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Distributed Energy Dispatch of Electrical Energy Storage Systems Using Consensus Control Approach
Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference on Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC Conference Modelling, Identification and Control of Nonlinear Systems Available online at www.sciencedirect.com Proceedings, 2nd IFAC Conference on Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018
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IFAC PapersOnLine 51-13 (2018) 229–234
Distributed Energy Dispatch of Electrical Distributed Energy Dispatch of Electrical Distributed Energy Dispatch of Electrical Energy Storage Systems Using Distributed Energy Dispatch ofConsensus Electrical Energy Storage Systems Using Consensus Energy Storage Systems Using Consensus Control Approach Energy Storage Systems Using Consensus Control Control Approach Approach Control Approach ∗,∗∗ Wenjing Xie ∗,∗∗∗ ∗,∗∗∗ Xiaohua Xia ∗,∗∗
Wenjing Xie ∗,∗∗∗ ∗,∗∗∗ Xiaohua Xia ∗,∗∗ Wenjing Xie ∗,∗∗∗ Xiaohua Xia ∗,∗∗ Wenjing Xie Xiaohua Xia ∗,∗∗ ∗ ∗ Department of Electrical, Electronic and Computer Engineering, Department of Electrical, Electronic and Computer Engineering, ∗ ∗ University of 0002, Africa. (e-mail: Department of Pretoria, Electrical,Pretoria Electronic andSouth Computer Engineering, ∗ University of Pretoria, Pretoria 0002, South Africa. (e-mail: Department of Pretoria, Electrical, Electronic andSouth Computer Engineering, University of Pretoria 0002, Africa. (e-mail: [email protected]). [email protected]). ∗∗ University of Pretoria, Pretoria 0002, South Africa. (e-mail: Northeastern University, Shenyang, 110004, China [email protected]). ∗∗ Shenyang, 110004, China ∗∗ Northeastern University, ∗∗∗ ∗∗ [email protected]). Computer and Information Science, Southwest University, Northeastern University, Shenyang, 110004, China ∗∗∗ School of ∗∗of Computer and Information Science, Southwest University, ∗∗∗ School ∗∗∗ Northeastern University, Shenyang, 110004, China Chongqing, 400715, China (e-mail: [email protected]) School of Computer and Information Science, Southwest University, Chongqing, 400715, China (e-mail: Science, [email protected]) ∗∗∗ School of Computer and Information Southwest University, Chongqing, 400715, China (e-mail: [email protected]) Chongqing, 400715, China (e-mail: [email protected]) Abstract: Energy Energy dispatch dispatch task task of of electrical energy energy storage storage systems systems (EESSs) (EESSs) contains contains the the state Abstract: Abstract: Energy dispatch tasksupply-demand of electrical electrical energy storage systems (EESSs) contains the state state of charge (SoC) balance and the balance, where the former ensures the maximum of charge (SoC) balance and the supply-demand balance, where the former ensures the maximum Abstract: Energy dispatch task of electrical energy storage systems (EESSs) contains the state power capacity of the group of EESSs, and the latter maintains the load operation and reduces of charge (SoC) balance and the supply-demand balance, where the former ensures the maximum power capacity of the group of EESSs, and the latter maintains theformer load operation and reduces of charge (SoC) balance and the supply-demand balance, where the ensures the maximum energy lost. A unified switching model is established for both charging and discharging processes power capacity of the group of EESSs, and the latter maintains the load operation and reduces energy lost. A unified switching model isand established both charging andoperation discharging power capacity of the group of EESSs, latterfor the load andprocesses reduces of EESSs. EESSs. EESSs areswitching assumed to share share athe common communication network. Dependent on energy lost.EESSs A unified model is established formaintains both charging and discharging processes of are assumed to a common communication network. Dependent on energy lost. A unified switching model is established for both charging and discharging processes of EESSs. EESSs are assumed to share a common communication network. Dependent on the communication between EESSs, three distributed energy dispatch algorithms are proposed the communication between EESSs, three adistributed dispatch algorithms are proposed of EESSs. EESSs are assumed to share common energy communication network. Dependent on based on approach, graph theory Lyapunov method. Without considering the communication between EESSs, three dispatch algorithms proposed based on consensus consensus control control approach, graphdistributed theory and andenergy Lyapunov method. Withoutare considering the communication between EESSs, three distributed energy dispatch algorithms are proposed constraints, the first algorithm is designed to determine the charging/discharging power reference based on consensus control approach, graph theory and Lyapunov method. Without considering constraints, the firstcontrol algorithm is designed to theory determine the charging/discharging power reference based on EESS consensus approach, graph and Lyapunov method. Without considering of every every as thealgorithm sum of of aisnominal nominal power value and an SoC-based correction term, where constraints, theas first designed to determine the charging/discharging power reference of EESS the sum a power value and an SoC-based correction term, where constraints, the first designed to determine thean charging/discharging power reference of every EESS as thealgorithm sum of aisnominal power value and SoC-based correction term, where the nominal term and the correction term respectively decrease the supply-demand and the SoC the nominal term and the correction termpower respectively decrease the supply-demand and the SoC of every EESS as the sum of a nominal value and anthe SoC-based correction term, where imbalances. By applying dynamic saturation function to first algorithm, aa second one is the nominal term and the a correction term respectively decrease the supply-demand and the SoC imbalances. By applying a dynamic saturation function to the first algorithm, second one is the nominal satisfying termapplying and the correction term respectively decrease the supply-demand and theone SoC constructed the charging/discharging mode switching constraint and the power limit imbalances. By a dynamic saturation function to the first algorithm, a second is constructed satisfying the charging/discharging mode switching constraint and the power limit imbalances. Bythe applying dynamic to first algorithm, a second one is constraint. On On basis ofacharging/discharging the secondsaturation algorithm,function third onethe is obtained obtained byand additionally taking constructed satisfying theof mode switching constraint the powertaking limit constraint. the basis the second algorithm, aaa third one is by additionally constructed satisfying the charging/discharging mode switching constraint and the power limit constraint. On the basis of the second algorithm, third one is obtained by additionally taking into account account the the SoC SoC limit. limit. Lyapunov Lyapunov stability stability analysis analysis shows shows that that the the third third algorithm algorithm