A SurrogateAssisted Adaptive Bat Algorithm for LargeScale Economic Dispatch
Abstract
:1. Introduction
 (1)
 We proposed an improved GRNN based on the SAMP sampling strategy for replacing the original objective function in optimization. First, we used GRNN to evaluate the fitness, which is constructed from the population that meets the constraint conditions and is randomly generated. The promising points ${x}_{p}$ randomly sampled from the SAMP search space are 10 percent of the number of the population in each iteration. Then, ${x}_{p}$ are taken to the database, and GRNN is finally updated according the database in every five generations;
 (2)
 We proposed an adaptive bat algorithm to perform LEDproblem optimization. First, by revealing the essence of ${r}_{d}$ in RCBA, we developed the ESE method to evaluate the relationship between the population distribution, fitness value, and ${r}_{d}$ of RCBA. ESE can improve the reliability of RCBA without increasing the algorithm’s complexity. Second, inspired by the principle of the evolutionary factor in ESE, we proposed an average evolutionary factor method to adaptively update ${r}_{d}$. Based on this, an adaptive bat algorithm was proposed, which eliminates the irrationality of the previous piecewise setting.
2. Problem Formulation
 Generation capacity constraints: the real active and reactive outputs of generators should be limited between their minimum and maximum, which means that generators should satisfy the following inequality constraint:$${P}_{j}^{min}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{j}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{j}^{max},{Q}_{j}^{min}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{Q}_{j}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{Q}_{j}^{max},$$
 Power balance constraint: the whole active power output should include the total load demand ${P}_{d}$ and total transmission line loss ${P}_{l}$.$$\sum _{j=1}^{{N}_{g}}{P}_{j}={P}_{d}+{P}_{l},$$The ${P}_{l}$ is calculated by [44]:$${P}_{j}{P}_{dj}{V}_{j}\sum _{k=1}^{{N}_{b}}{V}_{k}({G}_{jk}cos{\theta}_{jk}+{B}_{jk}sin{\theta}_{jk})=0,$$$${Q}_{j}{Q}_{dj}{V}_{j}\sum _{k=1}^{{N}_{b}}{V}_{k}({G}_{jk}sin{\theta}_{jk}{B}_{jk}cos{\theta}_{jk})=0,$$$${P}_{l}=\sum _{m=1}^{{N}_{\mathrm{line}}}{G}_{m}[{V}_{j}^{2}+{V}_{k}^{2}2{V}_{j}{V}_{k}cos{\theta}_{jk}],$$
 Voltage magnitude constraints: the voltage magnitude should be limited from the lower to upper bounds for secure operation.$${V}_{j}^{min}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{V}_{j}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{V}_{j}^{max},\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,{N}_{b}.$$
 Line flow constraints: the security constraint of the transmission line is limited by$${S}_{j}\le {S}_{j}^{max},j=1,\dots ,{N}_{\mathrm{line}\phantom{\rule{4.pt}{0ex}}},$$
 Ramp rate limits: the active output of the generators cannot be suddenly increased or decreased. Thus, it is limited by:$$\left\{\begin{array}{cc}{P}_{j}{P}_{j}^{0}\hfill & \le \phantom{\rule{3.33333pt}{0ex}}U{R}_{j},\hfill \\ {P}_{j}^{0}{P}_{j}\hfill & \le \phantom{\rule{3.33333pt}{0ex}}D{R}_{j},\hfill \end{array}\right.$$
 Prohibited operating zones: the thermal generator’s steam valve operation or bearing vibration makes the cost function discontinuous. Therefore, prohibited operating zones are considered as below:$$\left\{\begin{array}{c}{P}_{j}^{min}\le {P}_{j}\le {P}_{j,1}^{l},\hfill \\ {P}_{j,k1}^{u}\le {P}_{j}\le {P}_{j,k}^{l},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}k=2,\dots ,z,j=1,2\dots ,{N}_{g}\hfill \\ {P}_{j,z}^{u}\le {P}_{j}\le {P}_{j}^{max},\hfill \end{array}\right.$$
3. Related Technology
