68.714 137.42 103.06 927.62 923 4.62 1566.0 1397.1 2963.28 481.31 481.35 3925.9
20 103.40 206.80 310.21 68.937 137.87 103.41 930.64 926 4.63 1586.2 1398.2 2984.44 483.13 483.15 3950.7
21 105.30 210.60 315.90 70.202 140.40 105.30 947.72 943 4.71 1609.0 1420.7 3029.8 493.55 493.56 4016.9
22 106.53 213.06 319.59 71.020 142.04 106.53 958.78 954 4.77 1628.7 1442.5 3071.3 500.26 500.26 4071.8
23 107.09 214.18 321.26 71.394 142.78 107.09 963.81 959 4.79 1629.4 1467.1 3096.57 503.31 503.33 4103.2
24 116.13 232.26 348.40 77.422 154.84 116.14 1045.2 1040 5.20 1706.3 1646.2 3352.64 552.88 552.91 4458.4

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Table 11 The analysis of with VPL effect presence in 6 generating units
Time Thermal power PV power Wind Total Load Power loss With VPL
(h) power power demand (MW)
p1 p2 p3 P4 P5 P6 p (MW) (MW) Fuel cost Thermal emission Thermal PV cost Wind cost Total cost
($/h) cost (lb/h) cost ($/h) ($/kW) ($/kW) ($/h)

1 66.78 133.56 200.34 44.52 89.049 66.8 601.05 598 3 988.19 902.41 1890.6 365.93 366.04 2622.6
2 67 134.01 201.01 44.68 89.36 67.02 603.08 600 3.05 994.56 911.21 1905.77 367.02 367.1 2639.9
3 68.01 136.02 204.03 45.34 90.687 68.06 612.14 609 3.065 1024.2 916.76 1940.92 372.63 372.66 2686.2
4 73.93 147.85 221.78 49.29 98.578 73.96 665.39 662 3.34 1135.9 978.51 2114.4 405 405.13 2924.5
5 75.16 150.31 225.47 50.11 100.23 75.2 676.48 673 3.415 1152.7 999.3 2152.03 411.75 411.92 2975.7
6 75.94 151.87 227.81 50.63 101.25 75.97 683.46 680 3.43 1169.2 1005.6 2174.83 416.01 415.98 3006.8
7 78.95 157.9 236.86 52.65 105.29 78.97 710.62 707 3.585 1223.1 1042.8 2265.93 432.52 432.49 3130.9
8 81.18 162.37 243.55 54.13 108.26 81.24 730.73 727 3.675 1255.1 1073.4 2328.5 444.75 445 3218.2
9 82.08 164.15 246.23 54.72 109.45 82.09 738.72 735 3.705 1259.6 1107 2366.6 449.62 449.89 3266.1
10 82.97 165.94 248.91 55.32 110.64 82.98 746.76 743 3.735 1276.9 1111.3 2388.2 454.51 454.68 3297.4
11 83.64 167.28 250.92 55.77 111.54 83.66 752.82 749 3.765 1289.5 1111.4 2400.85 458.18 458.18 3317.2
12 83.87 167.73 251.6 55.91 111.83 83.88 754.82 751 3.775 1290.5 1133.7 2424.19 459.38 459.68 3343.2
13 84.2 168.4 252.6 56.13 112.27 84.22 757.82 754 3.82 1290.5 1133.7 2424.22 461.21 461.51 3346.9
14 84.87 169.74 254.61 56.59 113.19 84.92 763.91 760 3.85 1290.6 1154.4 2445.05 465.02 465.02 3375.1
15 86.88 173.76 260.64 57.92 115.84 86.89 781.93 778 3.9 1342.8 1171.5 2514.31 475.89 476.03 3466.2
16 88.33 176.66 264.99 58.9 117.79 88.38 795.05 791 4.005 1355.7 1195 2550.64 483.93 483.93 3518.5
17 92.46 184.92 277.39 61.66 123.31 92.51 832.25 828 4.18 1402.4 1260.5 2662.94 506.59 506.56 3676.1
18 95.14 190.29 285.43 63.44 126.89 95.19 856.38 852 4.3 1466.7 1294.1 2760.74 521.29 521.35 3803.4
19 103.1 206.14 309.21 68.73 137.46 103.1 927.74 923 4.655 1577.9 1405.5 2983.45 564.66 564.72 4112.8
20 103.4 206.81 310.22 68.95 137.89 103.5 930.72 926 4.68 1599.6 1408.7 3008.26 566.5 566.66 4141.4
21 105.3 210.61 315.92 70.2 140.41 105.4 947.8 943 4.775 1624 1430.2 3054.26 576.82 576.84 4207.9
22 106.5 213.06 319.6 71.03 142.07 106.6 958.86 954 4.82 1625.5 1463.4 3088.84 583.57 583.63 4256
23 107.1 214.18 321.28 71.4 142.79 107.1 963.85 959 4.845 1646.3 1464.2 3110.51 586.74 586.74 4284
24 116.1 232.27 348.41 77.44 154.88 116.2 1045.3 1040 5.23 1745.4 1628.3 3373.68 636.17 636.17 4646
S. Acharya et al.
 Table 12 The analysis of without VPL effect presence in 10 generating units
Time Thermal power PV power Wind power Total Load Power Without VPL
(h) power demand loss
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 p(MW) (MW) (MW) Fuel Thermal Thermal PV Wind Total cost
cost ($/ emission cost ($/ cost cost ($/ ($/h)
h) cost (lb/h) h) ($/ kW)
kW)

1 39.9 79.87 119.8 159.7 199.7 33.5 67 100.5 33.5 66.99 900.49 900 0.499 1522.87 1386.2 2909.08 1081.9 543.17 4534.203
2 49.4 98.76 148.1 197.5 246.9 41.3 82.63 123.95 41.3 82.62 1112.5 1110 1.526 1858.65 1736 3594.61 1310.2 657.3 5562.117
3 52.7 105.4 158.1 210.7 263.4 44.1 88.14 132.21 44.1 88.13 1186.9 1184 1.863 2015.11 1830.4 3845.46 1390.7 697.51 5933.639
4 55.9 111.8 167.8 223.7 279.6 46.8 93.65 140.48 46.8 93.64 1260.3 1258 1.995 2126.95 1960.6 4087.53 1471.1 737.73 6296.341
5 59.2 118.5 177.7 236.9 296.1 49.6 99.16 148.74 49.6 99.15 1334.6 1332 2.571 2267.27 2074.3 4341.62 1551.5 777.95 6671.075
6 62.5 125.1 187.6 250.1 312.7 52.3 104.7 157 52.3 104.7 1409 1406 2.865 2413.47 2185.5 4598.97 1632 818.17 7049.111
7 65.8 131.7 197.5 263.4 329.2 55.1 110.2 165.26 55.1 110.2 1483.4 1480 3.107 2536.24 2308.9 4845.17 1712.4 858.38 7415.934
8 65.8 131.7 197.5 263.4 329.2 55.1 110.2 165.26 55.1 110.2 1483.4 1480 3.135 2538.99 2309.7 4848.69 1712.4 858.39 7419.478
9 66.8 133.6 200.4 267.2 334 55.8 111.7 167.5 55.8 111.7 1504.5 1500 3.139 2539.34 2355.8 4895.11 1734.1 869.25 7498.467
10 69.1 138.3 207.4 276.6 345.7 57.8 115.7 173.53 57.8 115.7 1557.7 1554 3.492 2645.36 2436.4 5081.71 1792.8 898.6 7773.126
Optimization of cost and emission for dynamic load dispatch problem with...

