# American Institute of Mathematical Sciences

July  2021, 17(4): 1639-1661. doi: 10.3934/jimo.2020038

## Adjustable robust optimization in enabling optimal day-ahead economic dispatch of CCHP-MG considering uncertainties of wind-solar power and electric vehicle

 College of Electrical Engineering, Sichuan University, Chengdu, China

* Corresponding author: Yang Liu

Received  July 2019 Revised  October 2019 Published  July 2021 Early access  February 2020

At present, electric vehicles (EVs), small-scale wind power, and solar power have been increasingly integrated into modern power system via the combined cooling heating and power based microgrid (CCHP-MG). However, inside the microgrid the uncertainties of EVs charging, wind power, and solar power significantly impact the economy of CCHP-MG operation. Therefore to improve the economy deteriorated by the uncertainties, this paper presents a two-stage adjustable robust optimization to achieve the minimal operational cost for CCHP-MG. Before the realizations of the uncertainties, the day-ahead stage as the first stage decides an operational strategy that can withstand the worst-case uncertainties. As long as the uncertainties are observed, the real-time stage as the second stage adjusts the operational units to compensate the errors caused by the day-ahead operational strategy. Due to the difficulties of the model solution, this paper further adopts the duality theory, Big-M method, and column-and-constraint generation (C & CG) decomposition to convert the model into two tractable mixed integer linear programming (MILP) problems. Further, C & CG iteration algorithm is also employed to solve the MILPs, which can ultimately provide an optimal economic day-ahead dispatch strategy capable of handling uncertainties. The experimental results demonstrate the effectiveness of the presented approach.

Citation: Xianbang Chen, Yang Liu, Bin Li. Adjustable robust optimization in enabling optimal day-ahead economic dispatch of CCHP-MG considering uncertainties of wind-solar power and electric vehicle. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1639-1661. doi: 10.3934/jimo.2020038
##### References:
 [1] A. Ben-Tal, A. Goryashko, E. Guslitzer and A. Nemirovski, Adjustable robust solutions of uncertain linear programs, Math. Program., 99 (2004), 351-376.  doi: 10.1007/s10107-003-0454-y. [2] A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oper. Res., 23 (1998), 769-1024.  doi: 10.1287/moor.23.4.769. [3] C. Chen, Simulated annealing-based optimal wind-thermal coordination scheduling, IET Generation, Transmission & Distribution, 1 (2007), 447-455.  doi: 10.1049/iet-gtd:20060208. [4] C. M. Correa-Posada and P. Sánchez-Martín, Integrated power and natural gas model for energy adequacy in short-term operation, IEEE Transactions on Power Systems, 30 (2015), 3347-3355.  doi: 10.1109/TPWRS.2014.2372013. [5] C. Duan, L. Jiang, W. Fang and J. Liu, Data-driven affinely adjustable distributionally robust unit commitment, IEEE Transactions on Power Systems, 33 (2018), 1385-1398.  doi: 10.1109/TPWRS.2017.2741506. [6] C. Duan, L. Jiang, W. Fang, J. Liu and S. Liu, Data-driven distributionally robust energy-reserve-storage dispatch, IEEE Transactions on Industrial Informatics, 14 (2018), 2826-2836.  doi: 10.1109/TII.2017.2771355. [7] F. Fang, Q. H. Wang and Y. Shi, A novel optimal operational strategy for the CCHP system based on two operating modes, IEEE Transactions on Power Systems, 27 (2012), 1032-1041.  doi: 10.1109/TPWRS.2011.2175490. [8] F. Farmani, M. Parvizimosaed, H. Monsef and A. Rahimi-Kian, A conceptual model of a smart energy management system for a residential building equipped with CCHP system, Internat. J. Electrical Power Energy Systems, 95 (2018), 523-536.  doi: 10.1016/j.ijepes.2017.09.016. [9] H. Gao, J. Liu, L. Wang and Z. Wei, Decentralized energy management for networked microgrids in future distribution systems, IEEE Transactions on Power Systems, 33 (2018), 3599-3610.  