3.1. Original Hybrid Bat Algorithm RCBA
 (1)
 Initialize bat population, velocity, frequency, loudness, and pulse emission rate;
 (2)
 (3)
 (4)
 (5)
 Generate new fitness;
 (6)
 Update new fitness and position if the solution improves, or update if not;
 (7)
 (8)
 Repeat steps 3 to 7 until the stopping criterion is satisfied.
3.2. Evolutionary State Evaluation Method
 (1)
 Average distance is calculated by the Euclidean metric from the particle i to all the other particles, where ${N}_{p}$ is the population size and D is the number of dimensions, respectively.$${d}_{ip}=\frac{1}{N1}\sum _{j=1,j\ne i}^{{N}_{p}}\sqrt{\sum _{k=1}^{D}{({x}_{i}^{k}{x}_{j}^{k})}^{2}};$$
 (2)
 Evolutionary factor is denoted as the variation in the average distance of the global optimal particle during the optimization process, where ${d}_{\ast}$ is the average distance of the global optimal particle. In addition, the maximum and minimum average distances of all ${d}_{i}$ are defined as ${d}_{max}$ and ${d}_{min}$, respectively.$${f}_{ese}=\frac{{d}_{\ast}{d}_{min}}{{d}_{max}{d}_{min}}\in [0,1];$$
 (3)
 According to the concept of fuzzy classification, ${f}_{ese}$ is classified into four sets, namely ${S}_{1}$, ${S}_{2}$, ${S}_{3}$, and ${S}_{4}$, which represent the states of exploration, exploitation, convergence, and jumping out, respectively.
3.3. General Regression Neural Network
3.4. A SelfAdaptive “Minimizing the Predictor” Strategy
4. Proposed Method (GARCBA)
 (1)
 Initialization: The initial population is generated by pseudorandom number generators when meeting the constraints. The database is built through the initial population and used to build a GRNN for replacing objective function (1) or (2). In addition, the initial parameters include the maximum frequencies ${f}_{max}$, minimum frequencies ${f}_{min}$, velocity ${v}_{0}$, loudness ${A}_{0}$, pulse emission rate ${r}_{0}$, population size ${r}_{0}$, and system load.
 (2)
 (3)
 Update bat frequency, velocity, position, and local search by random black hole model: The frequency, velocity, and position are updated by Equations (12)–(14). The random black hole model is used for local search, which not only enhances the search ability but also increases convergence. Note that ${r}_{d}$ is set as a piecewise parameter that seriously effects the algorithm’s performance.
 (4)
 Evaluate the fitness by the GRNN: The GRNN is used to replace the real objective function for evaluating the fitness, which can greatly reduce the computational time.
 (5)
 Obtain the current best fitness and its position: The best fitness value and the corresponding bat position are predicted by the GRNN at each iteration, where the best fitness value is used for the SAMP sampling strategy of the GRNN, and the best bat position is used in the random black hole model (see Equation (14)).
 (6)
 (7)
 Update GRNN: Within the SAMP sampling strategy, the promising points of 10 percent of the population are randomly generated and evaluated by the real cost function. Then, the promising points are taken in the database. Finally, the GRNN is retrained using the database every five generations.
 (8)
 Estimate the relationship between evolutionary factor and fitness by ESE and adaptively update the effective radius ${r}_{d}$ of the random black hole: The ESE is introduced to clearly show the state of the bat position and fitness in every generation. According to the ESE, the average evolutionary factor is proposed to adaptively update ${r}_{d}$.
 (9)
 Repeat steps 3 to 7 until the stopping criterion is satisfied.
4.1. An Improved GRNN Base on SAMP Sampling Strategy
Algorithm 1 Pseudocode of updating GRNN by SAMP. 