11 72.5 144.9 217.4 289.8 362.3 60.6 121.2 181.79 60.6 121.2 1632.1 1628 3.627 2781.65 2558.5 5340.18 1873.2 938.81 8152.241
12 72.5 144.9 217.4 289.8 362.3 60.6 121.2 181.79 60.6 121.2 1632.1 1628 3.725 2811.19 2559.4 5370.63 1873.2 938.82 8182.695
13 75.9 151.9 227.8 303.7 379.7 63.5 127 190.5 63.5 127 1710.5 1706 3.814 2926 2664.6 5590.6 1958 981.21 8529.828
14 79.1 158.3 237.4 316.5 395.6 66.1 132.2 198.32 66.1 132.2 1781.8 1776 4.235 3020.35 2794.5 5814.82 2034.1 1019.3 8868.206
15 79.1 158.3 237.4 316.5 395.6 66.1 132.2 198.32 66.1 132.2 1781.9 1776 4.423 3027.79 2795.5 5823.34 2034.1 1019.3 8876.735
16 79.1 158.3 237.2 316.2 395.3 66.1 132.2 198.39 66.1 133 1781.9 1776 4.446 3028.06 2796 5824.04 2034.1 1019.3 8877.444
17 85.7 171.3 257 342.7 428.4 71.6 143.2 214.85 71.6 143.2 1929.6 1924 4.885 3300.92 3024 6324.95 2195 1099.7 9619.615
18 85.7 171.3 257 342.7 428.4 71.6 143.2 214.85 71.6 143.2 1929.6 1924 4.91 3301.27 3025.7 6326.96 2195 1099.7 9621.643
19 85.7 171.5 257.1 342.7 428.4 71.6 143.2 214.86 71.6 143.2 1930 1924 4.924 3301.28 3027 6328.24 2195 1099.7 9622.949
20 92.3 184.6 276.8 369.1 461.4 77.1 154.2 231.37 77.1 154.2 2078.3 2072 5.478 3545.29 3258.3 6803.61 2355.9 1180.1 10,339.6
21 92.3 184.6 277 369.4 461.5 77.1 154.2 231.37 77.1 154.2 2078.9 2072 5.482 3561.84 3259.4 6821.26 2355.9 1180.1 10,357.26
22 92.3 184.6 276.8 369.1 461.4 77.1 154.2 231.6 77.1 154.6 2078.9 2072 5.581 3561.87 3260.1 6821.96 2355.9 1180.1 10,357.98
23 95.7 191.3 287 382.6 478.3 79.9 159.8 239.63 79.9 159.7 2153.7 2146 5.798 3654.49 3414.4 7068.91 2436.3 1220.3 10,725.56
24 98.9 197.8 296.7 395.6 494.5 82.6 165.3 247.9 82.6 165.2 2227.1 2220 6.114 3655.97 3655.7 7311.7 2516.7 1260.6 11,088.99

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Table 13 The analysis of with VPL effect presence in 10 generating units
Time Thermal power PV power Wind power Total Load Power With VPL
(h) power demand loss
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 p(MW) (MW) (MW) Fuel Thermal Thermal PV Wind Total
cost emission cost cost ($/ cost ($/ cost $/ cost ($/
($/h) (lb/h) h) kW) kW) h)

1 40 80 120 160 200 33.5 67 100.5 33.5 67 901.5 900 2.447 1523.2 1388.7 2911.9 1119.3 554.7 4585.95
2 49.4 98.76 148.1 197.5 246.9 41.317 82.63 124 41.3 82.63 1112.6 1110 3.353 1872.1 1750.7 3622.7 1347.2 669.5 5639.41
3 52.7 105.4 158.1 210.7 263.4 44.072 88.14 132.2 44.1 88.13 1186.9 1184 3.724 1992 1876.8 3868.7 1428.2 708.8 6005.76
4 55.9 111.8 167.8 223.7 279.6 46.827 93.65 140.5 46.8 93.65 1260.3 1258 4.164 2115.1 1993.6 4108.7 1508.7 749.5 6366.87
5 59.2 118.5 177.7 236.9 296.1 49.581 99.16 148.7 49.6 99.16 1334.7 1332 4.427 2241.5 2103.5 4345 1589.1 790.1 6724.2
6 62.5 125.1 187.6 250.1 312.7 52.336 104.7 157 52.3 104.7 1409 1406 4.828 2385.2 2219.6 4604.8 1670.7 830.5 7105.98
7 65.9 131.8 197.7 263.6 329.5 55.089 110.2 165.3 55.1 110.2 1484.4 1480 5.071 2502.4 2334.7 4837.1 1749.3 870.4 7456.77
8 65.9 131.8 197.7 263.6 329.5 55.089 110.2 165.3 55.1 110.2 1484.4 1480 5.112 2507.9 2345.2 4853.1 1750.9 871.5 7475.45
9 66.7 133.5 200.2 266.9 333.7 55.833 111.7 167.5 55.8 111.7 1503.5 1500 5.186 2529.1 2377.8 4906.9 1771.8 881 7559.6
10 69.1 138.3 207.4 276.6 345.7 57.844 115.7 173.5 57.8 115.7 1557.8 1554 5.452 2634.1 2455.4 5089.6 1831.9 910.2 7831.66
11 72.5 145 217.6 290.1 362.6 60.599 121.2 181.8 60.6 121.2 1633.1 1628 5.657 2769.5 2576.7 5346.1 1911 951.3 8208.41
12 72.5 145 217.6 290.1 362.6 60.609 121.2 181.8 60.6 121.2 1633.2 1628 5.859 2770.5 2578.6 5349.1 1912.1 951.9 8213.12
13 75.9 151.9 227.8 303.7 379.7 63.502 127 190.5 63.5 127 1710.5 1706 6.026 2886.2 2705.7 5591.9 1995.1 993.6 8580.57
14 79.1 158.1 237.2 316.2 395.3 66.108 132.2 198.3 66.1 132.2 1780.9 1776 6.209 3022.1 2815.9 5838 2071.9 1031 8941.06
15 79.1 158.1 237.2 316.2 395.3 66.107 132.2 198.3 66.1 132.2 1780.9 1776 6.341 3024 2820.6 5844.5 2072.1 1031 8947.96
16 79.1 158.1 237.2 316.2 395.3 66.108 132.2 198.3 66.2 132.3 1781 1776 6.48 3024.1 2821.8 5845.9 2073.1 1032 8950.6
17 85.7 171.5 257.2 343 428.7 71.616 143.2 214.8 71.6 143.2 1930.6 1924 6.733 3277.3 3058.3 6335.6 2232.7 1113 9681.09
18 85.8 171.5 257.3 343 428.7 71.616 143.2 214.8 71.6 143.2 1930.8 1924 6.771 3277.9 3058.6 6336.5 2233.4 1113 9682.74
19 85.7 171.5 257.2 343 428.7 71.617 143.2 214.9 71.6 143.5 1930.9 1924 6.961 3279.6 3058.6 6338.2 2235 1113 9686.01
20 92.3 184.6 276.8 369.1 461.4 77.126 154.3 231.4 77.1 154.2 2078.4 2072 7.472 3527.9 3286.2 6814.2 2392.8 1192 10,399
21 92.3 184.7 277 369.4 461.7 77.125 154.2 231.4 77.1 154.2 2079.4 2072 7.527 3530.6 3300.2 6830.8 2392.9 1192 10,415.9
22 92.3 184.6 276.8 369.1 461.4 77.126 154.3 231.4 77.1 154.2 2078.3 2072 7.569 3542.2 3303.4 6845.6 2394.1 1192 10,432.2
23 95.6 191.2 286.8 382.4 477.9 79.879 159.8 239.6 79.9 159.7 2152.7 2146 7.78 3597.4 3459.1 7056.4 2473.7 1233 10,762.9
24 99 197.9 296.9 395.8 494.8 82.634 165.3 247.9 82.6 165.3 2228.1 2220 8.112 3639.5 3664.3 7303.8 2553.6 1275 11,131.9
S. Acharya et al.
 Table 14 The analysis of without VPL effect presence in 11 generating units
Time Thermal power PV power Wind power Total Load Power Without VPL
(h) power demand loss
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 p(MW) (MW) (MW) Fuel Thermal Thermal PV
cost emission cost cost ($/ cost ($/
($/h) (lb/h) h) kW)

