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doi: 10.3934/jimo.2022039
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Real-time pricing method for smart grid based on social welfare maximization model

School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

*Corresponding author: Shou-Qiang Du

Received  August 2021 Revised  February 2022 Early access March 2022

With the applications of big data, the research of real-time pricing method for smart grid has become increasingly important. Based on the demand side management and the real-time pricing model, the social welfare maximization model of smart grid is considered. We transform it by Karush-Kuhn-Tucker condition, then the social welfare maximization model is transformed into a nonsmooth equation by Fischer-Burmeister function. Then, taking advantage of simple calculation and small storage, we propose a new smoothing conjugate gradient method to solve real-time pricing problem for smart grid based on the social welfare maximization. Under general conditions, the global convergence of the new proposed method is proved. Finally, the numerical simulation results show the effectiveness of the proposed method for solving the real-time pricing problems for smart grid based on the social welfare maximization.

Citation: Yanxue Yang, Shou-Qiang Du, Yuanyuan Chen. Real-time pricing method for smart grid based on social welfare maximization model. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022039
References:
[1]

M. Al-BaaliA. CaliciottiG. Fasano and M. Roma, Exploiting damped techniques for nonlinear conjugate gradient methods, Math. Methods Oper. Res., 86 (2017), 501-522.  doi: 10.1007/s00186-017-0593-1.

[2]

J. BaiJ. LiF. Xu and H. Zhang, Generalized symmetric ADMM for separable convex optimization, Comput. Optim. Appl., 70 (2018), 129-170.  doi: 10.1007/s10589-017-9971-0.

[3]

D. P. Bertsekas and A. E. Ozdaglar, Pseudonormality and a Lagrange multiplier theory for constrained optimization, J. Optim. Theory Appl., 114 (2002), 287-343.  doi: 10.1023/A:1016083601322.

[4]

W. Cai, The marketing strategy of customer management in electric power companies, Jilin: Jilin University, (2006), 1–69.

[5]

B. ChenX. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function, Math. Program., 88 (2000), 211-216.  doi: 10.1007/PL00011375.

[6]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.

[7]

X. ChenL. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67 (1998), 519-540.  doi: 10.1090/S0025-5718-98-00932-6.

[8]

M. F. P. CostaR. B. FranciscoA. M. A. C. Rocha and E. M. G. P. Fernandes, Theoretical and practical convergence of a self-adaptive penalty algorithm for constrained global optimization, J. Optim. Theory Appl., 174 (2017), 875-893.  doi: 10.1007/s10957-016-1042-7.

[9] R. W. CottleJ.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Boston: Academic Press, 1992. 
[10]

Y. Dai and Y. Gao, Dynamic pricing decision based on distributed generation system, System Engineering, 34 (2016), 70-75. 

[11]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.

[12]

S. Du and Y. Chen, Global convergence of a modified spectral FR conjugate gradient method, Appl. Math. Comput., 202 (2008), 766-770.  doi: 10.1016/j.amc.2008.03.020.

[13]

M. Fahrioglu and F. Alvarado, Designing incentive compatible contracts for effective demand management, IEEE Tansactions on Power Systems, 15 (2000), 1255-1260.  doi: 10.1109/59.898098.

[14]

A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269-284.  doi: 10.1080/02331939208843795.

[15]

C. W. Gellings, The concept of demand-side management for electric utilities, Proceedings of the IEEE, 73 (1985), 1468-1470.  doi: 10.1109/PROC.1985.13318.

[16]

C. Gu and D. Zhu, Global and local convergence of a new affine scaling trust region algorithm for linearly constrained optimization, Acta Math. Sin., 32 (2016), 1203-1213.  doi: 10.1007/s10114-016-4513-8.

[17]

W. W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192.  doi: 10.1137/030601880.

[18]

J. Han, N. Xiu and H. Qi, Nonlinear Complementarity Theory and Algorithm, Shanghai: Shanghai Science and Technology Press, (2006).

[19]

X. Jiang and J. Jian, A sufficient descent Dai-Yuan type nonlinear conjugate gradient method for unconstrained optimization problems, Nonlinear Dynam., 72 (2013), 101-112.  doi: 10.1007/s11071-012-0694-6.

[20]

Y. Jin and Y. Gao, Optimal piecewise real-time pricing strategy for smart grid, Computer Simulation, 33 (2016), 171-205. 

[21]

C. Kanzow, Some nonlinear continuation methods for nonlinear complementarity problems, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 851-868.  doi: 10.1137/S0895479894273134.