4.2. An Adaptive Bat Algorithm
Algorithm 2 Adaptive bat algorithm. 

5. Simulation Results
5.1. Case 1: Simulation of Standard IEEE 118Bus System
5.1.1. Case 1.1: No ValvePoint Effects Are Included
5.1.2. Case 1.2: All Constraints Are Included
5.2. Case 2: Simulation of Standard IEEE 300Bus System
5.3. Case 3: Simulation of IEEE 40Unit Test System
5.3.1. Case 3.1: Standard of IEEE 40Unit Test System
5.3.2. Case 3.2: ValvePoint Effect and POZs Are Considered
6. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Items  a  b  c  e  f  ${\mathit{P}}_{\mathit{max}}$  ${\mathit{P}}_{\mathit{min}}$  Ramp Rate 

${P}_{1}$  0.0012  1.2420  0  120  0.073  900  100  305/60 
${P}_{2}$  0.0054  5.4050  0  50  0.032  90  10  18/60 
${P}_{3}$  0.0031  3.1250  0  120  0.073  300  30  1/1 
${P}_{4}$  0.0024  2.4150  0  120  0.073  400  40  80/60 
${P}_{5}$  0.0093  9.3460  0  25  0.026  10  1  2/60 
${P}_{6}$  0.0084  8.4030  0  25  0.026  23  3  5/60 
${P}_{7}$  0.0033  3.2890  0  120  00.073  240  30  48/60 
${P}_{8}$  0.0068  6.7570  0  30  0.051  50  5  10/60 
${P}_{8}$  0.0039  3.9220  0  120  0.073  200  20  40/60 
${P}_{10}$  0.0038  3.8460  0  120  0.073  200  20  40/60 
${P}_{11}$  0.0020  2.0370  0  120  0.073  400  90  130/60 
${P}_{12}$  0.0020  2.0320  0  120  0.073  400  90  130/60 
${P}_{13}$  0.0018  1.8180  0  120  0.073  500  50  200/60 
${P}_{14}$  0.0017  1.7330  0  120  0.073  600  50  120/60 
${P}_{15}$  0.0096  9.6150  0  25  0.026  5  1  1/60 
${P}_{16}$  0.0014  1.4140  0  120  0.073  700  50  150/60 
${P}_{17}$  0.0028  2.8410  0  120  0.073  300  30  60/60 
${P}_{18}$  0.0071  7.1430  0  30  0.051  50  5  10/60 
${P}_{19}$  0.0074  7.3530  0  30  0.048  40  4  8/60 
Items  a  b  c  ${\mathit{P}}_{\mathit{max}}$  ${\mathit{P}}_{\mathit{min}}$  Items  a  b  c  ${\mathit{P}}_{\mathit{max}}$  ${\mathit{P}}_{\mathit{min}}$  Items  a  b  c  ${\mathit{P}}_{\mathit{max}}$  ${\mathit{P}}_{\mathit{min}}$ 

${P}_{1}$  0.0018  1.818  0  500  50  ${P}_{20}$  0.001  1.12  0  1300  130  ${P}_{39}$  0.001  1.12  0  1350  135 
${P}_{2}$  0.0031  3.125  0  300  30  ${P}_{21}$  0.0017  1.733  0  600  60  ${P}_{40}$  0.0024  2.415  0  400  40 
${P}_{3}$  0.0024  2.415  0  400  40  ${P}_{22}$  0.0009  1.11  0  2100  210  ${P}_{41}$  0.0018  1.818  0  500  50 
${P}_{4}$  0.0039  3.922  0  200  20  ${P}_{23}$  0.0017  1.733  0  600  60  ${P}_{42}$  0.0018  1.818  0  500  50 
${P}_{5}$  0.0039  3.922  0  250  25  ${P}_{24}$  0.0024  2.415  0  400  40  ${P}_{43}$  0.0031  3.125  0  300  30 
${P}_{6}$  0.0009  1.11  0  2030  203  ${P}_{25}$  0.0039  3.922  0  200  20  ${P}_{44}$  0.0017  1.733  0  600  60 
${P}_{7}$  0.0024  2.415  0  400  40  ${P}_{26}$  0.0017  1.733  0  600  60  ${P}_{45}$  0.0017  1.733  0  600  60 
${P}_{8}$  0.0024  2.415  0  400  40  ${P}_{27}$  0.0031  3.125  0  350  35  ${P}_{46}$  0.0054  5.405  0  137  13.7 
${P}_{9}$  0.0012  1.242  0  800  80  ${P}_{28}$  0.0024  2.415  0  403  40.3  ${P}_{47}$  0.0008  1.1  0  2400  240 
${P}_{10}$  0.0039  3.922  0  200  20  ${P}_{29}$  0.0018  1.818  0  500  50  ${P}_{48}$  0.0054  5.405  0  145  14.5 
${P}_{11}$  0.0031  3.125  0  350  35  ${P}_{30}$  0.0024  2.415  0  400  40  ${P}_{49}$  0.0031  3.125  0  300  30 
${P}_{12}$  0.0039  3.922  0  250  25  ${P}_{31}$  0.0014  1.414  0  700  70  ${P}_{50}$  0.0018  1.818  0  500  50 
${P}_{13}$  0.0018  1.818  0  500  50  ${P}_{32}$  0.0031  3.125  0  350  35  ${P}_{51}$  0.0018  1.818  0  500  50 
${P}_{14}$  0.0031  3.125  0  350  35  ${P}_{33}$  0.0014  1.414  0  700  70  ${P}_{52}$  0.0039  3.922  0  250  25 
${P}_{15}$  0.0031  3.125  0  350  35  ${P}_{34}$  0.0014  1.414  0  700  70  ${P}_{53}$  0.001  1.12  0  1400  140 
${P}_{16}$  0.0031  3.125  0  350  35  ${P}_{35}$  0.0031  3.125  0  300  30  ${P}_{54}$  0.0012  1.31  0  800  80 
${P}_{17}$  0.0039  3.922  0  200  20  ${P}_{36}$  0.0039  3.922  0  200  20  ${P}_{55}$  0.0012  1.242  0  1000  100 
${P}_{18}$  0.0031  3.125  0  300  30  ${P}_{37}$  0.0017  1.733  0  600  60  ${P}_{56}$  0.0054  5.405  0  150  15 
${P}_{19}$  0.001  1.12  0  1300  130  ${P}_{38}$  0.0012  1.31  0  800  80  ${P}_{57}$  0.0054  5.405  0  108  10.8 
Items  a  b  c  e  f  ${\mathit{P}}_{\mathit{max}}$  ${\mathit{P}}_{\mathit{min}}$  Items  a  b  c  e  f  ${\mathit{P}}_{\mathit{max}}$  ${\mathit{P}}_{\mathit{min}}$ 

