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Discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res. 93 (2012) 93–105]
Electric Power Systems Research 108 (2014) 340–344
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Short communication
Discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res. 93 (2012) 93–105] H.R. Abdolmohammadi∗ , A. Kazemi Centre of Excellence for Power System Automation and Operation, Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 19 September 2012 Accepted 22 November 2012 Available online 17 December 2013 Keywords: Economic dispatch Prohibited operating zones Multi-fuel effects
a b s t r a c t This short communication presents a discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res.] 93 (2012) 93–105. The discussed paper presented economic dispatch problem by experimenting with five example systems considering 6, 10, 20, 40 and 140-unit test systems considering non-convex cost function with valve point loading effects, multi-fuel effects and prohibited operating zones. However, in the reported results for the 6 and 20-unit test system, the total generation, total loss and the cost quoted were different for the given generation schedule. In this communication, the corrected data of the 6-unit test system and clarification regarding power transmission losses and cost calculations are presented. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The authors are to be highly commended for presenting a very interesting paper [1]. They have provided an economic dispatch (ED) solution by considering valve-point loading effects, multi-fuel effects, prohibited operating zones (POZs) and power transmission loss which is denoted by the B-coefficient method. The feasibility of the continuous quick group search optimizer (CQGSO) algorithm to solve nonlinear and non-smooth ED problem is shown by experimenting with five test systems. However, by carefully examining the simulation results proposed in [1], we find that the system data and the reported results have some errors. In this paper we shall correct the system data and simulation results proposed in discussed paper.
to read as: Fi (Pi ) = ai + bi Pi + ci Pi2 where ai , bi , ci are cost function coefficients of ith unit. Also, when we studied and verified the results reported in the discussed paper for 6-unit test system, we found that there is a major difference between the total transmission loss reported in Table 5 of Ref. [1] and the result obtained by substituting the same values of power outputs of the generators. Coefficients of the Kron’s loss formula in per unit (with a 100 MVA base capacity) can be found in Appendix A of Ref. [2]. When the Boo value of 0.056 is changed to 0.0056 (after experimenting), obtained result is closer to the reported result in Ref. [1] than before. 3. Discussion
2. Correction of the system data The generators’ characteristics of the 6-unit test system are given in [2]. Table 1 shows these characteristics. In this table the units of the ai , bi , ci are reported as $, $/MW and $/MW2 , respectively. Therefore, it is necessary for the fuel input–power output characteristic of each unit in the 6-unit test system: Fi (Pi ) = ai Pi2 + bi Pi + ci
∗ Corresponding author. Tel.: +98 21 73225612; fax: +98 21 73225777. E-mail address: [email protected] (H.R. Abdolmohammadi). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.11.014
In the discussed paper [1], five case studies (6, 10, 20, 40 and 140 generators) of ED problems are performed to assess the efficiency of the proposed CQGSO approach. In these cases, valve-point loading effects, multi-fuel effects and prohibited operating zones are considered in the power system operation. In the test system I and III, some discrepancies are identified. 3.1. Test system I: 6-unit system with prohibited operating zones The first test system studied in the discussed paper [1] is 6unit test system. Table 1 gives the system parameters including fuel cost coefficients and generator capacities for each unit. These data are presented in [2]. The load demand is 1263 MW. From the
H.R. Abdolmohammadi, A. Kazemi / Electric Power Systems Research 108 (2014) 340–344
341
Table 1 Generating units’ characteristics of 6-unit test system (Table 1 of Ref. [2]). Unit
Pi0 (MW)
Pimin (MW)
Pimax (MW)
ai ($)
bi ($/MW)
ci ($/MW2 )
URi (MW/h)
DRi (MW/h)
Prohibited zones (MW)
1 2 3 4 5 6
440 170 200 150 190 110
150 150 20 20 150 50
455 455 130 130 470 120
240 200 220 200 220 190
7.0 10.0 8.5 11.0 10.5 12.0
0.0070 0.0095 0.0090 0.0090 0.0080 0.0075
80 50 65 50 50 50
120 90 100 90 90 90
[210, 240] [350, 380] [90, 110] [140, 160] [150, 170] [210, 240] [80, 90] [110, 120] [90, 110] [140, 150] [75, 85] [100, 105]
Table 2 Dispatch result of the GSO and CQGSO algorithm for 6-unit test system (load = 1263 MW). Unit
GSO [1]
CQGSO [1]
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total generation (MW) Total loss (MW) Exact total generation calculated by discussers (MW) Exact total loss calculated by discussers (MW) Power balance violation (MW)
263.9171a 173.1811 263.9171 139.0505 165.5743 86.6208 1275.415 12.4158 1092.2609 9.7885 180.5276
263.9079a 173.2418 263.9079 139.0529 165.6013 86.5357 1275.4163 12.4163 1092.2475 9.7884 180.5409
1082.4724b
1082.4591b
Pload = a b
Pi − PL (MW)
Violates the ramp rate constraint. Violates the power balance constraint.