not not into constraint. Onthe theSoC basis of the second algorithm, aanalysis third one isalso obtained additionally taking only asymptotically realizes the energy dispatch but satisfies the power and SoC into account limit. Lyapunov stability objective shows that thebythird algorithm not only asymptotically realizes the energy dispatch objective but also satisfies the power and SoC into account the SoC limit. Lyapunov stability analysis shows that the third algorithm not constraints, provided that the communication graph is undirected undirected and connected. Effectiveness only asymptotically realizes the energy dispatch objective but also satisfies the power and SoC constraints, provided that the communication graph is and connected. Effectiveness only asymptotically realizes the energy dispatch objective but also satisfies the power and SoC constraints, provided that the communication graph is undirected and connected. Effectiveness of the third third algorithm algorithm is is demonstrated by by simulation results. results. of constraints, the communication graph isresults. undirected and connected. Effectiveness of the the third provided algorithmthat is demonstrated demonstrated by simulation simulation © 2018, IFACalgorithm (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. of the third is demonstrated by simulation results. Keywords: Energy Energy storage storage systems, systems, energy energy dispatch, dispatch, distributed distributed control, control, consensus control. control. Keywords: Keywords: Energy storage systems, energy dispatch, distributed control, consensus consensus control. Keywords: Energy storage systems, energy dispatch, distributed control, consensus control. 1. INTRODUCTION INTRODUCTION the the dispatch dispatch task task of of EESSs: EESSs: the the supply-demand supply-demand balance balance 1. 1. INTRODUCTION the dispatch task of EESSs: the supply-demand balance and the SoC balance. The supply-demand balance is and the SoC balance. The supply-demand balance is that that 1. INTRODUCTION the dispatch task ofsupplied EESSs: the supply-demand balance and the SoCoutputs balance. The supply-demand balance is that the power from renewable generations Nowadays, electrical energy storage units are widely apthe power outputs supplied from renewable generations Nowadays, electrical energy storage units are widely ap- the the SoCoutputs balance. The supply-demand balance is that power supplied from renewable generations and EESSs need meet demands with aa low/zero Nowadays, electrical storagestorage, units are ap- and plied in in power power grids. energy With energy energy the widely imbalance and power EESSs outputs need to to supplied meet load load demands with generations low/zero plied grids. With storage, the imbalance the from renewable and EESSs need to meet load demands with a low/zero Nowadays, electrical energy storage units are widely apenergy lost (Xu et al., 2015). The SoC balance is plied in power grids. With energy generations storage, the and imbalance between intermittent renewable time- energy lost (Xu et al., 2015). The SoC balance is that that between intermittent renewable generations and timeand EESSs to al., meet load SoC demands a low/zero energy lost need (Xu et 2015). The value SoC with balance isetthat plied in power grids. With energy storage, the imbalance EESSs have the same dynamic (Morstyn al., between intermittent renewable generations and timevarying load demands can be reduced or vanished, deEESSs have the same dynamic SoC value (Morstyn et al., varying load demands renewable can be be reduced reduced or vanished, vanished, de- EESSs energy lost (Xu et al., 2015). The SoC balance iset that have the same dynamic SoC value (Morstyn al., between intermittent generations and time2015; Li et al., 2017). Reasons of balancing the SoCs varying load demands can or decreasing the usage of fossil fuel and electricity cost (Suberu 2015; Li et al., 2017). Reasons of balancing the SoCs creasing the usage of fossil fuel and electricity cost (Suberu EESSs have the dynamic SoC (Morstyn et al., Li etare al., same 2017). Reasons ofvalue balancing the SoCs varying load demands canfuel beand reduced or vanished, de- 2015; of as aspects in (Cai and creasing the Bragard usage of fossil electricity costdischarg(Suberu et al., al., 2014; 2014; et al., al., 2010). Charging and of EESSs EESSs areal.,summarized summarized as two two aspects in the (CaiSoCs and et Bragard et 2010). Charging and discharg2015; Li et 2017). Reasons of balancing of EESSs are summarized as two aspects in (Cai and creasing the usage of fossil fuel and electricity cost (Suberu Hu, 2016; Xu et al., 2017). Firstly, it protects EESSs et 2014;control Bragardofetsuch al., 2010). and dischargingal., power storageCharging units plays plays a crucial crucial Hu, 2016; Xu et al., 2017). Firstly, it protects EESSs ing power control ofetsuch such storage units of EESSs are two aspects in (Cai and 2016; Xu summarized et al., 2017).asFirstly, it aprotects EESSs et al., 2014; Bragard al., supporting 2010). Charging and dischargfrom overcharging or at higher ing power control of storage units plays aa crucial role in saving energy and load operation. So Hu, from 2016; overcharging or overdischarging overdischarging at aprotects higher system system role in saving energy and supporting load operation. So Hu, Xu et al., 2017). Firstly, it EESSs from overcharging or overdischarging at a higher system ing power control ofand such storage units a crucial Secondly, EESSs with SoCs have role in saving energy andcontrol supporting loadplays operation. So level. far, dispatch of storage device level. overcharging Secondly, the the EESSs with balancing balancing SoCssystem have far, energy energy dispatch and control of single single storage device from or EESSs overdischarging at the a higher Secondly,power the with balancing SoCs have role inbeen saving energyand and supporting load operation. So level. the maximum capacity. Without SoC balance far, energy dispatch control ofvia single storage device have extensively researched optimization and the maximum power capacity. Without the SoC balance have been extensively researched via optimization and level. Secondly, the EESSs with balancing SoCs have maximum some powerEESSs capacity. Without the SoC balance far, energy dispatchaiming andresearched control ofvia single storage device consideration, will stop when their have been extensively optimization and control techniques, to maximize maximize the renewable en- the consideration, some EESSs willWithout stop working working when their control techniques, aiming to the renewable enthe maximum power capacity. the SoC balance consideration, some EESSs will stop working when their have been extensively researched via optimization and SoCs reach the high or low limit. In this case, the power control techniques, aiming to maximize the renewable energy usage and simultaneously satisfy load demands and SoCs reach the high or low limit. In this case, the power ergy usage and simultaneously simultaneously satisfy load load demands and and consideration, stop working when their reach thesome high EESSs orEESSs low will limit. In this case, thea power control techniques, aiming to maximize the renewable en- SoCs capacity of may with worse ergy usage and satisfy demands constraints (Sichilalu et Tazvinga et capacity of the overall overall EESSs mayIndecrease decrease with a power worse constraints (Sichilalu et al., al., 2016; 2016; Tazvinga et al., al., 2015; 2015; SoCs reach high or low limit. this case, the capacity of the overall EESSs may decrease with a worse ergy usage and simultaneously satisfy load demands and system performance. Typical results incorporating the constraints (Sichilalu et al., Tazvinga et the al., devel2015; system performance. Typical results incorporating the two Zhu et et al., al., 2015; 2015; Sichilalu and2016; Xia, 2015). 