1 26.7 53.3 79.99 106.7 133.3 53.62 95.54 137.5 74.06 126.7 190.1 1077.43 1077 2.38 1786.7 1713.2 3499.9 1409.5 1902.3 6811.7
2 28.7 57.3 86.02 114.7 143.4 56.39 101.1 145.8 78.47 135.5 203.3 1150.66 1152 2.76 1931.2 1814.4 3745.7 1511.5 2031.1 7288.3
3 30.2 60.3 90.52 120.7 150.9 60.33 108 157.6 81.77 142.1 213.2 1215.62 1208 3.03 2030.3 1894.7 3925.1 1605.6 2127.3 7658
4 31 62 92.94 123.9 154.9 61.18 110.7 160.1 83.53 145.7 218.5 1244.35 1238 3.18 2095.5 1939.2 4034.6 1626.1 2178.9 7839.6
5 32.1 64.2 96.23 128.3 160.4 64.05 116.4 168.8 85.94 150.5 225.7 1292.52 1279 3.39 2148.4 2002.9 4151.3 1676.1 2249.3 8076.7
6 32.7 65.3 98 130.7 163.3 64.62 117.5 170.5 87.24 153.1 229.6 1312.57 1301 3.5 2191.8 2032.8 4224.6 1718.2 2287.1 8229.9
7 32.7 65.4 98.08 130.8 163.5 64.64 117.8 170.5 87.3 153.2 229.8 1313.64 1302 3.51 2206.1 2032.9 4238.9 1724 2288.8 8251.7
8 32.9 65.8 98.65 131.5 164.4 64.9 118.1 171.3 87.71 154 231 1320.26 1309 3.54 2216.5 2049.3 4265.8 1726.2 2300.8 8292.9
9 33.8 67.7 101.5 135.4 169.2 66.49 121.3 176.1 89.83 158.3 237.4 1357.01 1345 3.72 2274.6 2105.9 4380.4 1798.2 2362.7 8541.3
10 34.1 68.3 102.4 136.6 170.7 67.36 123 178.7 90.47 159.5 239.3 1370.5 1356 3.77 2284.8 2126.8 4411.6 1815 2381.6 8608.2
Optimization of cost and emission for dynamic load dispatch problem with...

11 34.3 68.7 103 137.3 171.6 68.01 124.3 180.6 90.88 160.4 240.6 1379.72 1363 3.81 2285.1 2144.4 4429.6 1838.4 2393.6 8661.6
12 46.3 92.7 139 185.3 231.7 91.91 171.1 252.3 117.2 213.1 319.6 1860.29 1811 6.05 3086.8 2853.9 5940.7 2510.7 3163.2 11,615
13 47 94 140.9 187.9 234.9 92.58 173.5 254.3 118.7 215.9 323.9 1883.48 1835 6.17 3088.9 2918.4 6007.3 2568.5 3204.5 11,780
14 47.8 95.7 143.5 191.3 239.2 93.97 176.2 258.5 120.5 219.7 329.5 1915.96 1867 6.33 3154.1 2968.7 6122.8 2579.4 3259.4 11,962
15 49 97.9 146.9 195.8 244.8 97.18 181.7 268.1 123 224.6 335.9 1964.94 1909 6.54 3233.3 3040.1 6273.4 2673.2 3331.6 12,278
16 52.3 105 157 209.4 261.7 103.1 194.4 285.8 130.4 239.4 359.2 2097.31 2035 7.17 3452.5 3228.8 6681.3 2848.9 3548.1 13,078
17 54.2 108 162.6 216.9 271.1 107.3 202.9 298.5 134.5 247.7 371.5 2175.62 2105 7.52 3569 3340.1 6909.2 2964.4 3668.3 13,542
18 55.1 110 165.3 220.4 275.5 107.9 204 300.2 136.5 251.6 377.3 2203.96 2138 7.69 3643.7 3387.6 7031.3 3006.4 3725 13,763
19 56 112 168 224 280 110.2 208.8 307.3 138.5 255.6 383.3 2243.78 2172 7.86 3690.2 3442 7132.1 3075.3 3783.4 13,991
20 57.6 115 172.9 230.5 288.1 113.4 215 316.7 142 262.6 393.9 2307.93 2232 8.16 3795.5 3553.9 7349.3 3141.7 3886.5 14,378
21 60.6 121 181.8 242.4 303 118.6 225.5 332.4 148.5 275.7 413.5 2423.13 2343 8.71 3941.7 3765.1 7706.8 3341 4077.2 15,125
22 61 122 182.9 243.9 304.8 119.5 227.3 335 149.4 277.3 416 2438.97 2357 8.78 3982.2 3782.2 7764.4 3354 4101.2 15,220
23 61 122 183 244 305 120.4 229.1 337.8 149.4 277.4 416.2 2445.25 2358 8.78 3996.6 3770.3 7766.9 3354.8 4102.9 15,225
24 64.8 130 194.4 259.2 324 127.5 243.4 359.2 157.8 294.1 441.2 2595.32 2500 9.5 4231.3 4007.3 8238.6 3572.6 4346.9 16,158