Table 3 Generating units’ characteristics of 20-unit test system (Table 2 of Ref. [3]). Unit
ai ($)
bi ($/MW)
ci ($/MW2 )
Pimin (MW)
Pimax (MW)
Unit
ai ($)
bi ($/MW)
ci ($/MW2 )
Pimin (MW)
Pimax (MW)
1 2 3 4 5 6 7 8 9 10
1000 970 600 700 420 360 490 660 765 770
18.19 19.26 19.80 19.10 18.10 19.26 17.14 18.92 18.27 18.92
0.00068 0.00071 0.00650 0.00500 0.00738 0.00612 0.00790 0.00813 0.00522 0.00573
150 50 50 50 50 20 25 50 50 30
600 200 200 200 160 100 125 150 200 150
11 12 13 14 15 16 17 18 19 20
800 970 900 700 450 370 480 680 700 850
16.69 16.76 17.36 18.70 18.70 14.26 19.14 18.92 18.47 19.79
0.00480 0.00310 0.00850 0.00511 0.00398 0.07120 0.00890 0.00713 0.00622 0.00773
100 150 40 20 25 20 30 30 40 30
300 500 160 130 185 80 85 120 120 100
Table 4 Comparison of generation costs for 10-unit test system with 2500 MW load. Method GSO CQGSO
Generation cost ($/h) reported by the authors in [1] 62,456.6332 62,456.6330
simulation results presented in the discussed paper [1] for 6-unit test system, authors mentioned the best costs as 15,442.6607 $/h and 15,442.6608 $/h for GSO and CQGSO algorithm, respectively. Although it may appear that the GSO and CQGSO algorithms outperform all other methods presented, the calculations for total generation and total loss, as made by the present discussers, reveal the discrepancies as given in Table 2. The authors convey that the obtained results reported in Table 5 of Ref. [1] satisfy the prohibited operating zones and ramp rate constraints. Also, transmission losses are considered in this test system. As can be seen from Table 1, the previous power output of unit 1 (P10 ) is 440 MW, the minimum and maximum generation capability is 150 and 455 MW, respectively, and the up-ramp limit (URi ) and down-ramp limit (DRi ) are 80 and 120 MW, respectively. With these data, the generator operation constraint for unit 1 can be defined as follows: Max(150, 440 − 120) ≤ P1 ≤
Min(455, 440 + 80)
Exact generation cost ($/h) found by the discussers 62,456.6364 62,456.6349
Power balance violation (MW) 0.0002 0.0001
This shows that the generation capability of unit 1 is from 320 to 455 MW, due to the ramp rate limits. But the results summarized in Table 2 for unit 1 are 263.9171 and 263.9079 MW for GSO and CQGSO algorithm, respectively. These values of power outputs of unit 1 violate the ramp rate constraint, and therefore these results do not validate the feasibility of the GSO and CQGSO algorithms. Furthermore, the total generation and total transmission loss reported in Table 2 are different from the authors’ values and therefore, the power balance constraint is not satisfied.