[22]

M. Kaucic, A multi-start opposition-based particle swarm optimization algorithm with adaptive velocity for bound constrained global optimization, J. Global Optim., 55 (2013), 165-188.  doi: 10.1007/s10898-012-9913-4.

[23]

L. R. Khaleel and B. A. Mitras, Hybrid whale optimization algorithm with modified conjugate gradient method to solve global optimization problems, Open Access Library Journal, 7 (2020), 1-18.  doi: 10.4236/oalib.1106459.

[24]

J. Kyparisis, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Math. Programming, 32 (1985), 242-246.  doi: 10.1007/BF01586095.

[25]

G. K. H. LarsenN. D. V. Foreest and J. M. A. Scherpen, Power supply-demand balance in a smart grid: An information sharing model for a market mechanism, Applied Mathematical Modelling, 38 (2014), 3350-3360.  doi: 10.1016/j.apm.2013.11.042.

[26]

Y. LiJ. LiY. Dang and Y. Gao, Smoothing Newton algorithm for real-time pricing of smart grid based on KKT conditions, Journal of Systems Science and Mathematical Sciences, 40 (2020), 646-656. 

[27]

Y. LiJ. LiJ. He and S. Zhang, The real-time optimization model of smart grid based on the utility function of the Logistic function, Energy, 224 (2021), 120172.  doi: 10.1016/j.energy.2021.120172.

[28]

J. LinB. XiaoH. ZhangX. Yang and P. Zhao, A novel multitype-users welfare equilibrium based real-time pricing in smart grid, Future Generation Computer System, 108 (2020), 145-160.  doi: 10.1016/j.future.2020.02.013.

[29]

J. Liu, S. Du and Y. Chen, A sufficient descent nonlinear conjugate gradient method for solving M-tensor equations, J. Comput. Appl. Math., 371 (2020), 112709, 11 pp. doi: 10.1016/j.cam.2019.112709.

[30]

P. LiuJ. Jian and X. Jiang, A new conjugate gradient projection method for convex constrained nonlinear equations, Complexity, 2020 (2020), 1-14.  doi: 10.1155/2020/8323865.

[31]

Z. LiuZ. LiP. Zhu and W. Chen, A parallel boundary search particle swarm optimization algorithm for constrained optimization problems, Struct. Multidiscip. Optim., 58 (2018), 1505-1522.  doi: 10.1007/s00158-018-1978-3.

[32]

W. MengX. Tong and M. Chen, Security energy-saving dispatch with $\alpha$-superquantile constraint in wind power integrated system, Power Electronics, 45 (2011), 111-116. 

[33]

A. H. Mohsenian-Rad and A. Leon-Garcia, Optimal residential load control with price prediction in real-time electricity pricing environments, IEEE Transactions on Smart Grid, 1 (2010), 120-133.  doi: 10.1109/TSG.2010.2055903.

[34]

A. H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich and R. Schober, Optimal and autonomous incentive-based energy consumption scheduling algorithm for smart grid, Innovative Smart Grid Technologies. Gaithersburg, MD, (2010). doi: 10.1109/ISGT.2010.5434752.

[35]

J. Nocedal and S. J. Wright, Numerical Optimization, New York: Springer, 1999. doi: 10.1007/b98874.

[36]

J.-S. Pang, Complementarity problem, Handbook of Global Optimization, 2 (1995), 271-338.  doi: 10.1007/978-1-4615-2025-2_6.

[37]

Y. Pei and D. Zhu, On the global convergence of a projective trust region algorithm for nonlinear equality constrained optimization, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1804-1828.  doi: 10.1007/s10114-018-7063-4.

[38]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112.  doi: 10.1016/0041-5553(69)90035-4.

[39]

B. Ramanathan and V. Vittal, A framework for evaluation of advanced direct load control with minimum disruption, IEEE Tansactions on Power Systems, 23 (2008), 1681-1688.  doi: 10.1109/TPWRS.2008.2004732.

[40]

P. SamadiA. H. Mohsenian-RedR. Schober and V. W. S. Wong, Optimal real-time pricing algorithm based on utility maximization for smart grid, IEEE International Conference on Smart Grid Communications, 54 (2010), 415-420.  doi: 10.1109/SMARTGRID.2010.5622077.