${P}_{1}$  0.00690  6.73  94.705  100  0.084  114  36  ${P}_{21}$  0.00298  6.63  785.96  300  0.035  550  254 
${P}_{2}$  0.00690  6.73  94.705  100  0.084  114  36  ${P}_{22}$  0.00298  6.63  785.96  300  0.035  550  254 
${P}_{3}$  0.02028  7.07  309.540  100  0.084  120  60  ${P}_{23}$  0.00284  6.66  794.53  300  0.035  550  254 
${P}_{4}$  0.00942  8.18  369.030  150  0.063  190  80  ${P}_{24}$  0.00284  6.66  794.53  300  0.035  550  254 
${P}_{5}$  0.01140  5.35  148.890  120  0.077  97  47  ${P}_{25}$  0.00277  7.10  801.32  300  0.035  550  254 
${P}_{6}$  0.01142  8.05  222.330  100  0.084  140  68  ${P}_{26}$  0.00277  7.10  801.32  300  0.035  550  254 
${P}_{7}$  0.00357  8.03  287.710  200  0.042  300  110  ${P}_{27}$  0.52124  3.33  1055.10  120  0.077  150  10 
${P}_{8}$  0.00492  6.99  391.980  200  0.042  300  135  ${P}_{28}$  0.52124  3.33  1055.10  120  0.077  150  10 
${P}_{9}$  0.00573  6.60  455.760  200  0.042  300  135  ${P}_{29}$  0.52124  3.33  1055.10  120  0.077  150  10 
${P}_{10}$  0.00605  12.9  722.820  200  0.042  300  130  ${P}_{30}$  0.01140  5.35  148.89  120  0.077  97  47 
${P}_{11}$  0.00515  12.9  635.200  200  0.042  375  94  ${P}_{31}$  0.00160  6.43  222.92  150  0.063  190  60 
${P}_{12}$  0.00569  12.8  654.690  200  0.042  375  94  ${P}_{32}$  0.00160  6.43  222.92  150  0.063  190  60 
${P}_{13}$  0.00421  12.5  913.400  300  0.035  500  125  ${P}_{33}$  0.00160  6.43  222.92  150  0.063  190  60 
${P}_{14}$  0.00752  8.84  1760.400  300  0.035  500  125  ${P}_{34}$  0.00010  8.95  107.87  200  0.042  200  90 
${P}_{15}$  0.00752  8.84  1760.400  300  0.035  500  125  ${P}_{35}$  0.00010  8.62  116.58  200  0.042  200  90 
${P}_{16}$  0.00752  8.84  1760.400  300  0.035  500  125  ${P}_{36}$  0.00010  8.62  116.58  200  0.042  200  90 
${P}_{17}$  0.00313  7.97  647.850  300  0.035  500  220  ${P}_{37}$  0.01610  5.88  307.45  80  0.098  110  25 
${P}_{18}$  0.00313  7.95  649.690  300  0.035  500  220  ${P}_{38}$  0.01610  5.88  307.45  80  0.098  110  25 
${P}_{19}$  0.00313  7.97  647.830  300  0.035  550  242  ${P}_{39}$  0.01610  5.88  307.45  80  0.098  110  25 
${P}_{20}$  0.00313  7.97  647.810  300  0.035  550  242  ${P}_{40}$  0.00313  7.97  647.83  300  0.035  150  242 
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Steps  [0, 50]  [50, 100]  [100, 200]  [200, 300]  [300, 400] 
${\mathit{r}}_{\mathit{d}}$  1 × 10${}^{1}$  1 × 10${}^{3}$  1 × 10${}^{4}$  1 × 10${}^{6}$  1 × 10${}^{9}$ 
Steps  [400, 500]  [500, 600]  [600, 700]  [700, 2 × 10${}^{4}$]  
${\mathit{r}}_{\mathit{d}}$  1 × 10${}^{12}$  1 × 10${}^{14}$  1 × 10${}^{17}$  1 × 10${}^{20}$ 
Items  ${\mathit{r}}_{\mathit{h}\mathit{T}1}$  ${\mathit{r}}_{\mathit{h}\mathit{T}2}$  ${\mathit{r}}_{\mathit{h}\mathit{T}3}$  ${\mathit{r}}_{\mathit{h}\mathit{T}4}$  ${\mathit{r}}_{\mathit{h}\mathit{T}5}$  ${\mathit{r}}_{\mathit{h}\mathit{T}6}$  ${\mathit{r}}_{\mathit{h}\mathit{T}7}$  ${\mathit{r}}_{\mathit{h}\mathit{T}8}$  ${\mathit{r}}_{\mathit{h}\mathit{T}9}$  ${\mathit{r}}_{\mathit{h}\mathit{T}10}$ 