3.2. Test system III: 20-unit system with prohibited operating zones In Ref. [1], GSO and CQGSO algorithms are applied to 20-unit test system considering transmission losses. Table 3 gives the system parameters including fuel cost coefficients and generator capacities for each unit. Coefficients of the Kron’s loss formula in per unit
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Table 5 Detailed cost calculation of GSO and CQGSO algorithms for 20-unit test system with 2500 MW load. Unit
Output power (MW) of GSO in [1]
Exact cost calculated by discussers ($/h)
Output power (MW) of CQGSO in [1]
Exact cost calculated by discussers ($/h)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
512.6382 169.1817 126.8581 102.8404 113.6931 73.4903 115.1345 116.4033 100.4915 106.0393 150.3287 292.7353 119.1552 30.8990 115.8099 36.2584 66.8621 87.9949 100.8210 54.3340
10,503.5914 4248.7615 3216.3947 2717.1324 2573.2399 1808.4762 2568.1274 2972.5097 2653.6941 2840.6936 3417.4598 6141.8949 3089.2169 1282.6901 2669.0246 980.6494 1799.5284 2400.0718 2625.3894 1948.0902
512.7303 169.0263 126.8806 102.8723 113.6836 73.5741 115.3037 116.4090 100.4303 106.0581 150.2337 292.7813 119.1165 30.8431 115.8179 36.2542 66.8611 87.9696 100.8088 54.3106
10,505.3310 4245.7312 3216.8773 2717.7745 2573.0520 1810.1656 2571.3355 2972.6284 2652.5118 2841.0721 3415.7372 6142.7493 3088.4667 1281.6271 2669.1816 980.5678 1799.5081 2399.5614 2625.1488 1947.6075
Total fuel cost ($/h)
62,456.6364
(with a 100 MVA base capacity) can be found in [3]. From Table 8 of Ref. [1] for 20-unit test system, although it may appear that the GSO and CQGSO algorithms outperforms all other methods presented, the calculations for generation costs, as made by the present discussers, reveal some discrepancies as given in Table 4. Detailed cost calculation of GSO and CQGSO algorithms for 2500 MW load demand is given in Table 5.
62,456.6349
of unit 1 violates the ramp rate limit constraint. In the 20-unit test system, the reported costs are not exactly correct. Therefore the solutions and costs for 6 and 20-unit test system given in [1] cannot be considered as true ones and they cannot measure the effectiveness of the GSO and CQGSO algorithms in solving the economic dispatch problems. References
4. Conclusions The errors in the 6-unit test system data and the solutions of ED problem for 6 and 20-unit test system in Ref. [1] have been pointed out. In the test system I, total power generation of the 6-unit test system is less than load demand, and thus the power balance constraint of the ED problem is not satisfied. Furthermore, generation
[1] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi, A. Rabiee, Continuous quick group search optimizer for solving non-convex economic dispatch problems, Electric Power Systems Research 93 (2012) 93–105. [2] S. Pothiya, I. Ngamroo, W. Kongprawechnon, Application of multiple Tabu search algorithm to solve dynamic economic dispatch considering generator constraints, Energy Conversion and Management 49 (2008) 506–516. [3] C.-T. Su, C.-T. Lin, New approach with a Hopfield modeling framework to economic dispatch, IEEE Transactions on Power Systems 15 (2000) 541–545.
Short communication
Response to the Discussion of “Continuous quick group search optimizer for solving non-convex economicdispatch problems” [Electr. Power Syst. Res. 93 (2012) 93–105] Mohammad Moradi-Dalvand, Behnam Mohammadi-Ivatloo, Arsalan Najafi, Abbas Rabiee Department of Electrical Engineering, Ashtian Branch, Islamic Azad University, Ashtian, Iran
Abstract This short communication provides a response to the Discussion of “Continuous quick group search optimizer for solving nonconvex economic dispatch problems” [Electr. Power Syst. Res. 93 (2012) 93–105]. 1. Introduction The authors are extremely thankful to discussers for their great interest on the authors’ work and discussion on the paper. In the discussed paper [1], a solution method has been proposed for solution of economic dispatch problems considering nonconvex fuel cost functions, prohibited operating zones and mutli fuel options. The proposed method was verified by implementing on 5 test systems. Respectful discussers had concerns regarding the results of 6-unit and 20-unit test cases. There concerns are replied and clarified in the following sections. 2. 6-Unit test system There was a typesetting error in Table 5 of [1], where in the second part, the third row (P3 ) was pasted instead of first row (P1 ). This typesetting error has been corrected by the publisher in [2]. The obtained optimal value for P1 using GSO and CQGSO algorithms are 447.072154 and 447.076770, respectively. While
E-mail address: [email protected] (M. Moradi-Dalvand). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.11.014
in [1] the corresponding values of P3 (263.9171 for GSO and 263.9079 for CQGSO) were pasted instead of them in typesetting process. Considering the corrected value of the P1 in [2], it can be observed that the obtained results are feasible and do not violate any constraints. The obtained results satisfy the ramp-rate (Max(150,440 − 120) ≤ P1 = 447.076770 ≤ Min(455,440 + 80))and load balance constraints (Pload = Pi − PL = 1263 MW). Table 1 shows the results of this test system with different precision levels, i.e., decimal places. In this Table the obtained results of MATLAB code for output of the generators and total cost (with 15 decimal places) are truncated to 4, 7 and 10 decimal places. 3. 20-Unit test system The respectful discussers have found that there are 0.0002 and 0.0001 MW power balance mismatch in the results of the20-unit test case. We should mention that these mismatches are raised from rounding the power outputs to 4 decimal places. However we have presented the obtained results with rounding the results to 7 and 10 decimal places in Table 1. As it can be observed from this table, the obtained results with the proposed algorithms do not have power mismatch and the presented total cost in [1] can be obtained by truncating the total cost to 4 decimal digits.