doi: 10.1109/TPWRS.2017.2773070. [10] W. Gu, S. Lu, Z. Wu, X. Zhang, J. Zhou, B. Zhao and J. Wang, Residential CCHP microgrid with load aggregator: Operation mode, pricing strategy, and optimal dispatch, Appl. Energy, 205 (2017), 173-186.  doi: 10.1016/j.apenergy.2017.07.045. [11] Y. Guo, J. Xiong, S. Xu and W. Su, Two-stage economic operation of microgrid-like electric vehicle parking deck, IEEE Transactions on Smart Grid, 7 (2016), 1703-1712.  doi: 10.1109/TSG.2015.2424912. [12] Z. Guo and X. Xiao, Wind power assessment based on a WRF wind simulation with developed power curve modeling methods, Abstract Appl. Anal., 2014 (2014), 1-15.  doi: 10.1155/2014/941648. [13] N. Haouas and P. R. Bertrand, Wind farm power forecasting, Math. Probl. Eng., 2013 (2013), 5pp. doi: 10.1155/2013/163565. [14] R. Hashemi, A developed offline model for optimal operation of combined heating and cooling and power systems, IEEE Transactions on Energy Conversion, 24 (2009), 222-229.  doi: 10.1109/TEC.2008.2002330. [15] S. Jin, Z. Mao, H. Li and W. Qi, Dynamic operation management of a renewable microgrid including battery energy storage, Math. Probl. Eng., 2018 (2018), 19pp. doi: 10.1155/2018/5852309. [16] Y. Lee and R. Baldick, A frequency-constrained stochastic economic dispatch model, IEEE Transactions on Power Systems, 28 (2013), 2301-2312.  doi: 10.1109/TPWRS.2012.2236108. [17] B. Li, X. Qian, J. Sun, K. L. Teo and C. Yue, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Appl. Math. Model., 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039. [18] B. Li, J. Sun and K. L. Teo, A distributionally robust approach to a class of three-stage stochastic linear programs, Pac. J. Optim., 15 (2019), 219-236. [19] B. Li, J. Sun, H. Xu and M Zhang, A class of two-stage distributionally robust games, J. Ind. Manag. Optim., 15 (2019), 387-400.  doi: 10.3934/jimo.2018048. [20] G. Li, G. Li and M. Zhou, Model and application of renewable energy accommodation capacity calculation considering utilization level of inter-provincial tie-line, Protection and Control of Modern Power Systems, 4 (2019), 1-1.  doi: 10.1186/s41601-019-0115-7. [21] G. Li, R. Zhang, T. Jiang, H. Chen, L. Bai, H. Cui and X. Li, Optimal dispatch strategy for integrated energy systems with cchp and wind power, Appl. Energy, 192 (2017), 408-419.  doi: 10.1016/j.apenergy.2016.08.139. [22] G. Li, R. Zhang, T. Jiang, H. Chen, L. Bai and X. Li, Security-constrained bi-level economic dispatch model for integrated natural gas and electricity systems considering wind power and power-to-gas process, Appl. Energy, 194 (2017), 696-704.  doi: 10.1016/j.apenergy.2016.07.077. [23] Y. Liu, Y. Liu, J. Liu, M. Li, T. Liu, G. Taylor and K. Zuo, A MapReduce based high performance neural network in enabling fast stability assessment of power systems, Math. Probl. Eng., 2017 (2017), 1-12.  doi: 10.1155/2017/4030146. [24] Y. Liu and N. C. Nair, A two-stage stochastic dynamic economic dispatch model considering wind uncertainty, IEEE Transactions on Sustainable Energy, 7 (2016), 819-829.  doi: 10.1109/TSTE.2015.2498614. [25] C. Marino, M. Marufuzzaman, M. Hu and M. D. Sarder, Developing a CCHP-microgrid operation decision model under uncertainty, Comput. Industrial Eng., 115 (2018), 354-367.  doi: 10.1016/j.cie.2017.11.021. [26] M. H. Sarparandeh and M. Ehsan, Pricing of vehicle-to-grid services in a microgrid by Nash bargaining theory, Math. Probl. Eng., 2017 (2017). doi: 10.1155/2017/1840140. [27] X. Shen, Y. Liu and Y. Liu, A multistage solution approach for dynamic reactive power optimization based on interval uncertainty, Math. Probl. Eng., 2018 (2018), 10pp. doi: 10.1155/2018/3854812. [28] R. Shi, C. Sun, Z. Zhou, L. Zhang, and Z. Liang, A robust economic dispatch of residential microgrid with wind power and electric vehicle integration, Chinese Control and Decision Conference (CCDC), 2016, 3672–3676. doi: 10.1109/CCDC.2016.7531621. [29] J. Soares, B. Canizes, M. A. F. Ghazvini, Z. Vale and G. K. Venayagamoorthy, Two-stage stochastic model using benders' decomposition for large-scale energy resource management in smart grids, IEEE Transactions on Industry Appl., 53 (2017), 5905-5914.  