Consensus-based Optimal Control Strategy for Multi-microgrid Systems with Battery Degradation Consideration

Tonghe Wang; Hong Liang; Bo He; Haochen Hua; Yuchao Qin; Junwei Cao

doi:10.17775/CSEEJPES.2021.03180

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Regular Paper | Open Access

Consensus-based Optimal Control Strategy for Multi-microgrid Systems with Battery Degradation Consideration

Tonghe Wang^¹, Hong Liang^², Bo He^³, Haochen Hua^⁴, Yuchao Qin^⁵, Junwei Cao^⁶()

1Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510640, China

2Meihua Holdings Group Co., Ltd., Langfang 065001, China

3Department of Astronautical Science and Technology, Space Engineering University, Beijing 101416, China

4College of Energy and Electrical Engineering, Hohai University, Nanjing, 211100, China

5Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

6Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China

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Abstract

Consensus has been widely used in distributed control, where distributed individuals need to share their states with their neighbors through communication links to achieve a common goal. However, the objectives of existing consensus-based control strategies for energy systems seldom address battery degradation cost, which is an important performance indicator to assess the performance and sustainability of battery energy storage (BES) systems. In this paper, we propose a consensus-based optimal control strategy for multi-microgrid systems, aiming at multiple control objectives including minimizing battery degradation cost. Distributed consensus is used to synchronize the ratio of BES output power to BES state-of-charge (SoC) among all microgrids while each microgrid is trying to reach its individual optimality. In order to reduce the pressure of communication links and prevent excessive exposure of local information, this ratio is the only state variable shared between microgrids. Since our complex nonlinear problem might be difficult to solve by traditional methods, we design a compressive sensing-based gradient descent (CSGD) method to solve the control problem. Numerical simulation results show that our control strategy results in a 74.24% reduction in battery degradation cost on average compared to the control method without considering battery degradation. In addition, the compressive sensing method causes an 89.12% reduction in computation time cost compared to the traditional Monte Carlo (MC) method in solving the control problem.

Keywords

Battery degradation compressive sensing consensus distributed control multi-microgrid

References

[1]

Razi

and Y.

Ali

, “Ranking renewable energy production methods based on economic and environmental criteria using multi-criteria decision analysis,” Environment Systems and Decisions, vol. 39, no. 2, pp. 198–213, Jun. 2019.

A. Abbreviations
BES	Battery energy storage.
CSGD	Compressive sensing-based gradient descent.
EI	Energy Internet.
MC	Monte Carlo.
MT	Microturbine.
PV	Photovoltaic.
SoC	State-of-charge.
WT	Wind turbine.
B. Variables, Constants, and Parameters in Microgrids
$k_{i}^{MT}$	Control gain of MT.
$M G_{i}$	The $i^{th}$ microgrid.
$P_{i}^{BES} (t)$	Working power of BES.
$P_{i}^{in} (t)$	Power transmitted to $M G_{i}$ .
$P_{i}^{Load} (t)$	Working power of load.
$P_{i}^{\max}$	Maximum working power of BES.
$P_{i}^{MT} (t)$	Working power of MT.
$P_{i}^{PV} (t)$	Working power of PV.
$Q_{i}$	BES battery capacity.
$r_{i} (t)$	Ratio of BES output power to BES SoC.
${SoC}_{i} (t)$	SoC of the BES.
${SoC}_{i}^{\min}$ , ${SoC}_{i}^{\max}$	Recommended lower and upper bounds for ${SoC}_{i} (t)$ .
$u_{i}^{MT} (t)$	Control signal applied to MT.
$W_{i}^{Load} (t)$	Scalar Wiener process that captures the uncertainty in the change of $P_{i}^{Load} (t)$ .
$W_{i}^{PV} (t)$	Scalar Wiener process that captures the uncertainty in the change of $P_{i}^{PV} (t)$ .
$η_{i}$	Charging/discharging coefficient of BES.
$η_{i}^{in}, η_{i}^{out}$	Charging and discharging coefficients of BES.
$ρ_{i}^{Load}, σ_{i}^{Load}$	System coefficients related to load.
$ρ_{i}^{MT}$	Constant related to MT.
$ρ_{i}^{PV}, σ_{i}^{PV}$	System coefficients related to PV.
C. Consensus Related
$𝒜 = {(a_{i j})}_{n \times n}$	Adjacent matrix of graph $𝒢$ .
$e_{i} (t)$	Consensus error.
$𝒢$	Undirected graph.
$K = {(k_{i j})}_{n \times n}$	Consensus coefficient matrix.
$𝒩 (i)$	Neighbor set of $M G_{i}$ .
$u_{i}^{r} (t)$	Local state feedback function during consensus update.
$𝒱, ℰ$	Vertex set and edge set of graph $𝒢$ .
$κ_{i}^{r}$	Constant in consensus update.
$(i, j)$	Communication link (edge) between $M G_{i}$ and $M G_{j}$ .
D. Control Objectives
$J_{bes}$	Performance indicator of BES working power overflow.
$J_{cons}$	Performance indicator of consensus error.
$J_{dc}$	Performance indicator of battery degradation cost.
$J_{soc}$	Performance indicator of SoC overflow.
$ϵ_{1}, \dots, ϵ_{4}$	Significant factors of each performance indicator.
$ϕ_{i}^{bes} (t)$	Penalty function for exceeding the maximum working power of BES.
$ϕ_{i}^{soc} (t)$	Penalty function for exceeding recommended SoC bounds.
E. CSGD Algorithm
$c_{j} (t)$	Sparse coefficient.
$x (t), λ (t)$	System state and conjugate state.
$ξ_{j}$	Independent identically distributed random variable sampled from the standard normal distribution.
$φ_{j}$	Orthonormal basis.
$ψ_{j} (𝝃)$	Hermite polynomial.