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M. Moradi-Dalvand et al. / Electric Power Systems Research 108 (2014) 340–344
Table 1a Representation of the obtained results for 6-unit test case with different precision levels. Unit
GSO Results presented in [1]
1* 2 3 4 5 6 Total power Total cost *
CQGSO Results with 7 decimal places
Results with 10 decimal places
Results presented in [2]
Results with 7 decimal places
Results with 10 decimal places
447.0722 173.1811 263.9171 139.0505 165.5743 86.6208
447.0721543 173.1811430 263.9170988 139.0504578 165.5743327 86.6207956
447.0721542514 173.1811429697 263.9170987534 139.0504577785 165.5743327268 86.6207955647
447.0768 173.2418 263.9079 139.0529 165.6013 86.5357
447.0767698 173.2417988 263.9078752 139.0528959 165.6012516 86.5356792
447.0767698000 173.2417987865 263.9078751509 139.0528959059 165.6012516494 86.5356791884
1275.416 15,442.6607
1275.4159821 15,442.6607086
1275.4159820445 15,442.6607085944
1275.4163 15,442.66
1275.416271 15,442.6608220
1275.4162704811 15,442.6608220157
With the corrected outputs presented in [2].
Table 1b Representation of the obtained results for 20-unit test case with different precision levels. Unit
GSO Results presented in [1]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total power Total cost
CQGSO Results with 7 decimal places
Results with 10 decimal places
Results presented in [1]
Results with 7 decimal places
Results with 10 decimal places
512.6382 169.1817 126.8581 102.8404 113.6931 73.4903 115.1345 116.4033 100.4915 106.0393 150.3287 292.7353 119.1552 30.899 115.8099 36.2584 66.8621 87.9949 100.821 54.334
512.6381732 169.1817262 126.8581232 102.8404059 113.6930530 73.4902667 115.1344654 116.4033252 100.4914805 106.0392906 150.3286856 292.7353022 119.1551886 30.8989622 115.8098505 36.2583896 66.8621270 87.9949245 100.8209794 54.3340017
512.6381732492 169.1817262284 126.8581231871 102.8404059008 113.6930529657 73.4902667228 115.1344654234 116.4033252408 100.4914805155 106.0392906006 150.3286856261 292.7353022227 119.1551885525 30.8989622299 115.8098505084 36.2583896331 66.8621270149 87.9949245013 100.8209794230 54.3340017424
512.7303 169.0263 126.8806 102.8723 113.6836 73.5741 115.3037 116.409 100.4303 106.0581 150.2337 292.7813 119.1165 30.8431 115.8179 36.2542 66.8611 87.9696 100.8088 54.3106
512.7302938 169.0262925 126.8805988 102.8722978 113.6835992 73.5740978 115.3036937 116.4089897 100.4302963 106.0580918 150.233693 292.7812959 119.116492 30.843095 115.817899 36.2541989 66.8610941 87.9695975 100.8087953 54.3105927
512.7302938375 169.0262924910 126.8805987774 102.8722978107 113.6835992100 73.5740978479 115.3036937115 116.4089896833 100.4302962502 106.0580917923 150.2336929572 292.7812958911 119.1164920477 30.8430949975 115.8178990058 36.2541988529 66.8610940848 87.9695975392 100.8087953142 54.3105926981
2591.9687 62,456.6332
2591.9687215 62,456.6332053
2591.9687214886 62,456.6332052574
2591.9650 62,456.6330
2591.9650048 62,456.6330164
2591.9650048003 62,456.6330164208
References [1] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi, A. Rabiee, Continuous quick group search optimizer for solving non-convex economic dispatch problems, Electric Power Systems Research 93 (December) (2012) 93–105.