doi: 10.1109/TIA.2017.2723339. [30] Y. Tan, Y. Cao, C. Li, Y. Li, J. Zhou and Y. Song, A two-stage stochastic programming approach considering risk level for distribution networks operation with wind power, IEEE Systems Journal, 10 (2016), 117-126.  doi: 10.1109/JSYST.2014.2350027. [31] L. Tian, S. Shi and Z. Jia, A statistical model for charging power demand of electric vehicles, Power System Technology, 11 (2010), 126-130.  doi: 10.13335/j.1000-3673.pst.2010.11.020. [32] T. A. Victoire and A. Jeyakumar, Hybrid PSO–CSQP for economic dispatch with valve-point effect, Electric Power Systems Research, 71 (2004), 51-59.  doi: 10.1016/j.epsr.2003.12.017. [33] D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Transactions on Power Systems, 8 (1993), 1325-1332.  doi: 10.1109/59.260861. [34] J. Wang, J. Wang, C. Liu and and J. Ruiz, Stochastic unit commitment with sub-hourly dispatch constraints, Appl. Energy, 105 (2013), 418-422.  doi: 10.1016/j.apenergy.2013.01.008. [35] P. Wei and Y. Liu, The integration of wind-solar-hydropower generation in enabling economic robust dispatch, Math. Probl. Eng., 2019 (2019), 12pp. doi: 10.1155/2019/4634131. [36] H. Wu, X Hou, B. Zhao and C. Zhu, Economical dispatch of microgrid considering plug-in electric vehicles, Automation of Electric Power Systems, 38 (2014), 77-84.  doi: 10.7500/AEPS20130911002. [37] T. Wu, Q. Yang, Z. Bao and W. Yan, Coordinated energy dispatching in microgrid with wind power generation and plug-in electric vehicles, IEEE Transactions on Smart Grid, 4 (2013), 1453-1463.  doi: 10.1109/TSG.2013.2268870. [38] W. Wu, J. Chen, B. Zhang and H. Sun, A robust wind power optimization method for look-ahead power dispatch, IEEE Transactions on Sustainable Energy, 5 (2014), 507-515.  doi: 10.1109/TSTE.2013.2294467. [39] Y. Xiang, J. Liu and Y. Liu, Robust energy management of microgrid with uncertain renewable generation and load, IEEE Transactions on Smart Grid, 7 (2016), 1034-1043.  doi: 10.1109/TSG.2014.2385801. [40] L. Xie, Y. Gu, X. Zhu and M. G. Genton, Short-term spatio-temporal wind power forecast in robust look-ahead power system dispatch, IEEE Transactions on Smart Grid, 5 (2014), 511-520.  doi: 10.1109/TSG.2013.2282300. [41] P. Xiong, P. Jirutitijaroen and C. Singh, A distributionally robust optimization model for unit commitment considering uncertain wind power generation, IEEE Transactions on Power Systems, 32 (2017), 39-49.  doi: 10.1109/TPWRS.2016.2544795. [42] P. Xiong and C. Singh, Distributionally robust optimization for energy and reserve toward a low-carbon electricity market, Electric Power Systems Res., 149 (2017), 137-145.  doi: 10.1016/j.epsr.2017.04.008. [43] Y. Yang, Practical robust optimization method for unit commitment of a system with integrated wind resource, Math. Probl. Eng., 2017 (2017), 13pp. doi: 10.1155/2017/5208290. [44] J. Yu, Q. Feng, Y. Li and J. Cao, Stochastic optimal dispatch of virtual power plant considering correlation of distributed generations, Math. Probl. Eng., 2015 (2015). doi: 10.1155/2015/135673. [45] B. Zeng and L. Zhao, Solving two-stage robust optimization problems using a column-and-constraint generation method, Oper. Res. Lett., 41 (2013), 457-461.  doi: 10.1016/j.orl.2013.05.003. [46] Y. Zhang, J. Meng, B. Guo and T. Zhang, An improved dispatch strategy of a grid-connected hybrid energy system with high penetration level of renewable energy, Math. Probl. Eng., 2014 (2014), 18pp. doi: 10.1155/2014/602063. [47] Y. Zhao, C. Li, M. Zhao, S. Xu, H. Gao and L. Song, Model design on emergency power supply of electric vehicle, Math. Probl. Eng., 2017 (2017), 6pp. doi: 10.1155/2017/9697051.