2015). With two Zhu Sichilalu and Xia, With the develcapacity of theobtained overall EESSs may decrease with athe worse performance. Typical results incorporating two constraints (Sichilalu et al., 2016; Tazvinga et al., 2015; system balances are in (Lu et al., 2014, 2015; Yang Zhu et al., 2015; Sichilalu andand Xia,sensor 2015).network, With the development of battery technology multiple balances are obtained in (Lu et al., 2014, 2015; Yang opment of battery battery technology and sensor network, multiple system performance. Typical results incorporating two are Cai obtained in 2016; (Lu etXual., 2014, 2015;the Yang Zhu et al., 2015; Sichilalu andand Xia, 2015).network, With the devel- balances et and et al., In (Lu opment of technology sensor multiple package-level electrical energy storage systems (EESSs) et al., al., 2017; 2017; Cai and Hu, Hu, 2016; Xual., et 2014, al., 2017). 2017). InYang (Lu package-level electrical energy storage systems (EESSs) balances are obtained in (Lu et 2015; et al., 2017; Cai and Hu, 2016; Xu et al., 2017). In (Lu opment of battery technology andstorage sensor network, multiple Yang et independent of package-level electrical energy systems (EESSs) also receive receive much attention. Every EESS is composed composed of et al., al., 2014, 2014, 2015; 2015; Yang et al., al., 2017), 2017), independent of also much attention. Every EESS is of al., andYang Hu, EESSs, 2016; et al.,independent 2017).adaptive In (Lu et al., 2017; 2014, Cai 2015; et al.,Xu 2017), of package-level electrical energy storage systems (EESSs) communications between decentralized also receive much attention. Every EESS is composed of series and/or parallel configured battery cells. communications between EESSs, decentralized adaptive series and/or parallel configured battery cells. et al., 2014, 2015; Yang et al., 2017), independent of betweenwere EESSs, decentralized adaptive also receive attention. Every EESScells. is composed of communications droop proposed to droop series and/ormuch parallel configured battery droop control control strategies strategies were proposed to tune tune the the droop communications between EESSs, decentralized adaptive For an an EESS, the configured charging/discharging power referrefer- droop control strategies were proposed to tune the droop series and/or parallel battery cells. coefficient according to SoC. In (Cai and Hu, 2016), for the For EESS, the charging/discharging power coefficient according to SoC. In (Cai and Hu, 2016), for the For EESS, thebycharging/discharging power droop control strategies weredesigned to tune the droop ence an is determined determined dispatch algorithm, algorithm, and and the refervolt- coefficient according to SoC. Inproposed (Cai and Hu, 2016), for the EESS with higher SoC, the distributed dispatch ence is by dispatch the voltEESS with higher SoC, the designed distributed dispatch For an EESS, the charging/discharging power reference is determined by dispatch algorithm, and the voltcoefficient according to SoC. In (Cai and Hu, 2016), for the age/current controller controller is is designed designed to to adjust adjust the the actual actual EESS withassigned higher SoC, the designed distributed dispatch algorithm more discharging (less charging) power age/current algorithm assigned morethe discharging (less charging) power ence istodetermined by dispatch algorithm, and theactual voltage/current controller is designed to adjust the EESS withassigned higher SoC, designed(less distributed dispatch power track the reference. There are two objectives in algorithm more discharging charging) power power to track track the reference. reference. There to areadjust two objectives objectives in age/current controller is designed the actual power to the There are two in algorithm assigned more discharging (less charging) power power to track the reference. There are two objectives in Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Federation of Automatic Control)
Proceedings, 2nd IFAC Conference on 229 Proceedings, 2nd IFAC Conference on 229 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference 229 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.283 Proceedings, 2nd IFAC Conference on 229 Systems Modelling, Identification and Control of Nonlinear Systems Modelling, and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018 Systems Identification
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than the estimated average value of desired total power, and for the one with lower SoC, assigned less discharging (more charging) power. In (Xu et al., 2017), a distributed power dispatch strategy was constructed by borrowing an economic dispatch algorithm. However, the determination of power reference in (Lu et al., 2014, 2015; Yang et al., 2017) had to rely on the information of all the EESSs and therefore were not distributed, constraints on power and SoC were not considered in (Cai and Hu, 2016), and in (Xu et al., 2017) the total charging/discharging power was only able to track the initial (not the real time) supply-demand mismatch. Based on these observations, energy dispatch of EESSs remains a hot topic of investigation. This paper concerns how to realize the energy dispatch task of EESSs with power and SoC constraints. The unified switching mode is firstly presented by using the sign function. The consensus control approach is applied to solve the dispatch problem. To guarantee the system stability under constraints, three algorithms are derived one by one. The first algorithm is designed in case of no constraints, then by additionally utilizing the dynamic saturation function, a second one is proposed in case of constrained power, and a final one is obtained in case of constrained power and SoC. Convergence of all the three algorithms is proved by Lyapunov method, under the assumption that the communication graph is undirected and connected. Simulation cases illustrate the effectiveness of the proposed third algorithm. The rest of this paper is organized as follows. Section 2 formulates the energy dispatch problem of multiple EESSs. Section 3 solves this problem using the consensus control approach. Simulation examples are provided in Section 4 to demonstrate the effectiveness of the proposed dispatch strategy. Finally, Section 5 concludes this work.
2.1 Simplified structure of EESSs in a microgrid generations
PG1
Bus P1
G PG2
P2
G PGm
P3
1
2
EESS
EESS
Pn-1 3
EESS communication network
Pn
PL
n-1
n
EESS
EESS