[41]

D. Sun and L. Qi, On NCP-Functions, Comput. Optim. Appl., 13 (1999), 201-220.  doi: 10.1023/A:1008669226453.

[42]

Q. TangK. YangD. ZhouY. Luo and F. Yu, A real-time dynamic pricing algorithm for smart grid with unstable energy providers and malicious users, IEEE Internet of Things Journal, 3 (2016), 554-562.  doi: 10.1109/JIOT.2015.2452960.

[43]

J. Tao and Y. Gao, Computing shadow prices with multiple Lagrange multipliers, J. Ind. Manag. Optim., 17 (2021), 2307-2329.  doi: 10.3934/jimo.2020070.

[44]

L. Tao and Y. Gao, Real-time pricing for smart grid with distributed energy and storage: A noncooperative game method considering spatially and temporally coupled constraints, International Journal of Electrical Power and Energy Systems, 115 (2020), 1-8.  doi: 10.1016/j.ijepes.2019.105487.

[45]

H. Wang and Y. Gao, Research on the real-time pricing of smart grid based on nonsmooth equations, Journal of Systems Engineering, 33 (2018), 320-327. 

[46]

H. Wang and Y. Gao, Real-time pricing method for smart grids based on complementarity problem, Journal of Modern Power Systems and Clean Energy, 7 (2019), 1280-1293.  doi: 10.1007/s40565-019-0508-7.

[47]

H. Wang and Y. Gao, Distributed real-time pricing method incorporating load uncertainty based on nonsmooth equations for smart grid, Math. Probl. Eng., 2019 (2019), Art. ID 1498134, 18 pp. doi: 10.1155/2019/1498134.

[48]

S. WangB. Xu and T. Liu, A conjugate gradient method for inverse problems of nonlinear coupled diffusion equations, Journal of Physics: Conference Series, 1634 (2020), 1-7. 

[49]

C. Wu, J. Zhang, Y. Lu and J. Chen, Signal reconstruction by conjugate gradient algorithm based on smoothing $l_1$-norm, Calcolo, 56 (2019), Paper No. 42, 26 pp. doi: 10.1007/s10092-019-0340-5.

[50]

Z. Ye and M. Sun, Bi-directional interation strategy household between demand side resource and electricity supplier, Electric Power Science and Engineering, 33 (2017), 1-6. 

[51]

G. YiX. TongP. ZhouY. Zhai and Z. Ye, Power system economic dispatch under CVaR and EVaR security operation risk management, Power System Protection and Control, 44 (2016), 49-56. 

[52]

K. ZhangK. ZhangX. Luo and X. Tong, Optimal control strategy for hybrid reactive power compensation system in transformer substation, Proceedings of the CSU-EPSA, 31 (2019), 137-142. 

[53]

L. Zhang and W. Zhou, On the global convergence of the Hager-Zhang conjugate gradient method with Armijo line search, Acta Math. Sci., 28 (2008), 840-845. 

[54]

N. ZhangY. LiH. XieR. Xu and C. You, A generalized conjugate gradient method for eigenvalue problems, Scientia Sinica Mathematica, 50 (2020), 1-24. 

[55]

G. Zhou, Convergence properties of a conjugate gradient method with Armijo-type line seareches, (Chinese) Gongcheng Shuxue Xuebao, 25 (2008), 405-410. 

[56]

H. ZhuY. GaoY. Hou and L. Tao, Multi-time slots real-time pricing strategy with power fluctuation caused by operating continuity of smart home appliances, Engineering Applications of Artificial Intelligence, 71 (2018), 166-174.  doi: 10.1016/j.engappai.2018.02.010.

[57]

H. ZhuY. GaoY. Hou and L. Tao, Real-time pricing considering different type of user based on Markov decision processes in smart grid, Systems Engineering-Theory and Practise, 38 (2018), 807-816. 

show all references

References:
[1]

M. Al-BaaliA. CaliciottiG. Fasano and M. Roma, Exploiting damped techniques for nonlinear conjugate gradient methods, Math. Methods Oper. Res., 86 (2017), 501-522.  doi: 10.1007/s00186-017-0593-1.

[2]

J. BaiJ. LiF. Xu and H. Zhang, Generalized symmetric ADMM for separable convex optimization, Comput. Optim. Appl., 70 (2018), 129-170.  doi: 10.1007/s10589-017-9971-0.

[3]

D. P. Bertsekas and A. E. Ozdaglar, Pseudonormality and a Lagrange multiplier theory for constrained optimization, J. Optim. Theory Appl., 114 (2002), 287-343.  doi: 10.1023/A:1016083601322.