Case 1.1  15  8  6  4  1  1 × 10${}^{3}$  1 × 10${}^{5}$  1 × 10${}^{7}$  1 × 10${}^{9}$  1 × 10${}^{13}$ 
Case 1.2  15  8  6  4  1  1 × 10${}^{3}$  1 × 10${}^{5}$  
Case 2  30  20  18  15  10  8  6  4  1 
Items  GARCBA  GA  PSO  CSO 

${P}_{1}$ (MW)  854.77  680.43  638.23  619.31 
${P}_{2}$ (MW)  10.00  86.29  90.00  88.42 
${P}_{3}$ (MW)  80.00  300.00  300.00  294.77 
${P}_{4}$ (MW)  212.89  400.00  322.34  383.81 
${P}_{5}$ (MW)  1.00  9.62  9.93  9.67 
${P}_{6}$ (MW)  3.00  23.00  23.00  22.78 
${P}_{7}$ (MW)  74.53  235.94  240.00  224.40 
${P}_{8}$ (MW)  5.00  48.01  50.00  47.84 
${P}_{8}$ (MW)  20.00  200.00  200.00  152.34 
${P}_{10}$ (MW)  22.08  199.98  176.26  180.89 
${P}_{11}$ (MW)  400.00  398.87  339.88  227.66 
${P}_{12}$ (MW)  324.97  99.69  325.48  185.34 
${P}_{13}$ (MW)  458.58  90.82  406.76  212.99 
${P}_{14}$ (MW)  552.33  263.09  244.77  444.31 
${P}_{15}$ (MW)  1.00  2.80  1.13  2.01 
${P}_{16}$ (MW)  651.13  424.18  525.18  461.15 
${P}_{17}$ (MW)  155.96  269.73  182.59  87.77 
${P}_{18}$ (MW)  5.00  22.49  48.80  18.17 
${P}_{19}$ (MW)  4.00  9.67  8.97  17.46 
${P}_{l}$ (MW)  168.26  96.64  465.38  41.78 
Time (s)  34.38  38.21  47.41  86.13 
Cost (USD)  10,179.52  10,485.59  10,440.37  10,446.03 
Algorithm  Fuel Cost (USD)  Mean Time (s)  

Minimum  Median  Maximum  Average  
GARCBA  10,179.52  10,209.21  10,297.29  10,234.96  36.24 
GA  10,485.59  10,553.75  10,637.39  10,556.12  41.93 
PSO  10,440.37  10,547.67  10,646.60  10,547.59  40.84 
CSO  10,445.87  10,569.72  10,628.48  10,559.04  82.57 
Items  GARCBA  GA  PSO  CSO 