[2] Erratum to “Continuous quick group search optimizer for solvingnon-convex economic dispatch problems” [Electr. Power Syst. Res. 93 (2012) 93–105], Electric Power Systems Research, in press.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Short communication
Discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res. 93 (2012) 93–105] H.R. Abdolmohammadi∗ , A. Kazemi Centre of Excellence for Power System Automation and Operation, Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 19 September 2012 Accepted 22 November 2012 Available online 17 December 2013 Keywords: Economic dispatch Prohibited operating zones Multi-fuel effects
a b s t r a c t This short communication presents a discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res.] 93 (2012) 93–105. The discussed paper presented economic dispatch problem by experimenting with five example systems considering 6, 10, 20, 40 and 140-unit test systems considering non-convex cost function with valve point loading effects, multi-fuel effects and prohibited operating zones. However, in the reported results for the 6 and 20-unit test system, the total generation, total loss and the cost quoted were different for the given generation schedule. In this communication, the corrected data of the 6-unit test system and clarification regarding power transmission losses and cost calculations are presented. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The authors are to be highly commended for presenting a very interesting paper [1]. They have provided an economic dispatch (ED) solution by considering valve-point loading effects, multi-fuel effects, prohibited operating zones (POZs) and power transmission loss which is denoted by the B-coefficient method. The feasibility of the continuous quick group search optimizer (CQGSO) algorithm to solve nonlinear and non-smooth ED problem is shown by experimenting with five test systems. However, by carefully examining the simulation results proposed in [1], we find that the system data and the reported results have some errors. In this paper we shall correct the system data and simulation results proposed in discussed paper.
to read as: Fi (Pi ) = ai + bi Pi + ci Pi2 where ai , bi , ci are cost function coefficients of ith unit. Also, when we studied and verified the results reported in the discussed paper for 6-unit test system, we found that there is a major difference between the total transmission loss reported in Table 5 of Ref. [1] and the result obtained by substituting the same values of power outputs of the generators. Coefficients of the Kron’s loss formula in per unit (with a 100 MVA base capacity) can be found in Appendix A of Ref. [2]. When the Boo value of 0.056 is changed to 0.0056 (after experimenting), obtained result is closer to the reported result in Ref. [1] than before. 3. Discussion
2. Correction of the system data The generators’ characteristics of the 6-unit test system are given in [2]. Table 1 shows these characteristics. In this table the units of the ai , bi , ci are reported as $, $/MW and $/MW2 , respectively. Therefore, it is necessary for the fuel input–power output characteristic of each unit in the 6-unit test system: Fi (Pi ) = ai Pi2 + bi Pi + ci
∗ Corresponding author. Tel.: +98 21 73225612; fax: +98 21 73225777. E-mail address: [email protected] (H.R. Abdolmohammadi). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.11.014
In the discussed paper [1], five case studies (6, 10, 20, 40 and 140 generators) of ED problems are performed to assess the efficiency of the proposed CQGSO approach. In these cases, valve-point loading effects, multi-fuel effects and prohibited operating zones are considered in the power system operation. In the test system I and III, some discrepancies are identified. 3.1. Test system I: 6-unit system with prohibited operating zones The first test system studied in the discussed paper [1] is 6unit test system. Table 1 gives the system parameters including fuel cost coefficients and generator capacities for each unit. These data are presented in [2]. The load demand is 1263 MW. From the
H.R. Abdolmohammadi, A. Kazemi / Electric Power Systems Research 108 (2014) 340–344
341
Table 1 Generating units’ characteristics of 6-unit test system (Table 1 of Ref. [2]). Unit
Pi0 (MW)
Pimin (MW)
Pimax (MW)
ai ($)
bi ($/MW)
ci ($/MW2 )
URi (MW/h)
DRi (MW/h)
Prohibited zones (MW)
1 2 3 4 5 6
440 170 200 150 190 110
150 150 20 20 150 50
455 455 130 130 470 120
240 200 220 200 220 190
7.0 10.0 8.5 11.0 10.5 12.0
0.0070 0.0095 0.0090 0.0090 0.0080 0.0075
80 50 65 50 50 50
120 90 100 90 90 90
[210, 240] [350, 380] [90, 110] [140, 160] [150, 170] [210, 240] [80, 90] [110, 120] [90, 110] [140, 150] [75, 85] [100, 105]
Table 2 Dispatch result of the GSO and CQGSO algorithm for 6-unit test system (load = 1263 MW). Unit
GSO [1]
CQGSO [1]
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total generation (MW) Total loss (MW) Exact total generation calculated by discussers (MW) Exact total loss calculated by discussers (MW) Power balance violation (MW)
263.9171a 173.1811 263.9171 139.0505 165.5743 86.6208 1275.415 12.4158 1092.2609 9.7885 180.5276
263.9079a 173.2418 263.9079 139.0529 165.6013 86.5357 1275.4163 12.4163 1092.2475 9.7884 180.5409
1082.4724b
1082.4591b
Pload = a b
Pi − PL (MW)
Violates the ramp rate constraint. Violates the power balance constraint.
