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

Xianbang Chen; Yang Liu; Bin Li

doi:10.3934/jimo.2020038

Article Contents

2021, Volume 17, Issue 4: 1639-1661. Doi: 10.3934/jimo.2020038

This issue Previous Article Worst-case analysis of Gini mean difference safety measure Next Article Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

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
^* Corresponding author: Yang Liu

Received: June 30, 2019

Revised: September 30, 2019

Early access: February 2020

Published: July 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. CCHP-MG system structure

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Figure 2. Details of data

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Figure 3. Intervals and stochastic scenarios of uncertainty sets with 30% prediction error

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Figure 4. Day-ahead dispatch decision of D-DED

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Figure 5. Day-ahead dispatch decision of S-DED

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Figure 6. Day-ahead dispatch decision of R-DED

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Figure 7. Day-ahead dispatch decision of A-DED

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Figure 8. RESs utilization results of A-DED with different budgets

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Figure 9. RESs utilization results of A-DED with different prediction errors

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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 $.

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Table 6. CGs parameters

CG	$P^{min}$	$P^{max}$	$R^{Up}$	$R^{Dn}$	$a$	$b$	$\lambda^{Up}$	$\lambda^{Dn}$
CG	(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

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Table 7. Penalty prices

$\lambda_{Wind}$ (＄/kWh)	$\lambda_{Solar}$ (＄/kWh)	$\lambda_{Load}$ (＄/kWh)
0.536	0.536	5

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Table 8. Electricity market prices

Hour	Day-ahead stage		Real-time stage
Hour	$\lambda_{Buy}^{DA}$	$\lambda_{Sell}^{DA}$	$\lambda_{Buy}^{RT}$	$\lambda_{Sell}^{RT}$
	(＄/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

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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)	$\eta_{ESS}^{Cha}/\eta_{ESS}^{Dis}$	$\delta_{ESS}$	(kWh)	(kWh)	(kWh)
0/200	0/200	0.9/0.9	0.001	480	120	120

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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)	$\eta_{HSS}^{Cha}/\eta_{HSS}^{Dis}$	$\delta_{HSS}$	(kWh)	(kWh)	(kWh)
0/200	0/200	0.9/0.9	0.01	600	0	0

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Table 2. Comparison of efficiencies and costs of different methods

Method	Time (s)	Day-ahead Cost(＄)			Expected	Actual (＄)
Method	Time (s)	$C_{MT}$	$C_{FC}$	$C_{DA}$	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

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Table 3. Comparison of electricity transactions of different methods

Method	Day-ahead Transaction			Actual Transaction
Method	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

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Table 4. Comparison of A-DED with different budgets

$\Gamma $	Iteration	Time (s)	Day-ahead Cost (＄)				Expected	Actual
$\Gamma $	Number	Time (s)	$C_{MT}$	$C_{FC}$	$C_{Grid}^{DA}$	$C_{DA}$	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

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Table 5. Comparison of A-DED with different budgets

Error	Iteration	Time	Day-ahead Cost (＄)				Expected	Actual (＄)
Error	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

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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

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