${P}_{1}$ (MW)  835.84  661.81  709.40  844.05 
${P}_{2}$ (MW)  10.97  90.00  90.00  51.81 
${P}_{3}$ (MW)  123.97  300.00  300.00  164.38 
${P}_{4}$ (MW)  275.65  400.00  400.00  251.63 
${P}_{5}$ (MW)  1.00  10.00  10.00  7.56 
${P}_{6}$ (MW)  3.00  23.00  23.00  14.94 
${P}_{7}$ (MW)  159.70  240.00  240.00  174.35 
${P}_{8}$ (MW)  5.00  50.00  50.00  27.46 
${P}_{8}$ (MW)  108.84  200.00  200.00  167.80 
${P}_{10}$ (MW)  23.58  200.00  198.33  137.47 
${P}_{11}$ (MW)  304.82  400.00  392.97  221.78 
${P}_{12}$ (MW)  229.12  397.96  320.45  341.58 
${P}_{13}$ (MW)  331.83  171.04  330.38  366.55 
${P}_{14}$ (MW)  550.53  312.60  207.95  424.08 
${P}_{15}$ (MW)  1.00  3.65  4.08  3.51 
${P}_{16}$ (MW)  660.99  492.14  263.21  263.52 
${P}_{17}$ (MW)  184.97  48.90  52.53  184.45 
${P}_{18}$ (MW)  58.80  7.72  25.46  19.41 
${P}_{19}$ (MW)  8.83  18.24  35.41  13.20 
${P}_{l}$ (MW)  160.5964  359.09  185.23  11.65 
Time (s)  35.37  37.99  39.62  86.13 
Cost (USD)  10,388.99  10,440.68  10,621.19  10,479.25 
Algorithm  Fuel Cost (USD)  Mean Time (s)  

Minimum  Median  Maximum  Average  
GARCBA  10,388.99  10,461.89  10,633.05  10,476.91  33.47 
GA  10,440.68  10,562.40  10,622.96  10,556.10  42.37 
PSO  10,621.19  10,852.95  11,017.31  10,843.59  44.86 
CSO  10,479.25  10,567.57  10,619.97  10,558.45  87.34 
Items  GARCBA  GA  PSO  CSO  Items  GARCBA  GA  PSO  CSO  Items  GARCBA  GA  PSO  CSO 

${P}_{1}$  354.58  393.51  359.43  190.39  ${P}_{20}$  1026.02  1281.87  1300.00  1283.13  ${P}_{39}$  1092.73  1214.48  246.08  1048.35 
${P}_{2}$  293.93  299.97  300.00  300.00  ${P}_{21}$  547.08  577.05  599.45  567.25  ${P}_{40}$  202.05  60.78  139.09  286.29 
${P}_{3}$  392.96  400.00  400.00  400.00  ${P}_{22}$  1493.32  1943.81  1526.57  1692.19  ${P}_{41}$  303.98  482.57  279.35  362.81 
${P}_{4}$  193.14  200.00  200.00  200.00  ${P}_{23}$  302.99  596.04  181.82  508.65  ${P}_{42}$  409.93  500.00  83.22  304.18 
${P}_{5}$  238.45  250.00  250.00  250.00  ${P}_{24}$  275.11  172.61  275.71  339.19  ${P}_{43}$  146.03  269.34  227.43  178.87 
${P}_{6}$  1742.67  2030.00  2030.00  2030.00  ${P}_{25}$  174.65  133.00  178.83  168.13  ${P}_{44}$  264.86  113.62  176.62  287.07 
${P}_{7}$  355.47  397.54  400.00  400.00  ${P}_{26}$  469.87  554.86  238.67  406.77  ${P}_{45}$  417.45  506.30  129.82  203.26 
${P}_{8}$  282.70  400.00  400.00  400.00  ${P}_{27}$  232.69  210.79  310.45  258.76  ${P}_{46}$  94.71  64.01  13.70  45.54 
${P}_{9}$  699.25  800.00  800.00  800.00  ${P}_{28}$  369.19  230.77  235.96  174.28  ${P}_{47}$  1317.24  2221.69  2400.00  1711.67 
${P}_{10}$  168.80  200.00  199.15  200.00  ${P}_{29}$  283.96  322.19  500.00  180.73  ${P}_{48}$  72.96  18.96  139.54  102.89 
${P}_{11}$  257.89  350.00  350.00  350.00  ${P}_{30}$  342.26  269.86  324.31  122.30  ${P}_{49}$  131.27  63.00  56.83  76.28 
${P}_{12}$  133.71  250.00  250.00  250.00  ${P}_{31}$  561.34  115.76  93.65  287.69  ${P}_{50}$  240.62  404.92  271.21  178.89 
${P}_{13}$  425.68  500.00  500.00  500.00  ${P}_{32}$  195.61  199.18  57.93  332.85  ${P}_{51}$  410.88  191.23  87.98  312.90 
${P}_{14}$  210.81  317.64  350.00  350.00  ${P}_{33}$  587.06  300.28  333.68  247.14  ${P}_{52}$  197.84  224.26  121.19  210.31 
${P}_{15}$  169.07  350.00  350.00  350.00  ${P}_{34}$  509.70  81.23  252.35  397.42  ${P}_{53}$  1073.70  1400.00  1385.45  171.85 
${P}_{16}$  233.67  350.00  350.00  350.00  ${P}_{35}$  153.20  212.95  282.33  135.61  ${P}_{54}$  444.97  498.45  299.31  633.26 
${P}_{17}$  104.35  200.00  200.00  200.00  ${P}_{36}$  92.08  147.21  148.11  140.97  ${P}_{55}$  756.39  692.88  900.89  611.65 
${P}_{18}$  199.68  300.00  300.00  300.00  ${P}_{37}$  464.22  516.29  144.46  380.14  ${P}_{56}$  138.40  65.99  134.94  105.02 
${P}_{19}$  1201.78  1300.00  1300.00  1268.31  ${P}_{38}$  677.32  689.86  152.15  444.33  ${P}_{57}$  58.05  68.15  90.23  29.95 
${P}_{l}$ (MW)  664.77  3,379.23  82.21  491.61  
Time (s)  86.09  150.99  199.23  306.92  
Cost (USD)  55,724.11  56,893.90  56,980.33  57,004.06 
Algorithm  Fuel Cost (USD)  Mean Time (s)  