Table 3 Generating units’ characteristics of 20-unit test system (Table 2 of Ref. [3]). Unit
ai ($)
bi ($/MW)
ci ($/MW2 )
Pimin (MW)
Pimax (MW)
Unit
ai ($)
bi ($/MW)
ci ($/MW2 )
Pimin (MW)
Pimax (MW)
1 2 3 4 5 6 7 8 9 10
1000 970 600 700 420 360 490 660 765 770
18.19 19.26 19.80 19.10 18.10 19.26 17.14 18.92 18.27 18.92
0.00068 0.00071 0.00650 0.00500 0.00738 0.00612 0.00790 0.00813 0.00522 0.00573
150 50 50 50 50 20 25 50 50 30
600 200 200 200 160 100 125 150 200 150
11 12 13 14 15 16 17 18 19 20
800 970 900 700 450 370 480 680 700 850
16.69 16.76 17.36 18.70 18.70 14.26 19.14 18.92 18.47 19.79
0.00480 0.00310 0.00850 0.00511 0.00398 0.07120 0.00890 0.00713 0.00622 0.00773
100 150 40 20 25 20 30 30 40 30
300 500 160 130 185 80 85 120 120 100
Table 4 Comparison of generation costs for 10-unit test system with 2500 MW load. Method GSO CQGSO
Generation cost ($/h) reported by the authors in [1] 62,456.6332 62,456.6330
simulation results presented in the discussed paper [1] for 6-unit test system, authors mentioned the best costs as 15,442.6607 $/h and 15,442.6608 $/h for GSO and CQGSO algorithm, respectively. Although it may appear that the GSO and CQGSO algorithms outperform all other methods presented, the calculations for total generation and total loss, as made by the present discussers, reveal the discrepancies as given in Table 2. The authors convey that the obtained results reported in Table 5 of Ref. [1] satisfy the prohibited operating zones and ramp rate constraints. Also, transmission losses are considered in this test system. As can be seen from Table 1, the previous power output of unit 1 (P10 ) is 440 MW, the minimum and maximum generation capability is 150 and 455 MW, respectively, and the up-ramp limit (URi ) and down-ramp limit (DRi ) are 80 and 120 MW, respectively. With these data, the generator operation constraint for unit 1 can be defined as follows: Max(150, 440 − 120) ≤ P1 ≤
Min(455, 440 + 80)
Exact generation cost ($/h) found by the discussers 62,456.6364 62,456.6349
Power balance violation (MW) 0.0002 0.0001
This shows that the generation capability of unit 1 is from 320 to 455 MW, due to the ramp rate limits. But the results summarized in Table 2 for unit 1 are 263.9171 and 263.9079 MW for GSO and CQGSO algorithm, respectively. These values of power outputs of unit 1 violate the ramp rate constraint, and therefore these results do not validate the feasibility of the GSO and CQGSO algorithms. Furthermore, the total generation and total transmission loss reported in Table 2 are different from the authors’ values and therefore, the power balance constraint is not satisfied.