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Table 15 The analysis of with VPL effect presence in 11 generating units
Time Thermal power PV power Wind power Total Load Power With VPL
(h) power demand loss
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 p(MW) (MW) (MW) Fuel Thermal Thermal PV Wind Total
cost emission cost ($/ cost cost ($/ cost
($/h) cost (lb/h) h) ($/ kW) ($/h)
kW)

1 26.9 53.73 80.59 107.5 134.3 53.44 96.28 136.1 74.32 127 190.57 1080.7 1077 2.59 1787 1728.7 3515.7 1646.7 1917.4 7079.8
2 28.9 57.75 86.62 115.5 144.4 57.22 103.8 147.5 78.73 135.9 203.8 1160 1152 2.96 1930.5 1830.9 3761.5 1761.4 2013 7535.9
3 30.4 60.75 91.12 121.5 151.9 59.72 108.8 155 82.03 142.5 213.69 1217.3 1208 3.24 2028.9 1911.9 3940.8 1847 2128.2 7916.1
4 31.2 62.36 93.54 124.7 155.9 61 111.4 158.8 83.79 146 218.98 1247.6 1238 3.39 2081.1 1969.4 4050.4 1892.9 2179.8 8123.1
5 32.3 64.55 96.83 129.1 161.4 63.44 116.5 166.1 86.21 150.8 226.22 1293.4 1279 3.6 2143.2 2023.9 4167.1 1955.6 2250.2 8372.9
6 32.9 65.73 98.6 131.5 164.3 65.31 120 171.7 87.5 153.4 230.1 1321.1 1301 3.71 2175.7 2064.7 4240.4 1989.2 2288.3 8517.9
7 32.9 65.79 98.68 131.6 164.5 65.37 120.1 171.9 87.56 153.5 230.27 1322.1 1302 3.71 2208.4 2046.4 4254.7 1990.7 2299.2 8544.7
8 33.1 66.16 99.24 132.3 165.4 65.4 121 172 87.97 154.3 231.51 1328.4 1309 3.75 2210.8 2070.8 4281.6 2001.4 2306.5 8589.5
9 34 68.09 102.1 136.2 170.2 66.89 123.2 176.5 90.09 158.6 237.86 1363.8 1345 3.92 2273 2133.3 4406.2 2056.5 2358.8 8821.5
10 34.3 68.68 103 137.4 171.7 68.34 126.1 180.8 90.74 159.9 239.81 1380.7 1356 3.98 2287.8 2139.7 4427.4 2073.3 2389.3 8890
11 34.5 69.06 103.6 138.1 172.6 68.55 126.5 181.4 91.15 160.7 241.04 1387.3 1363 4.02 2289.1 2171.2 4460.4 2084 2394.5 8938.9
12 46.5 93.07 139.6 186.1 232.7 91.29 172 249.7 117.5 213.4 320.11 1862 1811 6.26 3067.9 2888.6 5956.5 2769 3164.7 11,890
13 47.2 94.36 141.5 188.7 235.9 92.69 174.8 253.9 118.9 216.2 324.34 1888.5 1835 6.38 3081.7 2941.3 6023.1 2805.7 3205.6 12,034
14 48 96.07 144.1 192.1 240.2 95.09 179.6 261.1 120.8 220 329.99 1927.1 1867 6.54 3165 2973.5 6138.6 2854.6 3260.9 12,254
15 49.2 98.32 147.5 196.6 245.8 96.28 182 264.6 123.3 224.9 337.4 1965.9 1909 6.75 3232.3 3056.8 6289.2 2918.8 3332.5 12,540
16 52.5 105.1 157.6 210.2 262.7 103.6 196.6 286.6 130.7 239.8 359.64 2104.9 2035 7.38 3436.2 3260.9 6697.1 3111.5 3553.7 13,362
17 54.4 108.8 163.2 217.7 272.1 106.8 203.1 296.3 134.8 248 371.99 2177.2 2105 7.72 3547.6 3377.4 6925 3218.5 3683.5 13,827
18 55.3 110.6 165.9 221.2 276.5 108.1 205.6 300.2 136.7 251.9 377.82 2209.9 2138 7.89 3625.1 3422 7047.1 3269 3730.7 14,047
19 56.2 112.4 168.6 224.8 281 110.5 210.4 307.3 138.7 255.9 383.82 2249.7 2172 8.06 3649.6 3498.3 7147.9 3320.9 3790.1 14,259
20 57.8 115.6 173.5 231.3 289.1 113.3 216.1 315.8 142.3 262.9 394.41 2312.1 2232 8.36 3789.5 3575.6 7365.1 3412.7 3887.1 14,665
21 60.8 121.6 182.4 243.2 304 119.4 228.3 334.1 148.8 276 414 2432.5 2343 8.91 3949.2 3773.4 7722.6 3582.4 4078.6 15,384
22 61.2 122.3 183.5 244.7 305.8 120.2 229.7 336.3 149.6 277.6 416.47 2447.5 2357 8.98 3990.9 3789.2 7780.2 3603.8 4116.4 15,500
23 61.2 122.4 183.6 244.8 306 120.8 231 338.2 149.7 277.8 416.65 2452 2358 8.99 3991 3797.7 7788.7 3605.3 4170.6 15,565
24 65 130 195 260 325 127.8 245 359.2 158 294.5 441.71 2601.2 2500 9.7 4219.3 4035.1 8254.4 3822.4 4348.8 16,426
S. Acharya et al.
Optimization of cost and emission for dynamic load dispatch problem with...

Optimal Cost With VPL Optimal Cost Without VPL

3 1.6

2.5

1.5
Cost S/h

Cost S/h
2

1.5 1.4

0 50 100 150 200 0 50 100 150 200

Iteration Iteration

(a)
Optimal Emission With VPL Optimal Emission Without VPL

2.5 1.5
Cost S/h

2 Cost S/h
1.4

1.5

1.3
0 50 100 150 200 0 50 100 150 200
Iteration Iteration

(b)
Fig. 5 The 6 generating units (a) optimal fuel cost and (b) emission with and without VPL

where, the weight parameter is represented as x in the 1 Cp ðkÞ  q  r  Sw  Vw2

range between 0 and 1, c1 and c2 is represented as a scale twind ¼ tmec ¼ ð12Þ
2 k
factor as well as k1 and k2 is represented as shape factor.
where, k denotes the tip speed of the wind, r indicates the
3.1.2 Wind turbine model radius of turbine blades.
Uncertainty modeling of wind
Usually, the input energy of the wind turbine is expressed The Weibull Probability Density Function (PDF) is used
as: for modeling the wind speed and is expressed as,
 
 
k v k1 v k
Pm ¼ 0:5qSw Vw3 ð10Þ f ðv Þ ¼ exp  0v1 ð13Þ
c c c
The wind turbine’s output mechanical power is calcu-
lated by, where V is represented as wind speed random variables.