Minimum  Median  Maximum  Average  
GARCBA  55,724.11  58,106.37  60,709.82  57,996.27  88.29 
GA  56,893.91  58,668.93  64,007.34  60,351.17  164.47 
PSO  56,980.33  60,400.96  64,017.93  60,611.33  166.51 
CSO  57,004.06  59,227.10  63,667.94  60,274.10  319.06 
Items  GARCBA  BA [50]  BAPenalty [50]  Items  GARCBA  BA [50]  BAPenalty [50] 

${P}_{1}$  67.3344  113.1233  111.9952  ${P}_{21}$  522.0699  548.6068  523.2853 
${P}_{2}$  86.1292  111.4569  110.9453  ${P}_{22}$  546.3573  545.562  523.2868 
${P}_{3}$  118.6434  120  97.39597  ${P}_{23}$  496.7465  545.9307  523.2973 
${P}_{4}$  172.6246  179.9948  179.7417  ${P}_{24}$  501.5560  543.7959  514.5068 
${P}_{5}$  57.7019  97  88.92837  ${P}_{25}$  546.5400  549.7956  523.2821 
${P}_{6}$  130.0937  139.9736  105.4038  ${P}_{26}$  533.9505  543.9368  523.8991 
${P}_{7}$  292.4075  300  259.6279  ${P}_{27}$  121.3725  10  10.00444 
${P}_{8}$  251.1615  296.7893  284.6572  ${P}_{28}$  78.7507  10.04373  9.999218 
${P}_{9}$  260.2802  292.5603  284.6307  ${P}_{29}$  98.2577  10.00774  9.999577 
${P}_{10}$  245.6100  130.0603  131.9808  ${P}_{30}$  89.8568  96.83174  89.70938 
${P}_{11}$  215.7838  94  168.7988  ${P}_{31}$  183.2669  189.9952  110.7659 
${P}_{12}$  368.1091  94.1694  318.3965  ${P}_{32}$  76.9153  189.8675  191.6123 
${P}_{13}$  250  484.0661  375.8561  ${P}_{33}$  162.4044  190  191.5734 
${P}_{14}$  200  125.0045  394.2805  ${P}_{34}$  159.7818  199.9782  164.8092 
${P}_{15}$  425.2913  125.0941  125.0027  ${P}_{35}$  183.9541  199.9634  165.5802 
${P}_{16}$  214.1350  304.6026  394.2744  ${P}_{36}$  186.7423  200  164.9268 
${P}_{17}$  434.3557  489.5124  489.2821  ${P}_{37}$  101.6365  110  90.73679 
${P}_{18}$  469.7557  489.3235  489.3007  ${P}_{38}$  77.0279  110  111.304 
${P}_{19}$  503.7925  547.7208  511.2816  ${P}_{39}$  105.4998  110  111.1426 
${P}_{20}$  523.5192  549.9241  511.2772  ${P}_{40}$  433.9072  511.3088  511.3018 
Cost (USD)  121,563.2091  123,757.39  122,936.74 
Algorithm  Fuel Cost (USD)  Mean Time (s)  

