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
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C. DuanL. JiangW. FangJ. 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.

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F. FarmaniM. ParvizimosaedH. 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.

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G. LiG. 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.

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G. LiR. ZhangT. JiangH. ChenL. BaiH. 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.

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G. LiR. ZhangT. JiangH. ChenL. 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]

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[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]

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[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. SoaresB. CanizesM. A. F. GhazviniZ. 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. TanY. CaoC. LiY. LiJ. 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. TianS. 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. WangJ. WangC. 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. WuX HouB. 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. WuQ. YangZ. 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. WuJ. ChenB. 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. XiangJ. 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. XieY. GuX. 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. XiongP. 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-TalA. GoryashkoE. 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. DuanL. JiangW. 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. DuanL. JiangW. FangJ. 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. FangQ. 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. FarmaniM. ParvizimosaedH. 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. GaoJ. LiuL. 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. GuS. LuZ. WuX. ZhangJ. ZhouB. 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. GuoJ. XiongS. 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. LiX. QianJ. SunK. 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. LiJ. 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. LiJ. SunH. 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. LiG. 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. LiR. ZhangT. JiangH. ChenL. BaiH. 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. LiR. ZhangT. JiangH. ChenL. 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. LiuY. LiuJ. LiuM. LiT. LiuG. 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. MarinoM. MarufuzzamanM. 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. SoaresB. CanizesM. A. F. GhazviniZ. 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. TanY. CaoC. LiY. LiJ. 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. TianS. 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. WangJ. WangC. 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. WuX HouB. 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. WuQ. YangZ. 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. WuJ. ChenB. 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. XiangJ. 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. XieY. GuX. 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. XiongP. 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.

Figure 1.  CCHP-MG system structure
Figure 2.  Details of data
Figure 3.  Intervals and stochastic scenarios of uncertainty sets with 30% prediction error
Figure 4.  Day-ahead dispatch decision of D-DED
Figure 5.  Day-ahead dispatch decision of S-DED
Figure 6.  Day-ahead dispatch decision of R-DED
Figure 7.  Day-ahead dispatch decision of A-DED
Figure 8.  RESs utilization results of A-DED with different budgets
Figure 9.  RESs utilization results of A-DED with different prediction errors
Table 1.  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 $.
Table 6.  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
Table 7.  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
Table 8.  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
Table 9.  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
Table 10.  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
Table 2.  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 ($) RT SUM
D-DED 6.52 5422.48 2251.42 5972.69 5972.69 613.25 6585.95
S-DED 1695.69 5336.95 2099.66 6011.19 6449.07 566.70 6577.90
R-DED 6.92 5425.33 1146.89 6503.61 10916.15 457.79 6961.91
A-DED 8.37 5422.83 1796.67 6203.84 8305.34 367.47 6571.81
Table 3.  Comparison of electricity transactions of different methods
Method Day-ahead Transaction Actual Transaction
Revenue ($) 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
Table 4.  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
Table 5.  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
Table 11.  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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