Based on (2), the charging/discharging process of the ith EESS can be described by: Si (k + 1) = Si (k) − T ηi (k)Pi (k),
where k is the sampling time, T is the sampling interval, and 1 ηdi > 0, (discharging) ρi σi ζ Q = ρdiζ i Vi ηi (k) = Qi Vi i ci ηci > 0, (charging). Qi Vi Assume that all the EESSs have the same model, i.e., ηi (k) = η(k), thus Si (k + 1) = Si (k) − T η(k)Pi (k). (3)
Every EESS has three modes: charging, discharging and stop modes, and (3) is a switching system. Denote the global supply-demand mismatch by: (4) PM (k) = PL (k) − PG (k), where PL (k) and PG (k) are the total load demand and the total power generation respectively. If PM (k) > 0 (< 0), EESS i (i ∈ V) is in discharging or stop (charging or stop) operation with η(k) = ηd (ηc ) and Pi (k) ≥ 0 (≤ 0), where ηd = ηdi and ηc = ηci . If PM (k) = 0, the n EESSs stop working with Pi (k) = 0, which is not our consideration in this paper. Therefore, η(k) has the unified form: η(k) = 0.5sign(PM (k)) [sign(PM (k)) + 1] ηd
2. PROBLEM FORMULATION
G
ρi t (1) Ii (t)dt, i ∈ V, Si (t) = Si (0) − Qi 0 where V = {1, 2, · · · , n} is the index set of EESSs, Si (t) ∈ (0, 1) denotes the SoC of the ith EESS, and (ρi , Qi , Ii ) respectively represents the Coulombic efficiency, the capacity and the output current of the ith EESS. Coulombic efficiency ρi satisfies ρi = 1 in discharging mode and ρi ∈ (0, 1) in charging mode. Output current Ii is (Li et al., 2017): 1/ζdi , (discharging), Ii = σi Pi /Vi , σi = (charging), ζci , where Pi denotes the output power of the ith converter, Vi is the output voltage of the ith EESS, and σi is related to the converter efficiencies (ζdi , ζci ) of the ith converter during discharging and charging operations. It is assumed that Vi is time-invariant (Li et al., 2017). SoC equation (1) can be rewritten as: ρi σi t (2) Si (t) = Si (0) − Pi (t)dt, i ∈ V. Qi Vi 0
Loads
+ PM (k)
Fig. 1. Simplified structure of EESSs in microgrid. As shown in Fig. 1, renewable generations and EESSs are installed in the microgrid, in order to provide electrical energy to loads. The EESSs store the surplus power when the power generation is more than load demand, otherwise return the stored energy to loads. Suppose that there are n EESSs in the microgrid. 2.2 Unified model of charging/discharging process of EESSs The SoC of an EESS can be calculated by (Cai and Hu, 2016) 230
+ 0.5sign(PM (k)) [sign(PM (k)) − 1] ηc ,
(5)
− PM (k)
where sign(·) is the sign function. To make EESSs operate in appropriate modes, we propose the charging/discharging mode switching constraint: (6) PM (k)Pi (k) ≥ 0. Other two constraints for security of EESSs (Xu et al., 2015) are Pmin ≤ Pi (k) ≤ Pmax , (7) Smin ≤ Si (k) ≤ Smax ,
(8)
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where Pmax > 0, Pmin < 0, Pmax and Pmin are maximum power allowed to discharge and charge EESS i, and Smin and Smax are respectively the lower and the upper limits of SoC. Now, the unified switching model of EESSs is established as (3) and (5) with the constraints (6), (7) and (8). 2.3 Communication network of EESSs To make the EESSs cooperatively charge and discharge for their group objective, we suppose that the EESSs compose a communication network based on wireless sensors (Sudevalayam and Kulkarni, 2011). In this network, they can share information (e.g., Si (k) and ηi (k)) with each other, as shown in Fig. 1. Model the information flow between them as a communication graph G(V, E, A), where V = {1, 2, · · · , n} is the index set of EESSs, E ⊂ V ×V is an edge set of ordered pairs of EESSs, and A = [aij ] ∈ Rn×n is the adjacency matrix with entries aij = 1 or aij = 0 (Olfati-Saber and Murray, 2004). If EESS i can receive information from EESS j, then (j, i) ∈ E, aij = 1 and EESS j is called the communication neighbor of EESS i, denoted by j ∈ Ni , where Ni = {j ∈ V|aij = 1}. If EESS i cannot have access to the information of EESS j, then (j, i) ∈ / E, aij = 0 and j ∈ Ni . Self-connection is not considered for G, i.e., aii = 0, ∀i ∈ V (Olfati-Saber and Murray, 2004). A graph G is undirected if aij = aji for any i, j ∈ V. The Laplacian matrix L = [lij ] ∈ Rn×n is defined as: if j = i; ij , −a n lij = aij , if j = i. j=1
For undirected connected graph, L = LT has the following properties (Olfati-Saber and Murray, 2004): − → (9) 0 = λ1 < λ2 ≤ λ3 ≤ · · · ≤ λn , L 1 = 0, − → where (λ1 , λ2 , · · · , λn ) are the eigenvalues of L, and 1 = [1, 1, · · · , 1]T ∈ Rn . To achieve distributed dispatch, we make the following Assumption. Assumption 1. The communication graph G(V, E, A) of EESSs is undirected and connected. 2.4 Energy dispatch objective
The energy dispatch objective of EESSs considered in this paper is stated as (Cai and Hu, 2016): under Assumption 1, design a distributed dispatch strategy Pi (i ∈ V) for EESS i only using the information of its communication neighbors and itself, such that under constraints (6)-(8), the SoCs of the n EESSs reach a balance Si (k) − Sj (k) → 0, ∀i, j ∈ V, (10) and the power tracking error n Pe (k) = Pi (k) − PM (k) (11) i=1
is minimized after (10) is achieved. Objective (10) ensures the SoC balancing between EESSs, and minimizing Pe (k) guarantees the supply-demand balance as well as reduces energy lost in charging and discharging operations. Reasons and physical meanings of the SoC balance can be found in (Cai and Hu, 2016). For the general case of
231
231
Smin Si Smax , Pi (k) can be adjusted such that Pe (k) → 0. For the special cases that Si (k) is close or equal to Smax /Smin in charging/discharging mode, |Pi (k)| must be small or zero such that Si (k+1) remains in [Smin , Smax ], and Pe (k) can only be minimized rather than Pe (k) → 0.
The energy dispatch objective is asymptotically achieved if for any initial SoCs satisfying Si ∈ [Smin , Smax ], the SoC balancing errors (Si (k) − Sj (k), i, j ∈ V) are asymptotically convergent to zero, the power tracking errors Pe (k) is minimized after Si (k) − Sj (k) → 0, and the constraints (6)-(8) are met. 3. DISTRIBUTED ENERGY DISPATCH In this section, we employ multiagent consensus control idea to develop the first dispatch scheme without considering constraints, then extend it to the case of constraints, solving the energy dispatch problem of multiple EESSs. 3.1 Algorithm design without constraints For sake of simplicity, we assume that the power and SoC variables are not subject to constraint. Based on the consensus control idea in (Olfati-Saber et al., 2007), we propose the distributed dispatch algorithm ε aij [Si (k) − Sj (k)] , (12) Pi (k) = Pave (k) + η(k) j∈Ni
where ε > 0, the term Pave (k) = PMn(k) is the nominal power component corresponding to thesupply-demand aij [Si (k) − balance, and the SoC balance error j∈Ni Sj (k)] is correction term. Lemma 1. If Assumption 1 holds and ε is a sufficiently small positive constant, then the dispatch strategy (12) guarantees the SoC balance errors Si (k) − Sj (k) (i, j ∈ V) globally asymptotically converge to zero and Pe (k) = 0. Proof. Define S(k) [S1 (k), S2 (k), · · · , Sn (k)]T ,
P (k) [P1 (k), P2 (k), · · · , Pn (k)]T − → = Pave (k) 1 + εη −1 (k)LS(k),
(13)
− → where 1 = [1, 1, · · · , 1]T ∈ Rn . As Assumption 1 holds, n − →T − → − → − → 1 L = 0, and Pi (k) = 1 T P (k) = 1 T [Pave (k) 1 + i=1
εη −1 (k)LS(k)] = PM (k), implying Pe (k) = 0.