P m ¼ C p Pm 3.1.3 Start-up and shut-down constraints

ð11Þ
Pm ¼ 0:5qCp Sw Vw3
The start-up ramp state is given as,
where, q represents the density of air, the wind power
coefficient is denoted as Cp , Sw denoted the wind turbine 0  Pi ðtÞ  Pmin
i Ui ðtÞ ¼ 1; ristart ðtÞ ¼ 1; ð14Þ
blade’s swept area and Vw indicated the speed of the wind. Pi ðtÞ  Pi ðt  1Þ ¼ Pstart
i Ui ðtÞ ¼ 1 ð15Þ
The following equation evaluates the torque produced
by the turbine: where ristart ðtÞ is the variable of start-up ramp rate, while
Ui ðtÞ in the start-up state variable.

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 S. Acharya et al.

Optimal Cost With VPL Optimal Cost Without VPL

1.5
1.7

1.45
1.68
Cost S/h

Cost S/h
1.4
1.66

1.35
1.64

0 50 100 150 200 0 50 100 150 200

Iteration Iteration

(a)
Optimal Emission With VPL Optimal Emission Without VPL
0.75
0.78

0.7
0.76
Cost S/h

Cost S/h
0.65
0.74

0.6
0.72

0 50 100 150 200 0 50 100 150 200

Iteration Iteration

(b)
Fig. 6 The 10 generating units (a) optimal fuel cost and (b) emission with and without VPL

0  Pi ðtÞ  Pmin Vi ðtÞ ¼ 1; rishut ðtÞ ¼ 1; ð16Þ solution sets, the process of optimization is initiated. In the
i
recommended MVO algorithm, every candidate solution
Pi ðtÞ  Pi ðt þ 1Þ ¼ Pshut
i Vi ðtÞ ¼ 1 ð17Þ agrees with the cosmos, and the variables are handled as
where Vi ðtÞ is the state variable of the shut-down process gadgets in the universe. Similarly, to associate the results
and rishut ðtÞ is the ramp rate shut-down variable. and save the best one(s), the MVO has operators. Black and
White holes are randomly generated in the universes to
combine the solutions and cause objects’ movement. The
3.2 Multi-objective cost minimization using MVO
inflation rate is the objective value of the objective func-
algorithm
tion. The growing speed of a universe is defined as the
inflation rate and is evaluated proportionally to the objec-
MVO is considered in the evolutionary algorithm’s family
tive function. Sort the universes at every iteration based on
and is the population-based algorithm. With candidate
inflation rates, and roulette wheel selection (RWS) is used

 Pi ðxÞ  0; i ¼ 1; 2; . . .; m
min Fobj ð xÞ ¼ min Ftotal ; Ffuel ; Femission ; FScost ; Ftwind :::; Fm ; ð18Þ
hi ðxÞ ¼ 0; i ¼ 1; 2; . . .; p

123
Optimization of cost and emission for dynamic load dispatch problem with...

(a)

(b)
Fig. 7 The 11 generating units (a) optimal fuel cost and (b) emission with and without VPL

to get a white hole. Here, m is the number of parameters (variables) and n is

where n dimensional decision space to m dimensional the number of universes (candidate solutions). The matrix of
objective space is represented as F(x) and m denotes the G represents the complete population, that is, the group
decision space. Fi ðx1 Þ and Fi ðx2 Þ are the objective values population, which Pmn represents the element of the matrix G.
of the decision space x1 and x2, respectively. Pareto set
(PS) are the total potential Pareto-best vectors between 3.2.1 Initialize the positions of universe parameters
total feasible solutions in n-dimensional decision space.
Pareto front (PF) is the set of m dimensional objective The universe parameter positions are randomly created
results of PS, which creates the shape in the objective inside the search space by evenly distributed random
space. numbers among 1 and 0 in random initialization.
Step 1: The real power output for each generator is  max 
Pm min
i ðt  1Þ ¼ Pi þ Rm i Pi  Pmin
i ; i ¼ 1; 2; . . .; n; m
represented in each population. The number of generating ¼ 1; 2; . . .; npi
units is utilized for initialization purposes. The universe
parameters matrix is represented as G, which is described ð20Þ
as Eq. (19), A feasible outcome is provided; hence, the results fulfill
2 1 3
P1 P21    Pm 1
the ramp-rate limit condition, satisfy the actual power
6 P12 P22    Pm 7
6 2 7
G ¼ 6. .. .. .. 7 ð19Þ
4 .. . . . 5
P1n P2n  Pm
n

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 S. Acharya et al.

Fig. 8 Pareto effect of 6 generation units

balance constriction, and are placed inside the generator generation of each generator parameter, which is given in
capacity constraint. Eq. (21),

8  min 
min min m
< MaxPi ; Pi ðt  1Þ  DRi; Pi ; Pi ðtÞ\ max
> Pi ; Pi ðt  1Þ  DR
 max 
Pm
i ðtÞ ¼ Min PU i þ URi ; Pi
max
Pm
i ðtÞ\ min Pi ; Pi ðt  1Þ þ UR ð21Þ
>
:
Pmi ðtÞ otherwise

3.2.2 Handling of ramp-rate constraint The start-up and shut-down ramp constraints are con-
sidered for the operation. When the system is offline and is
The ramp-rate limit’s inequality constraints are taken started up, then the power trajectory is minimal, as stated in
 
during every movement of the group by regulating the the second condition Min PU max
i þ URi ; Pi Pmi ðtÞ\

123
Optimization of cost and emission for dynamic load dispatch problem with...

Fig. 9 Pareto effect of 10 generation units

  8 !
min Pmax
i ; Pi ðt  1Þ þ UR . Then on a shut-down con- >
> PLi;j 
>
> L
  if Pm L U m
> P
< i;j
r 1 i ðtÞ  Pi;j  Pi;j1  Pi ðtÞ
straint, the power trajectory will decrease from the point PU
 min  Pm
i;j1
!
Pm i ðtÞ ¼
i ðtÞ\ max Pi ; Pi ðt  1Þ  DR . This condition is sta- >
>
> PLi;j
> U
ted in the above equation at the first condition. At both the : Pi;j1 þ r 1  PU
> otherwise
i;j1
constraint, the power achieved is illustrated above.
ð22Þ
3.2.3 Handling of prohibited operating zones constraint Here r 2 ½0; 1 is a random number in uniform.

The inequality constraints are taken by eliminating the 3.2.4 Handling of constraints
prohibited operating zones by disturbing the ith generator’s
generation randomly at the time of t  1 movement of the Step 1: The equality constraint is taken to meet the demand
group given in Eq. (22), by randomly perturbing the generation, so the power bal-
ance condition is satisfied (Hemeida et al. 2022).