show all references

##### References:
 [1] A. Ben-Tal, A. Goryashko, E. Guslitzer and A. Nemirovski, Adjustable robust solutions of uncertain linear programs, Math. Program., 99 (2004), 351-376.  doi: 10.1007/s10107-003-0454-y. [2] A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oper. Res., 23 (1998), 769-1024.  doi: 10.1287/moor.23.4.769. [3] C. Chen, Simulated annealing-based optimal wind-thermal coordination scheduling, IET Generation, Transmission & Distribution, 1 (2007), 447-455.  doi: 10.1049/iet-gtd:20060208. [4] C. M. Correa-Posada and P. Sánchez-Martín, Integrated power and natural gas model for energy adequacy in short-term operation, IEEE Transactions on Power Systems, 30 (2015), 3347-3355.  doi: 10.1109/TPWRS.2014.2372013. [5] C. Duan, L. Jiang, W. Fang and J. Liu, Data-driven affinely adjustable distributionally robust unit commitment, IEEE Transactions on Power Systems, 33 (2018), 1385-1398.  doi: 10.1109/TPWRS.2017.2741506. [6] C. Duan, L. Jiang, W. Fang, J. Liu and S. Liu, Data-driven distributionally robust energy-reserve-storage dispatch, IEEE Transactions on Industrial Informatics, 14 (2018), 2826-2836.  doi: 10.1109/TII.2017.2771355. [7] F. Fang, Q. H. Wang and Y. Shi, A novel optimal operational strategy for the CCHP system based on two operating modes, IEEE Transactions on Power Systems, 27 (2012), 1032-1041.  doi: 10.1109/TPWRS.2011.2175490. [8] F. Farmani, M. Parvizimosaed, H. Monsef and A. Rahimi-Kian, A conceptual model of a smart energy management system for a residential building equipped with CCHP system, Internat. J. Electrical Power Energy Systems, 95 (2018), 523-536.  doi: 10.1016/j.ijepes.2017.09.016. [9] H. Gao, J. Liu, L. Wang and Z. Wei, Decentralized energy management for networked microgrids in future distribution systems, IEEE Transactions on Power Systems, 33 (2018), 3599-3610.  doi: 10.1109/TPWRS.2017.2773070. [10] W. Gu, S. Lu, Z. Wu, X. Zhang, J. Zhou, B. Zhao and J. Wang, Residential CCHP microgrid with load aggregator: Operation mode, pricing strategy, and optimal dispatch, Appl. Energy, 205 (2017), 173-186.  doi: 10.1016/j.apenergy.2017.07.045. [11] Y. Guo, J. Xiong, S. Xu and W. Su, Two-stage economic operation of microgrid-like electric vehicle parking deck, IEEE Transactions on Smart Grid, 7 (2016), 1703-1712.  doi: 10.1109/TSG.2015.2424912. [12] Z. Guo and X. Xiao, Wind power assessment based on a WRF wind simulation with developed power curve modeling methods, Abstract Appl. Anal., 2014 (2014), 1-15.  doi: 10.1155/2014/941648. [13] N. Haouas and P. R. Bertrand, Wind farm power forecasting, Math. Probl. Eng., 2013 (2013), 5pp. doi: 10.1155/2013/163565. [14] R. Hashemi, A developed offline model for optimal operation of combined heating and cooling and power systems, IEEE Transactions on Energy Conversion, 24 (2009), 222-229.  doi: 10.1109/TEC.2008.2002330. [15] S. Jin, Z. Mao, H. Li and W. Qi, Dynamic operation management of a renewable microgrid including battery energy storage, Math. Probl. Eng., 2018 (2018), 19pp. doi: 10.1155/2018/5852309. [16] Y. Lee and R. Baldick, A frequency-constrained stochastic economic dispatch model, IEEE Transactions on Power Systems, 28 (2013), 2301-2312.  doi: 10.1109/TPWRS.2012.2236108. [17] B. Li, X. Qian, J. Sun, K. L. Teo and C. Yue, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Appl. Math. Model., 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039. [18] B. Li, J. Sun and K. L. Teo, A distributionally robust approach to a class of three-stage stochastic linear programs, Pac. J. Optim., 15 (2019), 219-236. [19] B. Li, J. Sun, H. Xu and M Zhang, A class of two-stage distributionally robust games, J. Ind. Manag. Optim., 15 (2019), 387-400.  doi: 10.3934/jimo.2018048. [20] G. Li, G. Li and M. Zhou, Model and application of renewable energy accommodation capacity calculation considering utilization level of inter-provincial tie-line, Protection and Control of Modern Power Systems, 4 (2019), 1-1.  doi: 10.1186/s41601-019-0115-7. [21] G. Li, R. Zhang, T. Jiang, H. Chen, L. Bai, H. Cui and X. Li, Optimal dispatch strategy for integrated energy systems with cchp and wind power, Appl. Energy, 192 (2017), 408-419.  doi: 10.1016/j.apenergy.2016.08.139. [22] G. Li, R. Zhang, T. Jiang, H. Chen, L. Bai and X. Li, Security-constrained bi-level economic dispatch model for integrated natural gas and electricity systems considering wind power and power-to-gas process, Appl. Energy, 194 (2017), 696-704.  doi: 10.1016/j.apenergy.2016.07.077. [23] Y. Liu, Y. Liu, J. Liu, M. Li, T. Liu, G. Taylor and K. Zuo, A MapReduce based high performance neural network in enabling fast stability assessment of power systems, Math. Probl. Eng., 2017 (2017), 1-12.  doi: 10.1155/2017/4030146. [24] Y. Liu and N. C. Nair, A two-stage stochastic dynamic economic dispatch model considering wind uncertainty, IEEE Transactions on Sustainable Energy, 7 (2016), 819-829.  