Under algorithm (12), the closed-loop form of EESSs is − → (14) S(k + 1) = −η(k)Pave (k) 1 + (In − εL)S(k),
where In ∈ Rn×n is the identity matrix. Define the Lyapunov function (Olfati-Saber and Murray, 2004): (15) V (k) = S T (k)LS(k) ≥ 0, where L is positive semi-definite. Compute ∇V (k) = V (k + 1) − V (k) T − → = −η(k)Pave (k) 1 + (In − εL)S(k) (16) − → × L −η(k)Pave (k) 1 + (In − εL)S(k) − V (k) = S T(k)LT (−2εIn + ε2 L)LS(k),
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− → where L = LT and L 1 = 0 are used. Since ε is sufficiently small, −2εIn + ε2 L is guaranteed negative definite. So, ∇V (k) ≤ 0. Based on Assumption 1 and the negative definite property of −2εIn + ε2 L, one has − → ∇V (k) = 0 ⇔ LS(k) = 0 ⇔ S(k) ∈ span{ 1 }, thus n the largest invariant set in {S(k) ∈ R |∇V (k) = 0} − → is {S(k) ∈ span{ 1 }}. According to the discrete time version of LaSalle invariance principle (Alberto et al., 2007), S(k) globally asymptotically convergent to {S(k) ∈ − → span{ 1 }}, and the SoC balance errors are therefore globally asymptotically vanished to zero. 3.2 Generation to constrained power case With power constraints (6) and (7), it is required to make the following Assumption. Assumption 2. The number of EESSs is determined such that the nominal power is within the permissible power range Pmin < Pave (k) < Pmax . Improve the dispatch scheme (12) as (17) Pi (k) = Pave (k) + η −1 (k)f [εLi S(k)], aij [Si (k) − Sj (k)] , Li is the ith where Li S(k) = j∈Ni
row of matrix L, and the bounded dynamic saturation function f (·) is given by χ(k), if χ(k) ∈ [fmin (k), fmax (k)], f [χ(k)] = fmax (k), if χ(k) > fmax (k), fmin (k), if χ(k) < fmin (k), + fmin (k) = PM (k) [−η(k)Pave (k)] (18) − + PM (k) [η(k)(Pmin − Pave (k))] , − fmax (k) = PM (k) [−η(k)Pave (k)] + + PM (k) [η(k)(Pmax − Pave (k))] .
Remark 1. Under Assumption 2, the function f (·) in (18) has the following features: χ(k)f [χ(k)] ≥ f 2 [χ(k)] ≥ 0, (19) f [χ(k)] = 0 ⇔ χ(k) = 0,
which are proved as follows. When PM (k) > 0, we have Pave (k) > 0 and fmin (k) = −η(k)Pave (k) < 0, fmax (k) = η(k)[Pmax − Pave (k)] > 0. When PM (k) < 0, we get Pave (k) < 0 and fmin (k) = η(k)[Pmin − Pave (k)] < 0, fmax (k) = −η(k)Pave (k) > 0.
In conclusion, fmin (k) < 0 < fmax (k), so f [χ(k)] = 0 ⇔ χ(k) = 0. By (18), it follows χ(k)f [χ(k)] = 2 if χ(k) ∈ [fmin (k), fmax (k)], χ (k) = f 2 [χ(k)], χ(k)fmax (k) > f 2 [χ(k)], if χ(k) > fmax (k), χ(k)fmin (k) > f 2 [χ(k)], if χ(k) < fmin (k). Lemma 2. Under Assumptions 1-2 and ε > 0 is sufficiently small, the distributed dispatch scheme (17) satisfies the constraints (6) and (7), and ensures the SoC balance errors and Pe (k) globally asymptotically convergent to zero. 232
Proof. (I) To verify that (6) and (7) are satisfied. The following three cases are taken into account. (a) For the case of εLi S(k) ∈ [fmin (k), fmax (k)], the algorithm (17) takes the form of Pi (k) = Pave (k) + η −1 (k)εLi S(k). As εLi S(k) ∈ [fmin (k), fmax (k)], it follows that Pi (k) ≤ Pave (k) + η −1 (k)fmax (k) Pmax , if PM (k) > 0 (discharging), = 0, if PM (k) < 0 (charging), Pi (k) ≥ Pave (k) + η −1 (k)fmin (k) 0, if PM (k) > 0 (discharging), = Pmin , if PM (k) < 0 (charging). Thismeans that 0 ≤ Pi (k) ≤ Pmax , if PM (k) > 0 (discharging), Pmin ≤ Pi (k) ≤ 0, if PM (k) < 0 (charging).
(b) For the case of εLi S(k) > fmax (k), (17) becomes Pi (k) = Pave (k) + η −1 (k)fmax (k) Pmax , if PM (k) > 0 (discharging), = 0, if PM (k) < 0 (charging). (c) For the case of εLi S(k) < fmin (k), (17) shows that Pi (k) = Pave (k) + η −1 (k)fmin (k) 0, if PM (k) > 0 (discharging), = Pmin , if PM (k) < 0 (charging).
The above three cases imply that (6) and (7) are fulfilled. (II) To verify that the SoC balance errors and Pe (k) are tending to zero. − → By (17), we have P (k) = Pave (k) 1 + η −1 (k)f (εLS(k)), where f (εLS(k)) = [f (εL1 S(k)), · · · , f (εLn S(k))]T ∈ Rn . Compute ∇V (k) = V (k + 1) − V (k) as T
∇V (k) = [S(k) − f (εLS(k))] L[S(k) − f (εLS(k))] − V (k)
≤ f T (·)(L − 2ε−1 In )f (·), where f (·) = f (εLS(k)), and the first inequality in (19) is utilized. As ε > 0 is sufficiently small, L − 2ε−1 In is negative definite and ∇V (k) ≤ 0. Under Assumptions 1-2 and ε > 0, Remark 1 shows that f (εLS(k)) = 0 ⇔ S(k) ∈ − → span{ 1 }. So, S(k) globally asymptotically converges to − → {S(k) ∈ span{ 1 }} which is the largest invariant set in n {S(k) ∈ R |∇V (k) = 0} (Alberto et al., 2007). This implies that the SoC balance errors and Pe (k) globally asymptotically tend to zero. 3.3 Generation to the case of constrained power and SoC
Considering both the power and the SoC constraints (6)(8), we modify (17) as if Sˆi (k + 1) ∈ [Smin , Smax ], Pˆi (k), −1 Pi (k) = η (k)[Si (k) − Smax ], if Sˆi (k + 1) > Smax , −1 η (k)[Si (k) − Smin ], if Sˆi (k + 1) < Smin , (20) where Pˆi (k) = Pave (k) + η −1 (k)f [εLi S(k)], (21) Sˆi (k + 1) = Si (k) − η(k)Pˆi (k).
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Theorem 1. Suppose that Assumptions 1-2 hold and ε > 0 is sufficiently small, then for any initial SoCs satisfying Si (0) ∈ [Smin , Smax ], the distributed algorithm (20) guarantees that the constraints (6)-(8) are satisfied, the SoC balance errors of EESSs Si (k) − Sj (k) (i, j ∈ V) are asymptotically decayed to zero, and the power tracking error Pe (k) is minimized after Si (k) − Sj (k) → 0, i, j ∈ V. Proof. (I) To prove (8). The closed-loop system is Si (k + 1) = Si (k) − η(k)Pi (k) Sˆi (k + 1) ∈ [Smin , Smax ], if Sˆi (k + 1) ∈ [Smin , Smax ], = Smax , if Sˆi (k + 1) > Smax , Smin , if Sˆi (k + 1) < Smin . (II) To prove (6)-(7).