Scenario	Control Objective	Consensus Variable	Shared State Information	Reference
Economic dispatch	Social welfare maximization	Incremental cost	Estimated power mismatch	[16]
	Minimizing generation cost	Incremental cost	Power mismatch	[20]
	Social welfare maximization	Cost of consumed energy, cost of produced energy	Cost of consumed energy, cost of produced energy	[21]
Reactive and active power sharing	Voltage regulation	Critical bus voltage, cumulative time after triggering on load tap charger	Critical bus voltage, cumulative time interval after triggering on load tap charger, measured reactive power, predicted reactive power	[22]
	Voltage regulation	Reative power, voltage amplitude	Reative power, voltage amplitude	[23]
	Frequency regulation	Reative power, active power	Reative power, active power, control parameters of droop controllers	[24]
	Energy consumption reduction	Heat/electric power sharing error	Proportional change of heat/electric power and its first deviation, auxiliary control signal, heat/electric power sharing error	[25]
BES management	SoC balancing	Relative SoC variation	Estimated average of lumped consensus parameter	[26]
	power control	SoC, output power	SoC, output power, voltage, current	[27]
	Voltage regulation	Estimation of average SoC	Utilization ratio	[19]
	PV voltage control	Average bus voltage	Ratio of BES output power to SoC, estimated average voltage	[28]

$i$	1	2	3	4	5
$ρ_{i}^{Load}$	28.4	27.1	26.5	28.8	27
$ρ_{i}^{PV}$	16.9	17.1	18.7	17	16.2
$σ_{i}^{Load}$	0.4	0.26	0.35	0.28	0.3
$σ_{i}^{PV}$	0.29	0.31	0.27	0.3	0.25
$η_{i}^{in}$	0.95	0.98	0.97	0.92	0.95
$η_{i}^{out}$	0.96	0.90	0.92	0.90	0.97
$Q_{i}$ (kW $\cdot$ h)	110	140	95	160	72
${SoC}_{i}^{\min}$	0.17	0.25	0.2	0.15	0.3
${SoC}_{i}^{\max}$	0.72	0.86	0.75	0.85	0.9
$P_{i}^{\max}$ (kW)	175	210	200	273	300
$ρ_{i}^{TM}$ (s)	15	18	12.7	18.3	21
$k_{i}^{TM}$	7.9	9.2	10	8.5	8.6

Sample Number	Error of MC ( $\times 10^{- 2}$ )	Error of CSGD ( $\times 10^{- 2}$ )
20	3.43	8.91
40	2.95	1.23
60	2.43	0.46
80	1.56	0.04
100	1.21	0.03
1000	0.32	0.03
2000	0.03	0.03

Case	$M G_{1}$	$M G_{2}$	$M G_{3}$	$M G_{4}$	$M G_{5}$
With degradation cost control	0.0729	0.2025	0.5658	5.2239	1.7842
No degradation cost control	1.0305	1.9256	3.8677	7.6654	6.2784
Reduction (%)	92.93	89.48	85.37	31.85	71.58

Method	CSGD	MC
Time Cost (s)	312.87	2892.16