3.2. Test system III: 20-unit system with prohibited operating zones In Ref. [1], GSO and CQGSO algorithms are applied to 20-unit test system considering transmission losses. Table 3 gives the system parameters including fuel cost coefficients and generator capacities for each unit. Coefficients of the Kron’s loss formula in per unit
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Table 5 Detailed cost calculation of GSO and CQGSO algorithms for 20-unit test system with 2500 MW load. Unit
Output power (MW) of GSO in [1]
Exact cost calculated by discussers ($/h)
Output power (MW) of CQGSO in [1]
Exact cost calculated by discussers ($/h)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
512.6382 169.1817 126.8581 102.8404 113.6931 73.4903 115.1345 116.4033 100.4915 106.0393 150.3287 292.7353 119.1552 30.8990 115.8099 36.2584 66.8621 87.9949 100.8210 54.3340
10,503.5914 4248.7615 3216.3947 2717.1324 2573.2399 1808.4762 2568.1274 2972.5097 2653.6941 2840.6936 3417.4598 6141.8949 3089.2169 1282.6901 2669.0246 980.6494 1799.5284 2400.0718 2625.3894 1948.0902
512.7303 169.0263 126.8806 102.8723 113.6836 73.5741 115.3037 116.4090 100.4303 106.0581 150.2337 292.7813 119.1165 30.8431 115.8179 36.2542 66.8611 87.9696 100.8088 54.3106
10,505.3310 4245.7312 3216.8773 2717.7745 2573.0520 1810.1656 2571.3355 2972.6284 2652.5118 2841.0721 3415.7372 6142.7493 3088.4667 1281.6271 2669.1816 980.5678 1799.5081 2399.5614 2625.1488 1947.6075
Total fuel cost ($/h)
62,456.6364
(with a 100 MVA base capacity) can be found in [3]. From Table 8 of Ref. [1] for 20-unit test system, although it may appear that the GSO and CQGSO algorithms outperforms all other methods presented, the calculations for generation costs, as made by the present discussers, reveal some discrepancies as given in Table 4. Detailed cost calculation of GSO and CQGSO algorithms for 2500 MW load demand is given in Table 5.
62,456.6349
of unit 1 violates the ramp rate limit constraint. In the 20-unit test system, the reported costs are not exactly correct. Therefore the solutions and costs for 6 and 20-unit test system given in [1] cannot be considered as true ones and they cannot measure the effectiveness of the GSO and CQGSO algorithms in solving the economic dispatch problems. References
4. Conclusions The errors in the 6-unit test system data and the solutions of ED problem for 6 and 20-unit test system in Ref. [1] have been pointed out. In the test system I, total power generation of the 6-unit test system is less than load demand, and thus the power balance constraint of the ED problem is not satisfied. Furthermore, generation
[1] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi, A. Rabiee, Continuous quick group search optimizer for solving non-convex economic dispatch problems, Electric Power Systems Research 93 (2012) 93–105. [2] S. Pothiya, I. Ngamroo, W. Kongprawechnon, Application of multiple Tabu search algorithm to solve dynamic economic dispatch considering generator constraints, Energy Conversion and Management 49 (2008) 506–516. [3] C.-T. Su, C.-T. Lin, New approach with a Hopfield modeling framework to economic dispatch, IEEE Transactions on Power Systems 15 (2000) 541–545.
Short communication
Response to the Discussion of “Continuous quick group search optimizer for solving non-convex economicdispatch problems” [Electr. Power Syst. Res. 93 (2012) 93–105] Mohammad Moradi-Dalvand, Behnam Mohammadi-Ivatloo, Arsalan Najafi, Abbas Rabiee Department of Electrical Engineering, Ashtian Branch, Islamic Azad University, Ashtian, Iran
Abstract This short communication provides a response to the Discussion of “Continuous quick group search optimizer for solving nonconvex economic dispatch problems” [Electr. Power Syst. Res. 93 (2012) 93–105]. 1. Introduction The authors are extremely thankful to discussers for their great interest on the authors’ work and discussion on the paper. In the discussed paper [1], a solution method has been proposed for solution of economic dispatch problems considering nonconvex fuel cost functions, prohibited operating zones and mutli fuel options. The proposed method was verified by implementing on 5 test systems. Respectful discussers had concerns regarding the results of 6-unit and 20-unit test cases. There concerns are replied and clarified in the following sections. 2. 6-Unit test system There was a typesetting error in Table 5 of [1], where in the second part, the third row (P3 ) was pasted instead of first row (P1 ). This typesetting error has been corrected by the publisher in [2]. The obtained optimal value for P1 using GSO and CQGSO algorithms are 447.072154 and 447.076770, respectively. While
E-mail address: [email protected] (M. Moradi-Dalvand). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.11.014
in [1] the corresponding values of P3 (263.9171 for GSO and 263.9079 for CQGSO) were pasted instead of them in typesetting process. Considering the corrected value of the P1 in [2], it can be observed that the obtained results are feasible and do not violate any constraints. The obtained results satisfy the ramp-rate (Max(150,440 − 120) ≤ P1 = 447.076770 ≤ Min(455,440 + 80))and load balance constraints (Pload = Pi − PL = 1263 MW). Table 1 shows the results of this test system with different precision levels, i.e., decimal places. In this Table the obtained results of MATLAB code for output of the generators and total cost (with 15 decimal places) are truncated to 4, 7 and 10 decimal places. 3. 20-Unit test system The respectful discussers have found that there are 0.0002 and 0.0001 MW power balance mismatch in the results of the20-unit test case. We should mention that these mismatches are raised from rounding the power outputs to 4 decimal places. However we have presented the obtained results with rounding the results to 7 and 10 decimal places in Table 1. As it can be observed from this table, the obtained results with the proposed algorithms do not have power mismatch and the presented total cost in [1] can be obtained by truncating the total cost to 4 decimal digits.