123
 S. Acharya et al.

Fig. 10 Pareto effect of 11 generation units

Equation (23) evaluates the difference in mth member period.where, UðkÞ and VðkÞ denotes the start-up and shut-
power demand constraint, down process, respectively, at the period of k interval. The
X
N start-up and shut-down constraints are given below;
DPm m
Pm " #
demand ðtÞ ¼ Ploss ðtÞ þ Pdemand ðtÞ  i ðtÞ ð23Þ XDR XUR
i¼1 Pk  P vðkÞ  Uðk þ iÞ  Vðk  i þ 1Þ
i¼1 i¼1
The generators share the power demand randomly to
X
UR
fulfill the equality constraint. Until it is achieved, this þ PD ðiÞVðk  i þ 1Þ 8k
generation of updating is repeated. To meet this constraint, i¼1
the problems faced are the starting and shut-down time of 2K ð24Þ
the generator, which is a major concern for thermal " #
X
DR X
UR
generators. Pk  P vðkÞ  Uðk þ iÞ  Vðk  i þ 1Þ
The conditions involved are given below to handle the i¼1 i¼1
shut-down and start-up constraint. At start-up X
DR
PDR þ PU ðiÞVðk þ DR  i þ 1Þ 8k
i¼1 Uðk þ iÞ will be 1 at all hourly requirements except i¼1
at the shut-down time k. At shut-down, the condition is 2K ð25Þ
PUR
i¼1 Vðk  i þ 1Þ will be 1 except at time k on the start-up

123
Optimization of cost and emission for dynamic load dispatch problem with...

Table 16 Fuel cost coefficients of 6, 10, and 11 generator units

 .
Generator ai ð$=hÞ bi ð$=MWhÞ ci $ ðMWÞ2 h ei ðlb=MWhÞ fi ðlb=ðMWÞ2 hÞ Pmin
i ðMWÞ
Pmax
i ðMWÞ

6 generators
1 756.80 38.540 0.1525 0.3300 0.0042 10 125
2 451.32 46.160 0.1060 0.3300 0.0042 10 150
3 1050.0 40.400 0.0280 -0.5455 0.0068 35 225
4 1243.5 38.310 0.0355 -0.5455 0.0068 35 210
5 1658.5 36.328 0.0211 -0.5112 0.00046 130 325
6 1356.6 38.270 0.0180 -0.5112 0.0046 125 315
10 generators
1 1000.403 40.5407 0.12951 0.0174 36.0012 10 55
2 950.606 39.5804 0.10908 0.0178 350.0056 20 80
3 900.705 36.5104 0.12511 0.0162 330.0056 47 120
4 800.705 39.5104 0.12111 0.0168 330.0056 20 130
5 756.799 38.5390 0.15247 0.0148 13.8593 50 160
6 451.325 46.1592 0.10587 0.0163 13.8593 70 240
7 1243.531 38.3055 0.03546 0.0152 40.2669 60 300
8 1049.998 40.3965 0.02803 0.0128 40.2669 70 340
9 1658.569 36.3278 0.02111 0.0136 42.8955 135 470
10 1356.659 38.2704 0.01799 0.0141 42.8955 150 470
11 generators
1 387.85 192.699 0.00762 -0.67767 0.00419 20 250
2 441.62 211.969 0.00838 -0.69044 0.00419 20 210
3 422.57 219.196 0.00523 -0.67767 0.00419 20 250
4 552.50 201.983 0.00140 -0.54551 0.00683 60 300
5 557.75 212.181 0.00154 -0.40060 0.00751 20 210
6 562.18 191.528 0.00177 -0.54551 0.00683 60 300
7 568.39 210.681 0.00195 -0.40006 0.00751 20 215
8 682.93 199.138 0.00106 -0.51116 0.00355 100 455
9 741.22 199.802 0.00117 -0.56228 0.00417 100 455
10 617.83 212.352 0.00089 -0.41116 0.00355 110 460
11 674.61 210.487 0.00098 -0.56228 0.00417 110 465

The above two equations are set for the lower power X
DR
output limit. If a generator is not involved in both start-up Pk  PD ðiÞUðk þ DR  i þ 1Þ
and shut-down processes (processing the power genera- "
i¼1 #
X
DR
tion), then the last term of Eq. 17 and the first term of þ P vðkÞ  Uðk þ iÞ 8k
Eq. 16 are identical. i¼1

X
UR 2K ð27Þ
Pk  PU ðiÞVðk  i þ 1Þ
The above constraints are the start-up and shut-down
"
i¼1 #
X
UR process for higher power generation. Both constraints
þ P vðkÞ  Vðk  i þ 1Þ 8k induce maximum power for the shut-down and start-up
i¼1
process.
2K ð26Þ
Step2: The ith universe’s jth parameter is presented as
Eq. (28),
( j
j Pk R1 \NI ðGi Þ
Gi ¼ ð28Þ
Pij R1  NI ðGi Þ

123
 S. Acharya et al.

Table 17 Emission cost coefficients of 6, 10, and 11 generator units where WEP and TDR are the coefficients, R2 ; R3 ; and R4
are represented as random numbers, jth variable’s upper
Generator ai ðlb=hÞ bi ðlb=MWhÞ ci ððlb=ðMWÞ2 hÞ
bound is denoted as UBj , Pj indicates the jth parameter of
6 generators the best universe and LBj shows the lower bound of jth
1 13.860 0.3300 0.0042 variable. The proposed MVO algorithm flowchart is pre-
2 13.860 0.3300 0.0042 sented in Fig. 2.
3 40.267 -0.5455 0.0068 Over the iterations, TDR is improved compared to WEP
4 40.267 -0.5455 0.0068 to have high exact exploitation around the greatest attained
5 42.900 -0.5112 0.0046 universe. The WEP coefficient’s adaptive formula is
6 42.900 -0.5112 0.0046 determined as Eq. (30),
 
