# 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 & Management Optimization, 2021, 17 (4) : 1639-1661. doi: 10.3934/jimo.2020038
##### References:

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##### References:
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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