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M. Moradi-Dalvand et al. / Electric Power Systems Research 108 (2014) 340–344
Table 1a Representation of the obtained results for 6-unit test case with different precision levels. Unit
GSO Results presented in [1]
1* 2 3 4 5 6 Total power Total cost *
CQGSO Results with 7 decimal places
Results with 10 decimal places
Results presented in [2]
Results with 7 decimal places
Results with 10 decimal places
447.0722 173.1811 263.9171 139.0505 165.5743 86.6208
447.0721543 173.1811430 263.9170988 139.0504578 165.5743327 86.6207956
447.0721542514 173.1811429697 263.9170987534 139.0504577785 165.5743327268 86.6207955647
447.0768 173.2418 263.9079 139.0529 165.6013 86.5357
447.0767698 173.2417988 263.9078752 139.0528959 165.6012516 86.5356792
447.0767698000 173.2417987865 263.9078751509 139.0528959059 165.6012516494 86.5356791884
1275.416 15,442.6607
1275.4159821 15,442.6607086
1275.4159820445 15,442.6607085944
1275.4163 15,442.66
1275.416271 15,442.6608220
1275.4162704811 15,442.6608220157
With the corrected outputs presented in [2].
Table 1b Representation of the obtained results for 20-unit test case with different precision levels. Unit
GSO Results presented in [1]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total power Total cost
CQGSO Results with 7 decimal places
Results with 10 decimal places
Results presented in [1]
Results with 7 decimal places
Results with 10 decimal places
512.6382 169.1817 126.8581 102.8404 113.6931 73.4903 115.1345 116.4033 100.4915 106.0393 150.3287 292.7353 119.1552 30.899 115.8099 36.2584 66.8621 87.9949 100.821 54.334
512.6381732 169.1817262 126.8581232 102.8404059 113.6930530 73.4902667 115.1344654 116.4033252 100.4914805 106.0392906 150.3286856 292.7353022 119.1551886 30.8989622 115.8098505 36.2583896 66.8621270 87.9949245 100.8209794 54.3340017
512.6381732492 169.1817262284 126.8581231871 102.8404059008 113.6930529657 73.4902667228 115.1344654234 116.4033252408 100.4914805155 106.0392906006 150.3286856261 292.7353022227 119.1551885525 30.8989622299 115.8098505084 36.2583896331 66.8621270149 87.9949245013 100.8209794230 54.3340017424
512.7303 169.0263 126.8806 102.8723 113.6836 73.5741 115.3037 116.409 100.4303 106.0581 150.2337 292.7813 119.1165 30.8431 115.8179 36.2542 66.8611 87.9696 100.8088 54.3106
512.7302938 169.0262925 126.8805988 102.8722978 113.6835992 73.5740978 115.3036937 116.4089897 100.4302963 106.0580918 150.233693 292.7812959 119.116492 30.843095 115.817899 36.2541989 66.8610941 87.9695975 100.8087953 54.3105927
512.7302938375 169.0262924910 126.8805987774 102.8722978107 113.6835992100 73.5740978479 115.3036937115 116.4089896833 100.4302962502 106.0580917923 150.2336929572 292.7812958911 119.1164920477 30.8430949975 115.8178990058 36.2541988529 66.8610940848 87.9695975392 100.8087953142 54.3105926981
2591.9687 62,456.6332
2591.9687215 62,456.6332053
2591.9687214886 62,456.6332052574
2591.9650 62,456.6330
2591.9650048 62,456.6330164
2591.9650048003 62,456.6330164208
References [1] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi, A. Rabiee, Continuous quick group search optimizer for solving non-convex economic dispatch problems, Electric Power Systems Research 93 (December) (2012) 93–105.
[2] Erratum to “Continuous quick group search optimizer for solvingnon-convex economic dispatch problems” [Electr. Power Syst. Res. 93 (2012) 93–105], Electric Power Systems Research, in press.