If Sˆi (k + 1) ∈ [Smin , Smax ], it follows from Lemma 2 that Pi (k) = Pˆi (k) satisfies the conditions (6)-(7). If Sˆi (k + 1) > Smax , then EESS i is in charging/stop mode, and 0 ≥ Pi (k) = η −1 (k)[Si (k) − Smax ] = η −1 (k)[Sˆi (k + 1) − Smax ] + Pˆi (k) > Pˆi (k) ≥ Pmin ,
where Pˆi (k) ≥ Pmin has been proven by the proof of Lemma 2. If Sˆi (k+1) < Smax , then EESS i is in discharging mode, and 0 ≤ Pi (k) = η −1 (k)[Si (k) − Smin ] = η −1 (k)[Sˆi (k + 1) − Smin ] + Pˆi (k) < Pˆi (k) ≤ Pmax . Consequently, the power constraints (6) and (7) are satisfied. (III) To prove the convergence of SoC balance errors.
i=1 j∈Ni
ˆ + 1) = SˆT (k + 1)LS(k
By (22) and (23), it follows n {aij [Si (k + 1) − Sj (k + 1)]2 } V (k + 1) = i=1 j∈Ni
≤
n
i=1 j∈Ni
{aij [Sˆi (k + 1) − Sˆj (k + 1)]2 }
≤ V (k) + f T (·)(L − 2ε−1 In )f (·), which, together the proof of Lemma 2, shows that the SoC balance errors are asymptotically vanished to zero. (IV) To prove that Pe (k) is minimized after Si (k) − Sj (k) → 0.
Therefore, Si (k + 1) ∈ [Smin , Smax ].
According to the proof of Lemma 2, we have n {aij [Sˆi (k + 1) − Sˆj (k + 1)]2 }
233
(22)
≤ V (k) + f T (εLS(k))(L − 2ε−1 In )f (εLS(k)), ˆ + 1) = [Sˆ1 (k + 1), · · · , Sˆn (k + 1)]T . where S(k
If Sˆi (k + 1) and Sˆj (k + 1) are both in [Smin , Smax ], then [Si (k + 1) − Sj (k + 1)]2 = [Sˆi (k + 1) − Sˆj (k + 1)]2 .
If they do not belong to [Smin , Smax ], then [Si (k + 1) − Sj (k + 1)]2 = 0 ≤ [Sˆi (k + 1) − Sˆj (k + 1)]2 ,
where Si (k + 1) = Sj (k + 1) = Smax or Smin due to the fact that all the EESSs are in the same charging/stop or discharging/stop mode. If one of them (assumed as Sˆi (k + 1)) belongs to [Smin , Smax ] and the other one does not, then [Si (k + 1) − Sj (k + 1)]2 = [Smax − Sˆj (k + 1)]2 < [Sˆi (k + 1) − Sˆj (k + 1)]2 , (charging), [Si (k + 1) − Sj (k + 1)]2 = [Smin − Sˆj (k + 1)]2
< [Sˆi (k + 1) − Sˆj (k + 1)]2 , (discharging). Summarizing the above analysis yields: [Si (k + 1) − Sj (k + 1)]2 ≤ [Sˆi (k + 1) − Sˆj (k + 1)]2 . (23) 233
After Si (k) − Sj (k) → 0 is achieved, denote Si (k) = S ∗ (k), ∀i ∈ V, and all the EESSs have the same power if Sˆi (k + 1) ∈ [Smin , Smax ], Pave (k), −1 ∗ Pi (k) = η (k)[S (k) − Smax ], if Sˆi (k + 1) > Smax , −1 η (k)[S ∗ (k) − Smin ], if Sˆi (k + 1) < Smin . For the case of Sˆi (k + 1) ∈ [Smin , Smax ], one gets Pe (k) = 0. For the case of Sˆi (k + 1) > Smax , only Pi (k) ∈ [η −1 (k)(S ∗ (k)−Smin ), 0] can ensure Pi (k) and Si (k+1) fulfill (6)-(8). Note that Pave (k) < η −1 (k)(S ∗ (k) − Smin ) and n nPave (k) = PM (k), so under (6)-(8), Pe (k) = | Pi (k) − i=1
PM (k)| is minimized at Pi (k) = η −1 (k)(S ∗ (k) − Smin ). Analogously, for the case of Sˆi (k + 1) < Smin ), Pe (k) is minimized at Pi (k) = η −1 (k)[S ∗ (k) − Smin ]). 4. SIMULATION RESULTS
In this section, numerical simulations are implemented to demonstrate the effectiveness of the third proposed algorithm. Suppose that there are ten EESSs connected by the communication graph depicted in Fig. 2. The model parameters of EESSs are (Cai and Hu, 2016) Qi = 120Ah, Vi = 220V, ζdi = 0.92, ζci = 0.95, i = 1 ∼ 10. The SoC range is [Smin , Smax ] = [0.1, 0.9], and the power range is [Pmin , Pmax ] = [−5kW, 5kW]. Initial SoCs are chosen as S(0) = 0.01 × [28, 31, 34, 37, 40, 44, 46, 49, 52, 55]T , and ε = 0.2. The sampling interval is selected as T = 1 hour, and the operation period is two days. In the two days, the global supply-demand mismatch is periodic with period one day, shown in Tab. I. 1
2
3
4
5
10
9
8
7
6
Fig. 2. Communication graph. Carry out the third algorithm for two days. Fig. 3 shows the SoC and power trajectories of every EESS. It can be observed that the SoC variables are balanced to a common dynamic value, and SoC and power variables satisfy the constraints (6)-(8). Fig. 4 presents the power tracking error Pe (k), implying that Pe (k) is convergent to zero. 5. CONCLUSION In this paper, three energy dispatch algorithms are proposed to determine the charging/discharging power refer-
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Table 1. Global supply-demand mismatch PM (k) (kW) time 00 : 00 01 : 00 02 : 00 03 : 00 04 : 00 05 : 00 06 : 00 07 : 00
mismatch 7.56 7.56 7.56 7.56 7.56 8.40 12.08 1.904
time 08 : 00 09 : 00 10 : 00 11 : 00 12 : 00 13 : 00 14 : 00 15 : 00
mismatch −12.88 −24.32 −37.76 −35.616 −35.04 −25.28 −29.28 −25.28
time 16 : 00 17 : 00 18 : 00 19 : 00 20 : 00 21 : 00 22 : 00 23 : 00
mismatch −11.84 −0.32 12.656 16.80 16.80 16.80 14.00 11.20
SoC Si (k) (%)
100 80 60
EESS 2
20
EESS 3
0
Power Pi (k) (Kw)
EESS 1
40
EESS 4 0
5
10
15
20 25 time (h)
30
35
40
45
EESS 5 EESS 6
6
EESS 7
4
EESS 8 EESS 9
2
EESS 10
0 −2 −4
0
5
10
15
20 25 time (h)
30
35
40
45
Power tracking error Pe (k) (Kw)
Fig. 3. SoC and power trajectories under the third algorithm. 1 0.8 0.6 0.4 0.2 0
0
5
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15
20
25 time (h)
30
35
40
45
Fig. 4. Power tracking error Pe (k) under the third algorithm. ence for every EESS with the objectives of SoC and supplydemand balances. Consensus control approach, Lyapunov method and graph theory are employed in designing these algorithms. The third one asymptotically realizes the energy dispatch task under SoC and power constraints. Future work may lie on the distributed energy dispatch problem of EESSs with different model parameters. REFERENCES Alberto, L.F., Calliero, T.R., and Martins, A.C. (2007). An invariance principle for nonlinear discrete autonomous dynamical systems. IEEE Transactions on Automatic Control, 52(4), 692–697. Bragard, M., Soltau, N., Thomas, S., and De Doncker, R.W. (2010). The balance of renewable sources and user demands in grids: Power electronics for modular battery energy storage systems. IEEE Transactions on Power Electronics, 25(12), 3049–3056. Cai, H. and Hu, G. (2016). Distributed control scheme for package-level state-of-charge balancing of gridconnected battery energy storage system. IEEE Transactions on Industrial Informatics, 12(5), 1919–1929. 234