doi: 10.1109/TSTE.2015.2498614. [25] C. Marino, M. Marufuzzaman, M. Hu and M. D. Sarder, Developing a CCHP-microgrid operation decision model under uncertainty, Comput. Industrial Eng., 115 (2018), 354-367.  doi: 10.1016/j.cie.2017.11.021. [26] M. H. Sarparandeh and M. Ehsan, Pricing of vehicle-to-grid services in a microgrid by Nash bargaining theory, Math. Probl. Eng., 2017 (2017). doi: 10.1155/2017/1840140. [27] X. Shen, Y. Liu and Y. Liu, A multistage solution approach for dynamic reactive power optimization based on interval uncertainty, Math. Probl. Eng., 2018 (2018), 10pp. doi: 10.1155/2018/3854812. [28] R. Shi, C. Sun, Z. Zhou, L. Zhang, and Z. Liang, A robust economic dispatch of residential microgrid with wind power and electric vehicle integration, Chinese Control and Decision Conference (CCDC), 2016, 3672–3676. doi: 10.1109/CCDC.2016.7531621. [29] J. Soares, B. Canizes, M. A. F. Ghazvini, Z. Vale and G. K. Venayagamoorthy, Two-stage stochastic model using benders' decomposition for large-scale energy resource management in smart grids, IEEE Transactions on Industry Appl., 53 (2017), 5905-5914.  doi: 10.1109/TIA.2017.2723339. [30] Y. Tan, Y. Cao, C. Li, Y. Li, J. Zhou and Y. Song, A two-stage stochastic programming approach considering risk level for distribution networks operation with wind power, IEEE Systems Journal, 10 (2016), 117-126.  doi: 10.1109/JSYST.2014.2350027. [31] L. Tian, S. Shi and Z. Jia, A statistical model for charging power demand of electric vehicles, Power System Technology, 11 (2010), 126-130.  doi: 10.13335/j.1000-3673.pst.2010.11.020. [32] T. A. Victoire and A. Jeyakumar, Hybrid PSO–CSQP for economic dispatch with valve-point effect, Electric Power Systems Research, 71 (2004), 51-59.  doi: 10.1016/j.epsr.2003.12.017. [33] D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Transactions on Power Systems, 8 (1993), 1325-1332.  doi: 10.1109/59.260861. [34] J. Wang, J. Wang, C. Liu and and J. Ruiz, Stochastic unit commitment with sub-hourly dispatch constraints, Appl. Energy, 105 (2013), 418-422.  doi: 10.1016/j.apenergy.2013.01.008. [35] P. Wei and Y. Liu, The integration of wind-solar-hydropower generation in enabling economic robust dispatch, Math. Probl. Eng., 2019 (2019), 12pp. doi: 10.1155/2019/4634131. [36] H. Wu, X Hou, B. Zhao and C. Zhu, Economical dispatch of microgrid considering plug-in electric vehicles, Automation of Electric Power Systems, 38 (2014), 77-84.  doi: 10.7500/AEPS20130911002. [37] T. Wu, Q. Yang, Z. Bao and W. Yan, Coordinated energy dispatching in microgrid with wind power generation and plug-in electric vehicles, IEEE Transactions on Smart Grid, 4 (2013), 1453-1463.  doi: 10.1109/TSG.2013.2268870. [38] W. Wu, J. Chen, B. Zhang and H. Sun, A robust wind power optimization method for look-ahead power dispatch, IEEE Transactions on Sustainable Energy, 5 (2014), 507-515.  doi: 10.1109/TSTE.2013.2294467. [39] Y. Xiang, J. Liu and Y. Liu, Robust energy management of microgrid with uncertain renewable generation and load, IEEE Transactions on Smart Grid, 7 (2016), 1034-1043.  doi: 10.1109/TSG.2014.2385801. [40] L. Xie, Y. Gu, X. Zhu and M. G. Genton, Short-term spatio-temporal wind power forecast in robust look-ahead power system dispatch, IEEE Transactions on Smart Grid, 5 (2014), 511-520.  doi: 10.1109/TSG.2013.2282300. [41] P. Xiong, P. Jirutitijaroen and C. Singh, A distributionally robust optimization model for unit commitment considering uncertain wind power generation, IEEE Transactions on Power Systems, 32 (2017), 39-49.  doi: 10.1109/TPWRS.2016.2544795. [42] P. Xiong and C. Singh, Distributionally robust optimization for energy and reserve toward a low-carbon electricity market, Electric Power Systems Res., 149 (2017), 137-145.  doi: 10.1016/j.epsr.2017.04.008. [43] Y. Yang, Practical robust optimization method for unit commitment of a system with integrated wind resource, Math. Probl. Eng., 2017 (2017), 13pp. doi: 10.1155/2017/5208290. [44] J. Yu, Q. Feng, Y. Li and J. Cao, Stochastic optimal dispatch of virtual power plant considering correlation of distributed generations, Math. Probl. Eng., 2015 (2015). doi: 10.1155/2015/135673. [45] B. Zeng and L. Zhao, Solving two-stage robust optimization problems using a column-and-constraint generation method, Oper. Res. Lett., 41 (2013), 457-461.  doi: 10.1016/j.orl.2013.05.003. [46] Y. Zhang, J. Meng, B. Guo and T. Zhang, An improved dispatch strategy of a grid-connected hybrid energy system with high penetration level of renewable energy, Math. Probl. Eng., 2014 (2014), 18pp. doi: 10.1155/2014/602063. [47] Y. Zhao, C. Li, M. Zhao, S. Xu, H. Gao and L. Song, Model design on emergency power supply of electric vehicle, Math. Probl. Eng., 2017 (2017), 6pp. doi: 10.1155/2017/9697051.