Minimum  Maximum  Average  
GARCBA  121,563.2091  122,089.762  122,432.1055  32.5984 
BA [50]  123,757.39  128,510.43  125,979.26  NA 
BAPenalty [50]  122,936.74  129,218.58  126,093.09  NA 
MBA [51]  121,578.4856  121,601.0042  121,583.3047  NA 
ESO [52]  122,122.1600  123,143.0700  122,558.4565  NA 
Items  GARCBA  NGWO [10]  PSOLRS [53]  Items  GARCBA  NGWO [10]  PSOLRS [53] 

${P}_{1}$  95.3933  111.3177  111.9858  ${P}_{21}$  522.0699  526.1137  523.4072 
${P}_{2}$  91.3479  112.7551  110.5273  ${P}_{22}$  546.3573  532.1443  523.4599 
${P}_{3}$  106.9435  118.6377  98.5560  ${P}_{23}$  496.7465  536.8421  523.4756 
${P}_{4}$  164.2468  183.3649  182.9622  ${P}_{24}$  501.5560  524.4669  523.7032 
${P}_{5}$  84.5461  91.8097  87.7254  ${P}_{25}$  546.5400  525.2461  523.7854 
${P}_{6}$  121.5171  104.3697  139.9933  ${P}_{26}$  533.9505  529.3289  523.2757 
${P}_{7}$  233.2553  297.6533  259.6628  ${P}_{27}$  121.3725  9.9500  10.0000 
${P}_{8}$  297.5397  289.4349  297.7912  ${P}_{28}$  78.7507  9.9500  10.6251 
${P}_{9}$  271.7046  298.4044  284.8459  ${P}_{29}$  98.2577  9.9500  10.0727 
${P}_{10}$  266.2047  129.3500  130.0000  ${P}_{30}$  89.8568  88.4106  51.3321 
${P}_{11}$  215.7838  241.9702  94.6741  ${P}_{31}$  183.2669  188.9088  189.8048 
${P}_{12}$  368.1091  166.9113  94.3734  ${P}_{32}$  76.9153  188.8126  189.7386 
${P}_{13}$  250  214.8490  214.7369  ${P}_{33}$  162.4044  186.9624  189.9122 
${P}_{14}$  200  215.6690  394.1370  ${P}_{34}$  159.7818  195.0897  199.3258 
${P}_{15}$  425.2913  305.6922  483.1816  ${P}_{35}$  183.9541  171.5047  199.3065 
${P}_{16}$  214.1350  394.6479  304.5381  ${P}_{36}$  186.7423  176.1085  192.8977 
${P}_{17}$  434.3557  494.7618  489.2139  ${P}_{37}$  101.6365  89.5297  109.8628 
${P}_{18}$  469.7557  493.1559  489.6154  ${P}_{38}$  77.0279  89.3589  111.304 
${P}_{19}$  503.7925  512.7416  511.1782  ${P}_{39}$  105.4998  109.3222  92.8751 
${P}_{20}$  523.5192  520.8929  511.7336  ${P}_{40}$  433.9072  512.5412  511.6883 
Cost (USD)  121,768.1229  121,881.81  122,035.7946 
Algorithm  Fuel Cost (USD)  Mean Time (s)  

Minimum  Maximum  Average  
GARCBA  121,768.1229  121,801.0585  121,864.9455  28.3488 
NGWO [10]  121,881.81  NA  122,787.77  NA 
PSOLRS[53]  122,035.7946  NA  122,558.4565  NA 
IGA [54]  121,915.93  NA  122,811.41  NA 
PSO [55]  123,930.45  123,143.0700  124,154.49  933.39 
CJAYA [53]  121,799.88  NA  122,581.85  NA 
CPSO [56]  121,865.23  NA  122,100.87  114.65 
DECSQP [57]  121,741.9793  122,981.5913  122,295.1278  386.1809 
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Pang, A.; Liang, H.; Lin, C.; Yao, L. A SurrogateAssisted Adaptive Bat Algorithm for LargeScale Economic Dispatch. Energies 2023, 16, 1011. https://doi.org/10.3390/en16021011
Pang A, Liang H, Lin C, Yao L. A SurrogateAssisted Adaptive Bat Algorithm for LargeScale Economic Dispatch. Energies. 2023; 16(2):1011. https://doi.org/10.3390/en16021011
Chicago/Turabian StylePang, Aokang, Huijun Liang, Chenhao Lin, and Lei Yao. 2023. "A SurrogateAssisted Adaptive Bat Algorithm for LargeScale Economic Dispatch" Energies 16, no. 2: 1011. https://doi.org/10.3390/en16021011