[4]

W. Cai, The marketing strategy of customer management in electric power companies, Jilin: Jilin University, (2006), 1–69.

[5]

B. ChenX. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function, Math. Program., 88 (2000), 211-216.  doi: 10.1007/PL00011375.

[6]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.

[7]

X. ChenL. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67 (1998), 519-540.  doi: 10.1090/S0025-5718-98-00932-6.

[8]

M. F. P. CostaR. B. FranciscoA. M. A. C. Rocha and E. M. G. P. Fernandes, Theoretical and practical convergence of a self-adaptive penalty algorithm for constrained global optimization, J. Optim. Theory Appl., 174 (2017), 875-893.  doi: 10.1007/s10957-016-1042-7.

[9] R. W. CottleJ.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Boston: Academic Press, 1992. 
[10]

Y. Dai and Y. Gao, Dynamic pricing decision based on distributed generation system, System Engineering, 34 (2016), 70-75. 

[11]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.

[12]

S. Du and Y. Chen, Global convergence of a modified spectral FR conjugate gradient method, Appl. Math. Comput., 202 (2008), 766-770.  doi: 10.1016/j.amc.2008.03.020.

[13]

M. Fahrioglu and F. Alvarado, Designing incentive compatible contracts for effective demand management, IEEE Tansactions on Power Systems, 15 (2000), 1255-1260.  doi: 10.1109/59.898098.

[14]

A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269-284.  doi: 10.1080/02331939208843795.

[15]

C. W. Gellings, The concept of demand-side management for electric utilities, Proceedings of the IEEE, 73 (1985), 1468-1470.  doi: 10.1109/PROC.1985.13318.

[16]

C. Gu and D. Zhu, Global and local convergence of a new affine scaling trust region algorithm for linearly constrained optimization, Acta Math. Sin., 32 (2016), 1203-1213.  doi: 10.1007/s10114-016-4513-8.

[17]

W. W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192.  doi: 10.1137/030601880.

[18]

J. Han, N. Xiu and H. Qi, Nonlinear Complementarity Theory and Algorithm, Shanghai: Shanghai Science and Technology Press, (2006).

[19]

X. Jiang and J. Jian, A sufficient descent Dai-Yuan type nonlinear conjugate gradient method for unconstrained optimization problems, Nonlinear Dynam., 72 (2013), 101-112.  doi: 10.1007/s11071-012-0694-6.

[20]

Y. Jin and Y. Gao, Optimal piecewise real-time pricing strategy for smart grid, Computer Simulation, 33 (2016), 171-205. 

[21]

C. Kanzow, Some nonlinear continuation methods for nonlinear complementarity problems, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 851-868.  doi: 10.1137/S0895479894273134.

[22]

M. Kaucic, A multi-start opposition-based particle swarm optimization algorithm with adaptive velocity for bound constrained global optimization, J. Global Optim., 55 (2013), 165-188.  doi: 10.1007/s10898-012-9913-4.

[23]

L. R. Khaleel and B. A. Mitras, Hybrid whale optimization algorithm with modified conjugate gradient method to solve global optimization problems, Open Access Library Journal, 7 (2020), 1-18.  doi: 10.4236/oalib.1106459.

[24]

J. Kyparisis, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Math. Programming, 32 (1985), 242-246.  doi: 10.1007/BF01586095.

[25]

G. K. H. LarsenN. D. V. Foreest and J. M. A. Scherpen, Power supply-demand balance in a smart grid: An information sharing model for a market mechanism, Applied Mathematical Modelling, 38 (2014), 3350-3360.  doi: 10.1016/j.apm.2013.11.042.

[26]

Y. LiJ. LiY. Dang and Y. Gao, Smoothing Newton algorithm for real-time pricing of smart grid based on KKT conditions, Journal of Systems Science and Mathematical Sciences, 40 (2020), 646-656. 

[27]

Y. LiJ. LiJ. He and S. Zhang, The real-time optimization model of smart grid based on the utility function of the Logistic function, Energy, 224 (2021), 120172.  doi: 10.1016/j.energy.2021.120172.

[28]

J. LinB. XiaoH. ZhangX. Yang and P. Zhao, A novel multitype-users welfare equilibrium based real-time pricing in smart grid, Future Generation Computer System, 108 (2020), 145-160.  doi: 10.1016/j.future.2020.02.013.