CCHP-MG system structure
Details of data
Intervals and stochastic scenarios of uncertainty sets with 30% prediction error
RESs utilization results of A-DED with different budgets
RESs utilization results of A-DED with different prediction errors
Steps of C & CG iteration algorithm
 C & CG Iteration Algorithm Step 1 (Initialization): Set an initial scenario ${\boldsymbol{u}_1}$ and convergence gap $\delta$. Initialize upper bound $U_0=+\infty$, lower bound $L_0=-\infty$, and iteration number $k = 1$. Step 2 (Solve MP): Input the scenario set $\boldsymbol{u}_1$ into (95) to solve MP. Record the optimal solution ($\boldsymbol{x}_k$, $\boldsymbol{y}_l$), the optimal value $\alpha$ of objective, and $\boldsymbol{c}^{\top}\boldsymbol{x}$. Update the lower bound $L_k = \alpha$, $l = 1, 2, \ldots, k$. Step 3 (Solve SP): Input $\boldsymbol{x}_k$ into (98) to solve SP. Record the optimal solution ($\boldsymbol{u}_{k}^{0}$, $\boldsymbol{y}_{k}^{0}$) and the optimal value $\beta$ of objective. Set the worst scenario $\boldsymbol{u}_{k+1}$ to $\boldsymbol{u}_{k}^{0}$. Update the upper bound $U_k=\beta +\boldsymbol{c}^{\top}\boldsymbol{x}$. Step 4 (Check Convergence): If $U_k - L_k\leq d$, terminate the algorithm and record the optimal value $\nu$ as the expected cost. Otherwise, add constraints (99) and real-time adjustment variables $\boldsymbol{y}_{k+1}$ correspondingly to $\boldsymbol{u}_{k+1}$; return to Step 2 and set $k = k+1$.
 C & CG Iteration Algorithm Step 1 (Initialization): Set an initial scenario ${\boldsymbol{u}_1}$ and convergence gap $\delta$. Initialize upper bound $U_0=+\infty$, lower bound $L_0=-\infty$, and iteration number $k = 1$. Step 2 (Solve MP): Input the scenario set $\boldsymbol{u}_1$ into (95) to solve MP. Record the optimal solution ($\boldsymbol{x}_k$, $\boldsymbol{y}_l$), the optimal value $\alpha$ of objective, and $\boldsymbol{c}^{\top}\boldsymbol{x}$. Update the lower bound $L_k = \alpha$, $l = 1, 2, \ldots, k$. Step 3 (Solve SP): Input $\boldsymbol{x}_k$ into (98) to solve SP. Record the optimal solution ($\boldsymbol{u}_{k}^{0}$, $\boldsymbol{y}_{k}^{0}$) and the optimal value $\beta$ of objective. Set the worst scenario $\boldsymbol{u}_{k+1}$ to $\boldsymbol{u}_{k}^{0}$. Update the upper bound $U_k=\beta +\boldsymbol{c}^{\top}\boldsymbol{x}$. Step 4 (Check Convergence): If $U_k - L_k\leq d$, terminate the algorithm and record the optimal value $\nu$ as the expected cost. Otherwise, add constraints (99) and real-time adjustment variables $\boldsymbol{y}_{k+1}$ correspondingly to $\boldsymbol{u}_{k+1}$; return to Step 2 and set $k = k+1$.