10 generators max  min
1 360.0012 -3.9864 0.04702 WEP ¼ min þq  ð30Þ
Q
2 350.0056 -3.9524 0.04652
3 330.0056 -3.9023 0.04652 Here, min and max are constants, q indicates the current
4 330.0056 -3.9023 0.04652 iteration and Q shows the maximum iterations. The adap-
5 13.8593 0.3277 0.00420 tive formula for TDR coefficient is determined as Eq. (31),
6 13.8593 0.3277 0.00420 q1=z
7 40.2669 -0.5455 0.00680
TDR ¼ 1  ð31Þ
Q1=z
8 40.2669 -0.5455 0.00680
9 42.8955 -0.5112 0.00460
As z rises, the exploitation/local search’s accuracy also
improves. The sooner and more precise local search/ex-
10 42.8955 -0.5112 0.00460
ploitation is obtained in higher z. According to the results,
11 generators
the adaptive values are recommended, and the TDR and
1 33.93 -0.67767 0.00419
WEP are considered constants.
2 24.62 -0.69044 0.00461
Step 4: MVO algorithm optimization procedure initiates
3 33.93 -0.67767 0.00419
with setting many random variables. In the iteration process,
4 27.14 -0.54551 0.00683
objects with huge inflation levels shift to the universe with
5 24.15 -0.40060 0.00751
reduced inflation rates through white/black holes. Each
6 27.14 -0.54551 0.00683
universe leads its objects toward the great universe, and this
7 24.15 -0.40060 0.00751
iteration continues until the stopping criterion is satisfied. In
8 30.45 -0.51116 0.00355
every iteration, the sorting universe is activated, and the
9 25.59 -0.56228 0.00417
quicksort algorithm is adopted, which have oðn2 Þ and
10 30.45 -0.41116 0.00355
oðn log nÞ complexity in the worst and best case, corre-
11 25.59 -0.56228 0.00417
spondingly. Based on the implementation, the RWS is rep-
resented as oðnÞ or oðlog nÞ. Hence, the entire estimation
where Pij indicates the optimal solution of Gi shows ith complexity is given as Eqs. (32) and (33),
universe, NI ðGi Þ is normalized the ith universe’s objective OðMVOÞ ¼ OðLðOðquick sortÞ þ N  D  ðOðRWÞÞÞÞ
function, a random number in [0, 1] is represented as R1 , and
ð32Þ
Pkj denotes the jth parameter of kth universe. Depending on
the normalized objective function, the determination and OðMVOÞ ¼ Oðqðn2 þ N  f  log NÞÞ ð33Þ
selection of white holes are activated using RWS. The white/ Here, D represents the number of objects, a number of
black hole tunnels produce a higher probability of sending universes is indicated as N, and L represents the high
objects and a lower inflation rate. For the maximization iterations.
issues, NI it should be changed to NI [34]. With a set of random universes, the optimization process
Step 3: By employing wormholes, the formulation of is initiated in the MVO algorithm. The white/black holes
this mechanism is introduced for enhancing the objective move the objects in the universes with maximum to low
function is given by Eq. (17), inflation rates in every iteration. In the meantime, all single
8    
< Pj þ TDR   UBj  LBj  R4 þ lbj ; R3 \0:5 universe faces random teleportation toward the best uni-
; R2 \WEP
Pij ¼ Pj  TDR  UBj  LBj  R4 þ lbj ; R3  0:5
: j
Pi R2  WEP
verse in its objects through wormholes. Depending on the
universe sorting mechanism, RWS mechanism, number of
ð29Þ
universes, and number of iterations, the introduced
approach’s computational complexity is evaluated. The

123
Optimization of cost and emission for dynamic load dispatch problem with...

Table 18 Best, worst, and average cost for 6 units

Population size 10 20 30 40 50 100

Without RES and with transmission power loss

Worst 98,150.5 82,819.12 82,225.30 79,012.9 75,678.80 71,234.30
Average 83,678 82,901 76,234 77,567 72,890 70,987
Best 75,890 72,123 69,579 65,013 58,057 56,091
Without RES and transmission power loss
Worst 45,012 42,978 37,654 34,321 29,012 30,345
Average 43,345 39,678 35,234 32,567 27,890 25,123
Best 39,678 35,901 32,987 30,456 25,802 20,468
With RES and with transmission power loss
Worst 58,024 55,680 52,135 49,791 48,012 45,345
Average 56,678 53,923 49,456 47,789 45,013 42,456
Best 54,789 51,012 48,456 42,789 39,012 37,345
With RES and without transmission power loss
Worst 46,021 44,574 38,101 35,890 32,432 29,543
Average 39,589 35,023 32,234 29,098 26,109 25,210
Best 33,054 31,876 29,567 24,765 21,876 19,123

parameters setting of the MOMVO algorithm is given in the load demand is determined, and the generator generates
Table 2. The Pseudo code of the MVO algorithm is the power for the load without any losses. If any losses are
depicted in Table 3. present, the generator should generate additional power.
Also the corresponding fuel and emission cost is evaluated
based on the presence of the valve point effect. The gen-
4 Results and discussion erated power from the presented 6, 10, and 11 generators is
illustrated in Fig. 3.
This section discussed the proposed scheme’s viability, The cost analysis of the 6, 10, and 11 generator units are
performance, and applicability for real applications that considered in Tables 4, 5, 6, 7, 8, and 9 for its corre-
have been tested over two different power system cases. sponding load demand. The fuel and emission cost values
The cases 6, 10, and 11 generator systems comprised here are changed on time, which is varied at the interval of 24 h,
with and without VPL effect. Case 1 comprises only the proving the effectiveness of the proposed technique. The
thermal power generation system, and the next comprises following sub-section presents the optimal cost function
the thermal and PV generation systems. The loss coeffi- and power losses based fitness in the combination of PV
cient scheme evaluates the network loss by calculating the and thermal generator systems.
ramp rate restrictions, power balance constraints and the
VPL effect. For the same model, the results of cost and 4.2 Case 2: DELD with combination of PV, wind
emission are compared with other algorithms to illustrate and thermal generator systems
the proposed system’s competitiveness. Simultaneously,
assume the solar power cost function data as Mcost ¼ In this case, a PV and wind turbine model power generation
0:016ð$=kWÞ Icost ¼ 5000ð$=kWÞ, N ¼ 20 years; and r ¼ with 6, 10, and 11 generation units and its generated power
0:09: The proposed algorithm has been implemented in are presented in Fig. 4. In this PV power generation, the
MATLAB R2016a with 2 GHz core 2duo processor and emission is less, so the generation cost is also very low
4 GB RAM. The results are based on the performance of compared to the previous case. But the installation and
loss, cost, and emissions. maintenance costs will be presented. Based on this, the
installation cost may be higher than other generators.
4.1 Case 1: DELD with thermal generators The cost analysis of the 6, 10, and 11 generator units are
considered in Tables 10, 11, 12, 13, 14, and 15 for its
In this case, the traditional system with thermal generators corresponding load demand. The fuel and emission cost
is used to evaluate the generated power, losses, and cost values are changed on time, which is varied at the interval
analysis with the presence and absence of VPE. Initially, of 24 h, proving the effectiveness of the proposed