Li, C., Coelho, E.A.A., Dragicevic, T., Guerrero, J.M., and Vasquez, J.C. (2017). Multiagent-based distributed state of charge balancing control for distributed energy storage units in AC microgrids. IEEE Transactions on Industry Applications, 53(3), 2369–2381. Lu, X., Sun, K., Guerrero, J.M., Vasquez, J.C., and Huang, L. (2015). Double-quadrant state-of-charge-based droop control method for distributed energy storage systems in autonomous DC microgrids. IEEE Transactions on Smart Grid, 6(1), 147–157. Lu, X., Sun, K., Guerrero, J.M., Vasquez, J.C., and Huang, L. (2014). State-of-charge balance using adaptive droop control for distributed energy storage systems in DC microgrid applications. IEEE Transactions on Industrial Electronics, 61(6), 2804–2815. Morstyn, T., Hredzak, B., and Agelidis, V.G. (2015). Distributed cooperative control of microgrid storage. IEEE Transactions on Power Systems, 30(5), 2780– 2789. Olfati-Saber, R., Fax, J.A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215–233. Olfati-Saber, R. and Murray, R.M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Sichilalu, S., Tazvinga, H., and Xia, X. (2016). Optimal control of a fuel cell/wind/pv/grid hybrid system with thermal heat pump load. Solar Energy, 135, 59–69. Sichilalu, S.M. and Xia, X. (2015). Optimal power dispatch of a grid tied-battery-photovoltaic system supplying heat pump water heaters. Energy Conversion and Management, 102, 81–91. Suberu, M.Y., Mustafa, M.W., and Bashir, N. (2014). Energy storage systems for renewable energy power sector integration and mitigation of intermittency. Renewable and Sustainable Energy Reviews, 35, 499–514. Sudevalayam, S. and Kulkarni, P. (2011). Energy harvesting sensor nodes: Survey and implications. IEEE Communications Surveys & Tutorials, 13(3), 443–461. Tazvinga, H., Zhu, B., and Xia, X. (2015). Optimal power flow management for distributed energy resources with batteries. Energy Conversion and Management, 102, 104–110. Xu, Y., Li, Z., Zhao, J., and Zhang, J. (2017). Distributed robust control strategy of grid-connected inverters for energy storage system’s state-of-charge balancing. IEEE Transactions on Smart Grid, published online. Xu, Y., Zhang, W., Hug, G., Kar, S., and Li, Z. (2015). Cooperative control of distributed energy storage systems in a microgrid. IEEE Transactions on Smart Grid, 6(1), 238–248. Yang, H., Qiu, Y., Li, Q., and Chen, W. (2017). A selfconvergence droop control of no communication based on double-quadrant state of charge in dc microgrid applications. Journal of Renewable and Sustainable Energy, 9(3), 034102. Zhu, B., Tazvinga, H., and Xia, X. (2015). Switched model predictive control for energy dispatching of a photovoltaic-diesel-battery hybrid power system. IEEE Transactions on Control Systems Technology, 23(3), 1229–1236.
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IFAC PapersOnLine 51-13 (2018) 229–234
Distributed Energy Dispatch of Electrical Distributed Energy Dispatch of Electrical Distributed Energy Dispatch of Electrical Energy Storage Systems Using Distributed Energy Dispatch ofConsensus Electrical Energy Storage Systems Using Consensus Energy Storage Systems Using Consensus Control Approach Energy Storage Systems Using Consensus Control Control Approach Approach Control Approach ∗,∗∗ Wenjing Xie ∗,∗∗∗ ∗,∗∗∗ Xiaohua Xia ∗,∗∗
Wenjing Xie ∗,∗∗∗ ∗,∗∗∗ Xiaohua Xia ∗,∗∗ Wenjing Xie ∗,∗∗∗ Xiaohua Xia ∗,∗∗ Wenjing Xie Xiaohua Xia ∗,∗∗ ∗ ∗ Department of Electrical, Electronic and Computer Engineering, Department of Electrical, Electronic and Computer Engineering, ∗ ∗ University of 0002, Africa. (e-mail: Department of Pretoria, Electrical,Pretoria Electronic andSouth Computer Engineering, ∗ University of Pretoria, Pretoria 0002, South Africa. (e-mail: Department of Pretoria, Electrical, Electronic andSouth Computer Engineering, University of Pretoria 0002, Africa. (e-mail: [email protected]). [email protected]). ∗∗ University of Pretoria, Pretoria 0002, South Africa. (e-mail: Northeastern University, Shenyang, 110004, China [email protected]). ∗∗ Shenyang, 110004, China ∗∗ Northeastern University, ∗∗∗ ∗∗ [email protected]). Computer and Information Science, Southwest University, Northeastern University, Shenyang, 110004, China ∗∗∗ School of ∗∗of Computer and Information Science, Southwest University, ∗∗∗ School ∗∗∗ Northeastern University, Shenyang, 110004, China Chongqing, 400715, China (e-mail: [email protected]) School of Computer and Information Science, Southwest University, Chongqing, 400715, China (e-mail: Science, [email protected]) ∗∗∗ School of Computer and Information Southwest University, Chongqing, 400715, China (e-mail: [email protected]) Chongqing, 400715, China (e-mail: [email protected]) Abstract: Energy Energy dispatch dispatch task task of of electrical energy energy storage storage systems systems (EESSs) (EESSs) contains contains the the state Abstract: Abstract: Energy dispatch tasksupply-demand of electrical electrical energy storage systems (EESSs) contains the state state of charge (SoC) balance and the balance, where the former ensures the maximum of charge (SoC) balance and the supply-demand balance, where the former ensures the maximum Abstract: Energy dispatch task of electrical energy storage systems (EESSs) contains the state power capacity of the group of EESSs, and the latter maintains the load operation and reduces of charge (SoC) balance and the supply-demand balance, where the former ensures the maximum power capacity of the group of EESSs, and the latter maintains theformer load operation and reduces of charge (SoC) balance and the supply-demand balance, where the ensures the maximum energy lost. A unified switching model is established for both charging and discharging processes power capacity of the group of EESSs, and the latter maintains the load operation and reduces energy lost. A unified switching model isand established both charging andoperation discharging power capacity of the group of EESSs, latterfor the load andprocesses reduces of EESSs. EESSs. EESSs areswitching assumed to share share athe common communication network. Dependent on energy lost.EESSs A unified model is established formaintains both charging and discharging processes of are assumed to a common communication network. Dependent on energy lost. A unified switching model is established for both charging and discharging processes of EESSs. EESSs are assumed to share a common communication network. Dependent on the communication between EESSs, three distributed energy dispatch algorithms are proposed the communication between EESSs, three adistributed dispatch algorithms are proposed of EESSs. EESSs are assumed to share common energy communication network. Dependent on based on approach, graph theory Lyapunov method. Without considering the communication between EESSs, three dispatch algorithms proposed based on consensus consensus control control approach, graphdistributed theory and andenergy Lyapunov method. Withoutare considering the communication between EESSs, three distributed energy dispatch algorithms are proposed constraints, the