CGs parameters
 CG $P^{min}$ $P^{max}$ $R^{Up}$ $R^{Dn}$ $a$ $b$ $\lambda^{Up}$ $\lambda^{Dn}$ (kW) (kW) (kW/min) (kW/min) (＄/kWh) (＄/kWh) (＄/kWh) (＄/kWh) MT 50 550 6 6 0.67 0 2.5 1.5 FC 50 240 2 2 0.60 0 2.5 1.5 EB 20 500 5 4 - - 0.5 0.5
 CG $P^{min}$ $P^{max}$ $R^{Up}$ $R^{Dn}$ $a$ $b$ $\lambda^{Up}$ $\lambda^{Dn}$ (kW) (kW) (kW/min) (kW/min) (＄/kWh) (＄/kWh) (＄/kWh) (＄/kWh) MT 50 550 6 6 0.67 0 2.5 1.5 FC 50 240 2 2 0.60 0 2.5 1.5 EB 20 500 5 4 - - 0.5 0.5
Penalty prices
 $\lambda_{Wind}$ (＄/kWh) $\lambda_{Solar}$ (＄/kWh) $\lambda_{Load}$ (＄/kWh) 0.536 0.536 5
 $\lambda_{Wind}$ (＄/kWh) $\lambda_{Solar}$ (＄/kWh) $\lambda_{Load}$ (＄/kWh) 0.536 0.536 5
Electricity market prices
 Hour Day-ahead stage Real-time stage $\lambda_{Buy}^{DA}$ $\lambda_{Sell}^{DA}$ $\lambda_{Buy}^{RT}$ $\lambda_{Sell}^{RT}$ (＄/kWh) (＄/kWh) (＄/kWh) (＄/kWh) (00:00-08:00) 1.35 1.04 2.70 0.11 (08:00-09:00, 12:00-19:00) 0.90 0.69 1.80 0.07 (09:00-12:00, 19:00-24:00) 0.50 0.39 1.00 0.04
 Hour Day-ahead stage Real-time stage $\lambda_{Buy}^{DA}$ $\lambda_{Sell}^{DA}$ $\lambda_{Buy}^{RT}$ $\lambda_{Sell}^{RT}$ (＄/kWh) (＄/kWh) (＄/kWh) (＄/kWh) (00:00-08:00) 1.35 1.04 2.70 0.11 (08:00-09:00, 12:00-19:00) 0.90 0.69 1.80 0.07 (09:00-12:00, 19:00-24:00) 0.50 0.39 1.00 0.04
Energy storage system parameters
 $P_{Cha}^{min}/P_{Cha}^{max}$ $P_{Dis}^{min}/P_{Dis}^{max}$ $\eta_{ESS}^{Cha}/\eta_{ESS}^{Dis}$ $\delta_{ESS}$ $E_{ESS}^{max}$ $E_{ESS}^{min}$ $E_{ESS}(0)$ (kW) (kW) (kWh) (kWh) (kWh) 0/200 0/200 0.9/0.9 0.001 480 120 120
 $P_{Cha}^{min}/P_{Cha}^{max}$ $P_{Dis}^{min}/P_{Dis}^{max}$ $\eta_{ESS}^{Cha}/\eta_{ESS}^{Dis}$ $\delta_{ESS}$ $E_{ESS}^{max}$ $E_{ESS}^{min}$ $E_{ESS}(0)$ (kW) (kW) (kWh) (kWh) (kWh) 0/200 0/200 0.9/0.9 0.001 480 120 120
Heat storage system parameters
 $Q_{Cha}^{min}/Q_{Cha}^{max}$ $Q_{Dis}^{min}/Q_{Dis}^{max}$ $\eta_{HSS}^{Cha}/\eta_{HSS}^{Dis}$ $\delta_{HSS}$ $E_{HSS}^{max}$ $E_{HSS}^{min}$ $E_{HSS}(0)$ (kW) (kW) (kWh) (kWh) (kWh) 0/200 0/200 0.9/0.9 0.01 600 0 0
 $Q_{Cha}^{min}/Q_{Cha}^{max}$ $Q_{Dis}^{min}/Q_{Dis}^{max}$ $\eta_{HSS}^{Cha}/\eta_{HSS}^{Dis}$ $\delta_{HSS}$ $E_{HSS}^{max}$ $E_{HSS}^{min}$ $E_{HSS}(0)$ (kW) (kW) (kWh) (kWh) (kWh) 0/200 0/200 0.9/0.9 0.01 600 0 0
Comparison of efficiencies and costs of different methods
 Method Time (s) Day-ahead Cost(＄) Expected Actual (＄) $C_{MT}$ $C_{FC}$ $C_{DA}$ Cost (＄) RT SUM D-DED 6.52 5422.48 2251.42 5972.69 5972.69 613.25 6585.95 S-DED 1695.69 5336.95 2099.66 6011.19 6449.07 566.70 6577.90 R-DED 6.92 5425.33 1146.89 6503.61 10916.15 457.79 6961.91 A-DED 8.37 5422.83 1796.67 6203.84 8305.34 367.47 6571.81