[29]

J. Liu, S. Du and Y. Chen, A sufficient descent nonlinear conjugate gradient method for solving M-tensor equations, J. Comput. Appl. Math., 371 (2020), 112709, 11 pp. doi: 10.1016/j.cam.2019.112709.

[30]

P. LiuJ. Jian and X. Jiang, A new conjugate gradient projection method for convex constrained nonlinear equations, Complexity, 2020 (2020), 1-14.  doi: 10.1155/2020/8323865.

[31]

Z. LiuZ. LiP. Zhu and W. Chen, A parallel boundary search particle swarm optimization algorithm for constrained optimization problems, Struct. Multidiscip. Optim., 58 (2018), 1505-1522.  doi: 10.1007/s00158-018-1978-3.

[32]

W. MengX. Tong and M. Chen, Security energy-saving dispatch with $\alpha$-superquantile constraint in wind power integrated system, Power Electronics, 45 (2011), 111-116. 

[33]

A. H. Mohsenian-Rad and A. Leon-Garcia, Optimal residential load control with price prediction in real-time electricity pricing environments, IEEE Transactions on Smart Grid, 1 (2010), 120-133.  doi: 10.1109/TSG.2010.2055903.

[34]

A. H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich and R. Schober, Optimal and autonomous incentive-based energy consumption scheduling algorithm for smart grid, Innovative Smart Grid Technologies. Gaithersburg, MD, (2010). doi: 10.1109/ISGT.2010.5434752.

[35]

J. Nocedal and S. J. Wright, Numerical Optimization, New York: Springer, 1999. doi: 10.1007/b98874.

[36]

J.-S. Pang, Complementarity problem, Handbook of Global Optimization, 2 (1995), 271-338.  doi: 10.1007/978-1-4615-2025-2_6.

[37]

Y. Pei and D. Zhu, On the global convergence of a projective trust region algorithm for nonlinear equality constrained optimization, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1804-1828.  doi: 10.1007/s10114-018-7063-4.

[38]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112.  doi: 10.1016/0041-5553(69)90035-4.

[39]

B. Ramanathan and V. Vittal, A framework for evaluation of advanced direct load control with minimum disruption, IEEE Tansactions on Power Systems, 23 (2008), 1681-1688.  doi: 10.1109/TPWRS.2008.2004732.

[40]

P. SamadiA. H. Mohsenian-RedR. Schober and V. W. S. Wong, Optimal real-time pricing algorithm based on utility maximization for smart grid, IEEE International Conference on Smart Grid Communications, 54 (2010), 415-420.  doi: 10.1109/SMARTGRID.2010.5622077.

[41]

D. Sun and L. Qi, On NCP-Functions, Comput. Optim. Appl., 13 (1999), 201-220.  doi: 10.1023/A:1008669226453.

[42]

Q. TangK. YangD. ZhouY. Luo and F. Yu, A real-time dynamic pricing algorithm for smart grid with unstable energy providers and malicious users, IEEE Internet of Things Journal, 3 (2016), 554-562.  doi: 10.1109/JIOT.2015.2452960.

[43]

J. Tao and Y. Gao, Computing shadow prices with multiple Lagrange multipliers, J. Ind. Manag. Optim., 17 (2021), 2307-2329.  doi: 10.3934/jimo.2020070.

[44]

L. Tao and Y. Gao, Real-time pricing for smart grid with distributed energy and storage: A noncooperative game method considering spatially and temporally coupled constraints, International Journal of Electrical Power and Energy Systems, 115 (2020), 1-8.  doi: 10.1016/j.ijepes.2019.105487.

[45]

H. Wang and Y. Gao, Research on the real-time pricing of smart grid based on nonsmooth equations, Journal of Systems Engineering, 33 (2018), 320-327. 

[46]

H. Wang and Y. Gao, Real-time pricing method for smart grids based on complementarity problem, Journal of Modern Power Systems and Clean Energy, 7 (2019), 1280-1293.  doi: 10.1007/s40565-019-0508-7.

[47]

H. Wang and Y. Gao, Distributed real-time pricing method incorporating load uncertainty based on nonsmooth equations for smart grid, Math. Probl. Eng., 2019 (2019), Art. ID 1498134, 18 pp. doi: 10.1155/2019/1498134.

[48]

S. WangB. Xu and T. Liu, A conjugate gradient method for inverse problems of nonlinear coupled diffusion equations, Journal of Physics: Conference Series, 1634 (2020), 1-7. 