123
 S. Acharya et al.

Table 19 Best, worst, and average cost for 10 units

Population size 10 20 30 40 50 100

Without RES and with transmission power loss

Worst 1,36,171 1,31,012 1,30,456 1,28,789 1,26,876 1,14,543
Average 1,30,210 12,89,987 1,26,654 1,24,321 1,22,098 1,10,765
Best 1,25,432 1,24,109 1,22,876 1,20,543 1,18,210 1,02,012
Without RES and transmission power loss population size
Worst 60,345 59,678 58,909 55,876 53,543 51,210
Average 57,987 55,654 52,321 50,012 49,345 48,678
Best 54,912 52,345 49,678 46,901 43,234 45,567
With RES and with transmission power loss
Worst 79,890 76,098 75,765 72,432 70,109 69,876
Average 74,543 73,210 70,987 68,654 66,321 65,098
Best 71,765 70,432 68,109 66,012 64,345 63,678
With RES and without transmission power loss
Worst 59,901 57,234 55,567 53,890 51,123 48,456
Average 55,789 53,012 52,345 51,678 49,987 46,654
Best 53,321 52,098 50,765 49,432 46,109 44,876

technique. In addition, the wind power generation and cost generator is compared with a proposed and existing tech-
values are evaluated in this case. The following sub-section nique like an Oppositional based Chaotic Grasshopper
presents the optimal cost function and power losses based Optimization Algorithm (OCGOA) (Mandal and Roy
fitness in the combination of thermal, PV and wind gen- 2021), Particle Swarm Optimization (PSO) (Vaisakh et al.
erator systems. 2012), Differential Evolution (DE) (Vaisakh et al. 2012),
Estimating fuel charge and discharge repeated 200 times Bacterial Foraging Optimization Algorithm (BFOA) (Vai-
and attained the least values for both with and without sakh et al. 2012), Bacterial Foraging PSO-DE (BPSO-DE)
VPL. One common feature from the convergence curves of (Vaisakh et al. 2012), which is given in Table 21. Table 22
different conditions can be noticed from the 0th to 200th shows the cost for 10 unit generator comparison with a
iteration: the process starts with lower efficiency and proposed and existing technique like OCGOA (Mandal and
maximizes at the end. An excellent property of these Roy 2021), Hybrid differential evolution-based-chemical
graphs is almost zero cost and emission at the final itera- reaction optimization (HCRO) (Roy and Bhui 2016),
tion, demonstrated in Figs. 5, 6, and 7. MRGA (Zhu et al. 2016), Differential harmony search
Figures 8, 9, and 10 show that the Pareto optimality and (DHS) (Li et al. 2019) and Improved Harmony Search
impression of dominance are explained from the given 6, (HIS) (Li et al. 2019).
10, and 11 generation units. The emission function reduces
two objective functions in the domination concept, which
dominates the cost solution if the objective function for 5 Conclusion
emission is better than the cost. The Pareto optimal front or
optimal solution defines that the test will be applied in the In this paper, the dynamic load dispatch issue is optimized
entire search space for all individuals in the population to by the MOMVO algorithm and produces the best cost and
appear with a set of solutions in optimization. The cost and emission control. This algorithm considers the two objec-
emission values obtained in the proposed work are men- tive functions: fuel cost and emission. Renewable energy,
tioned in red. The fuel and emission coefficients data for 6, such as thermal, solar, and wind systems, have been uti-
10, and 11 generating units are given in Tables 16 and 17. lized in this work. Here, the linear and nonlinear con-
The corresponding fuel and emission costs of all the gen- straints are also satisfied as the power generation limits are
erating units are presented based on the two cases. The generated. The 6, 10, and 11 generation units are consid-
best, worst, and average cost for 6, 10, and 11 units is given ered in this system to optimize the emission and cost values
in Tables 18, 19, and 20. Moreover, the cost for 6 unit based on with and without VPL. Thus, the result has shown

123
Optimization of cost and emission for dynamic load dispatch problem with...

Table 20 Best, worst, and average cost for 11 units

Population size 10 20 30 40 50 100

Without RES and with transmission power loss

Worst 69,098 65,321 63,654 60,987 59,890 59,567
Average 66,789 64,012 61,345 59,678 59,901 57,234
Best 65,456 63,123 61,210 60,543 58,876 55,219
Without RES and transmission power loss population size
Worst 43,019 41,876 39,543 37,219 35,876 33,543
Average 41,234 39,901 37,678 36,345 34,012 31,109
Best 40,432 38,765 367,098 35,321 33,654 30,987
With RES and with transmission power loss
Worst 67,210 65,543 63,876 61,109 59,432 57,765
Average 65,098 63,901 61,678 59,345 57,012 55,789
Best 63,456 61,123 59,890 57,567 55,234 53,901
With RES and without transmission power loss
Worst 49,678 47,345 45,012 43,789 41,456 39,123
Average 47,210 45,543 43,876 41,109 39,432 37,765
Best 45,098 43,321 41,654 39,987 37,210 35,543

Table 21 Comparison of cost for 6 unit generator

Algorithms Maximum cost ($/24 h) Minimum cost ($/24 h) Average cost ($/24 h) Simulation time (s)

OCGOA (Mandal and Roy 2021) 313,928.8713 313,906.6535 313,912.2647 19.74

PSO (Vaisakh 2012) NR 314,351.47 NR NR
DE (Vaisakh 2012) NR 314,162.58 NR NR
BFOA (Vaisakh 2012) NR 314,081.65 NR NR
BPSO-DE (Vaisakh 2012) 314,351.47 314,025.37 314,144.52 21.89
MOMVO 312,852.82 312,140.12 312,206.02 19.90

Table 22 Comparison of cost for 10 unit generator

Algorithms Maximum cost ($/24 h) Minimum cost ($/24 h) Average cost ($/24 h) Simulation time (s)

OCGOA (Mandal and Roy 2021) 2478 378.097 2478 342.746 2478 353.127 22.89
HCRO (Roy and Bhui 2016) NR 2,479,931.38 2,479,962 NR
MRGA (Zhu 2016) NR 2,497,000 NR NR
DHS (Li 2019) NR 2500 827.66 NR NR
IHS (Li 2019) NR 2481 884.498 NR NR
MOMVO 2452 476.456 2378 510.630 2393 904.524 21.47

better performance in VPL cases than without. The power intelligence-based technique will be used to optimize the
loss is calculated, and it attained less power loss from all cost and emission for lower and higher-level generators.
the expressions, that the proposed system is better than the
existing one. Compared with existing methods, the pro-
posed approach test results are quite effective and Author contributions All authors read and approved the final
manuscript.
promising, with good emission and a better quality solution
produced by the generation cost. In future, the artificial Funding No funding is provided for the preparation of manuscript.

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 S. Acharya et al.

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(2020) New genetic algorithm for economic dispatch of stand- jurisdictional claims in published maps and institutional affiliations.
alone three-modular microgrid in DongAo Island. Appl Energy
263:114508 Springer Nature or its licensor (e.g. a society or other partner) holds
Zhu Z, Wang J, Baloch MH (2016) Dynamic economic emission exclusive rights to this article under a publishing agreement with the
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Ziane I, Benhamida F, Graa A (2017) Simulated annealing algorithm terms of such publishing agreement and applicable law.
for combined economic and emission power dispatch using
max/max price penalty factor. Neural Comput Appl
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Authors and Affiliations

Srinivasa Acharya1 • S. Ganesan2 • D. Vijaya Kumar1 • S. Subramanian3

& Srinivasa Acharya 2

Department of Electrical and Electronics Engineering,
srinivasaacharya90@gmail.com Government College of Engineering, Salem 636 011, India
3
Department of Electrical Engineering, Faculty of Engineering
1
Department of Electrical and Electronics Engineering, Aditya and Technology, Annamalai University, Annamalai Nagar,
Institute of Technology and Management, Tekkali, Chidambaram 608 002, India
Andhra Pradesh, India

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