 Method Time (s) Day-ahead Cost(＄) Expected Actual (＄) $C_{MT}$ $C_{FC}$ $C_{DA}$ Cost (＄) RT SUM D-DED 6.52 5422.48 2251.42 5972.69 5972.69 613.25 6585.95 S-DED 1695.69 5336.95 2099.66 6011.19 6449.07 566.70 6577.90 R-DED 6.92 5425.33 1146.89 6503.61 10916.15 457.79 6961.91 A-DED 8.37 5422.83 1796.67 6203.84 8305.34 367.47 6571.81
Comparison of electricity transactions of different methods
 Method Day-ahead Transaction Actual Transaction Revenue (＄) Loss (＄) State Revenue (＄) Loss (＄) State D-DED 1701.19 - Profit 1303.22 - Profit S-DED 1425.36 - Profit 1051.32 - Profit R-DED - 68.61 Loss - 275.65 Loss A-DED 1015.67 - Profit 671.63 - Profit
 Method Day-ahead Transaction Actual Transaction Revenue (＄) Loss (＄) State Revenue (＄) Loss (＄) State D-DED 1701.19 - Profit 1303.22 - Profit S-DED 1425.36 - Profit 1051.32 - Profit R-DED - 68.61 Loss - 275.65 Loss A-DED 1015.67 - Profit 671.63 - Profit
Comparison of A-DED with different budgets
 $\Gamma$ Iteration Time (s) Day-ahead Cost (＄) Expected Actual Number $C_{MT}$ $C_{FC}$ $C_{Grid}^{DA}$ $C_{DA}$ Cost (＄) SUM (＄) 4 1 17.5 5422.8 1945.8 -1249.8 6118.6 7580.2 6568.5 8 2 28.7 5422.8 1596.3 -715.1 6304.1 8975.5 6753.4 12 3 36.2 5422.8 1332.3 -323.9 6431.6 9981.2 6880.8 16 5 37.1 5422.8 1237.5 -203.4 6457.3 10594.5 6900.3 20 6 42.0 5422.8 1180.7 -118.4 6485.0 10809.4 6928.3
 $\Gamma$ Iteration Time (s) Day-ahead Cost (＄) Expected Actual Number $C_{MT}$ $C_{FC}$ $C_{Grid}^{DA}$ $C_{DA}$ Cost (＄) SUM (＄) 4 1 17.5 5422.8 1945.8 -1249.8 6118.6 7580.2 6568.5 8 2 28.7 5422.8 1596.3 -715.1 6304.1 8975.5 6753.4 12 3 36.2 5422.8 1332.3 -323.9 6431.6 9981.2 6880.8 16 5 37.1 5422.8 1237.5 -203.4 6457.3 10594.5 6900.3 20 6 42.0 5422.8 1180.7 -118.4 6485.0 10809.4 6928.3
Comparison of A-DED with different budgets
 Error Iteration Time Day-ahead Cost (＄) Expected Actual (＄) Number (s) $C_{MT}$ $C_{FC}$ $C_{Grid}^{DA}$ $C_{DA}$ Cost (＄) RT SUM 10% 1 15.3 5422.5 2098.4 -1500.2 6020.2 7152.4 103.5 6123.7 20% 3 36.3 5422.8 1715.5 -914.3 6224.8 8562.3 235.7 6460.5 30% 3 29.2 5422.8 1332.7 -323.9 6431.4 9987.2 449.4 6880.8 40% 4 30.3 5409.2 1050.6 176.8 6636.0 11425.6 479.9 7385.9 50% 8 36.4 5306.4 937.1 752.9 6996.3 13021.7 1023.3 8019.6
 Error Iteration Time Day-ahead Cost (＄) Expected Actual (＄) Number (s) $C_{MT}$ $C_{FC}$ $C_{Grid}^{DA}$ $C_{DA}$ Cost (＄) RT SUM 10% 1 15.3 5422.5 2098.4 -1500.2 6020.2 7152.4 103.5 6123.7 20% 3 36.3 5422.8 1715.5 -914.3 6224.8 8562.3 235.7 6460.5 30% 3 29.2 5422.8 1332.7 -323.9 6431.4 9987.2 449.4 6880.8 40% 4 30.3 5409.2 1050.6 176.8 6636.0 11425.6 479.9 7385.9 50% 8 36.4 5306.4 937.1 752.9 6996.3 13021.7 1023.3 8019.6
Energy conversion coefficients
 $\eta_{MT}^{EH}$ $\eta_{EB}^{EH}$ $\eta_{MT}^{HC}$ $\eta_{EB}^{HC}$ $\eta_{HSS}^{HC}$ 0.8 0.8 0.8 0.8 0.8
 $\eta_{MT}^{EH}$ $\eta_{EB}^{EH}$ $\eta_{MT}^{HC}$ $\eta_{EB}^{HC}$ $\eta_{HSS}^{HC}$ 0.8 0.8 0.8 0.8 0.8
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