[49]

C. Wu, J. Zhang, Y. Lu and J. Chen, Signal reconstruction by conjugate gradient algorithm based on smoothing $l_1$-norm, Calcolo, 56 (2019), Paper No. 42, 26 pp. doi: 10.1007/s10092-019-0340-5.

[50]

Z. Ye and M. Sun, Bi-directional interation strategy household between demand side resource and electricity supplier, Electric Power Science and Engineering, 33 (2017), 1-6. 

[51]

G. YiX. TongP. ZhouY. Zhai and Z. Ye, Power system economic dispatch under CVaR and EVaR security operation risk management, Power System Protection and Control, 44 (2016), 49-56. 

[52]

K. ZhangK. ZhangX. Luo and X. Tong, Optimal control strategy for hybrid reactive power compensation system in transformer substation, Proceedings of the CSU-EPSA, 31 (2019), 137-142. 

[53]

L. Zhang and W. Zhou, On the global convergence of the Hager-Zhang conjugate gradient method with Armijo line search, Acta Math. Sci., 28 (2008), 840-845. 

[54]

N. ZhangY. LiH. XieR. Xu and C. You, A generalized conjugate gradient method for eigenvalue problems, Scientia Sinica Mathematica, 50 (2020), 1-24. 

[55]

G. Zhou, Convergence properties of a conjugate gradient method with Armijo-type line seareches, (Chinese) Gongcheng Shuxue Xuebao, 25 (2008), 405-410. 

[56]

H. ZhuY. GaoY. Hou and L. Tao, Multi-time slots real-time pricing strategy with power fluctuation caused by operating continuity of smart home appliances, Engineering Applications of Artificial Intelligence, 71 (2018), 166-174.  doi: 10.1016/j.engappai.2018.02.010.

[57]

H. ZhuY. GaoY. Hou and L. Tao, Real-time pricing considering different type of user based on Markov decision processes in smart grid, Systems Engineering-Theory and Practise, 38 (2018), 807-816. 

Figure 1.  The utility function of users $ \left(\eta = 5\right) $
Figure 2.  Electricity consumption of Numerical Simulation Experiment $ 1 $ based on Algorithm 3.1
Figure 3.  Electricity consumption of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 4.  Optimal electricity price of Numerical Simulation Experiment $ 1 $ based on Algorithm 3.1
Figure 5.  Optimal electricity price of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 6.  Utility of optimal electricity price of Numerical Simulation Experiment $ 1 $ based on Algorithm 3.1
Figure 7.  Utility of optimal electricity price of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 8.  Cost of optimal electricity price of Numerical Simulation Experiment $ 1 $ based on Algorithm 3.1
Figure 9.  Cost of optimal electricity price of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 10.  Social welfare value of Numerical Simulation Experiment $ 1 $ based on Algorithm 3.1
Figure 11.  Social welfare value of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 12.  Electricity consumption of Numerical Simulation Experiment $ 2 $ based on Algorithm 3.1
Figure 13.  Electricity consumption of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 14.  Optimal electricity price of Numerical Simulation Experiment $ 2 $ based on Algorithm 3.1
Figure 15.  Optimal electricity price of Numerical Simulation Experiment $ 1 $ based on algorithm in [53]
Figure 16.  Utility of optimal electricity price of Numerical Simulation Experiment $ 2 $ based on Algorithm 3.1
Figure 17.  Utility of optimal electricity price of Numerical Simulation Experiment $ 2 $ based on algorithm in [53]
Figure 18.  Cost of optimal electricity price of Numerical Simulation Experiment $ 2 $ based on Algorithm 3.1
Figure 19.  Cost of optimal electricity price of Numerical Simulation Experiment $ 2 $ based on algorithm in [53]
Figure 20.  Social welfare value of Numerical Simulation Experiment $ 2 $ based on Algorithm 3.1
Figure 21.  Social welfare value of Numerical Simulation Experiment $ 2 $ based on algorithm in [53]
Figure 22.  Electricity consumption of Numerical Simulation Experiment $ 1 $ based on Newton method in [26]
Figure 23.  Optimal electricity price of Numerical Simulation Experiment $ 1 $ based on Newton method in [26]
Figure 24.  Electricity consumption of Numerical Simulation Experiment $ 1 $ based on Newton method in [26]
Figure 25.  Optimal electricity price of Numerical Simulation Experiment $ 1 $ based on Newton method in [26]
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