first algorithm is designed to determine the charging/discharging power reference based on consensus control approach, graph theory and Lyapunov method. Without considering constraints, the firstcontrol algorithm is designed to theory determine the charging/discharging power reference based on EESS consensus approach, graph and Lyapunov method. Without considering of every every as thealgorithm sum of of aisnominal nominal power value and an SoC-based correction term, where constraints, theas first designed to determine the charging/discharging power reference of EESS the sum a power value and an SoC-based correction term, where constraints, the first designed to determine thean charging/discharging power reference of every EESS as thealgorithm sum of aisnominal power value and SoC-based correction term, where the nominal term and the correction term respectively decrease the supply-demand and the SoC the nominal term and the correction termpower respectively decrease the supply-demand and the SoC of every EESS as the sum of a nominal value and anthe SoC-based correction term, where imbalances. By applying dynamic saturation function to first algorithm, aa second one is the nominal term and the a correction term respectively decrease the supply-demand and the SoC imbalances. By applying a dynamic saturation function to the first algorithm, second one is the nominal satisfying termapplying and the correction term respectively decrease the supply-demand and theone SoC constructed the charging/discharging mode switching constraint and the power limit imbalances. By a dynamic saturation function to the first algorithm, a second is constructed satisfying the charging/discharging mode switching constraint and the power limit imbalances. Bythe applying dynamic to first algorithm, a second one is constraint. On On basis ofacharging/discharging the secondsaturation algorithm,function third onethe is obtained obtained byand additionally taking constructed satisfying theof mode switching constraint the powertaking limit constraint. the basis the second algorithm, aaa third one is by additionally constructed satisfying the charging/discharging mode switching constraint and the power limit constraint. On the basis of the second algorithm, third one is obtained by additionally taking into account account the the SoC SoC limit. limit. Lyapunov Lyapunov stability stability analysis analysis shows shows that that the the third third algorithm algorithm not not into constraint. Onthe theSoC basis of the second algorithm, aanalysis third one isalso obtained additionally taking only asymptotically realizes the energy dispatch but satisfies the power and SoC into account limit. Lyapunov stability objective shows that thebythird algorithm not only asymptotically realizes the energy dispatch objective but also satisfies the power and SoC into account the SoC limit. Lyapunov stability analysis shows that the third algorithm not constraints, provided that the communication graph is undirected undirected and connected. Effectiveness only asymptotically realizes the energy dispatch objective but also satisfies the power and SoC constraints, provided that the communication graph is and connected. Effectiveness only asymptotically realizes the energy dispatch objective but also satisfies the power and SoC constraints, provided that the communication graph is undirected and connected. Effectiveness of the third third algorithm algorithm is is demonstrated by by simulation results. results. of constraints, the communication graph isresults. undirected and connected. Effectiveness of the the third provided algorithmthat is demonstrated demonstrated by simulation simulation © 2018, IFACalgorithm (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. of the third is demonstrated by simulation results. Keywords: Energy Energy storage storage systems, systems, energy energy dispatch, dispatch, distributed distributed control, control, consensus control. control. Keywords: Keywords: Energy storage systems, energy dispatch, distributed control, consensus consensus control. Keywords: Energy storage systems, energy dispatch, distributed control, consensus control. 1. INTRODUCTION INTRODUCTION the the dispatch dispatch task task of of EESSs: EESSs: the the supply-demand supply-demand balance balance 1. 1. INTRODUCTION the dispatch task of EESSs: the supply-demand balance and the SoC balance. The supply-demand balance is and the SoC balance. The supply-demand balance is that that 1. INTRODUCTION the dispatch task ofsupplied EESSs: the supply-demand balance and the SoCoutputs balance. The supply-demand balance is that the power from renewable generations Nowadays, electrical energy storage units are widely apthe power outputs supplied from renewable generations Nowadays, electrical energy storage units are widely ap- the the SoCoutputs balance. The supply-demand balance is that power supplied from renewable generations and EESSs need meet demands with aa low/zero Nowadays, electrical storagestorage, units are ap- and plied in in power power grids. energy With energy energy the widely imbalance and power EESSs outputs need to to supplied meet load load demands with generations low/zero plied grids. With storage, the imbalance the from renewable and EESSs need to meet load demands with a low/zero Nowadays, electrical energy storage units are widely apenergy lost (Xu et al., 2015). The SoC balance is plied in power grids. With energy generations storage, the and imbalance between intermittent renewable time- energy lost (Xu et al., 2015). The SoC balance is that that between intermittent renewable generations and timeand EESSs to al., meet load SoC demands a low/zero energy lost need (Xu et 2015). The value SoC with balance isetthat plied in power grids. With energy storage, the imbalance EESSs have the same dynamic (Morstyn al., between intermittent renewable generations and timevarying load demands can be reduced or vanished, deEESSs have the same dynamic SoC value (Morstyn et al., varying load demands renewable can be be reduced reduced or vanished, vanished, de- EESSs energy lost (Xu et al., 2015). The SoC balance iset that have the same dynamic SoC value (Morstyn al., between intermittent generations and time2015; Li et al., 2017). Reasons of balancing the SoCs varying load demands can or decreasing the usage of fossil fuel and electricity cost (Suberu 2015; Li et al., 2017). Reasons of balancing the SoCs creasing the usage of fossil fuel and electricity cost (Suberu EESSs have the dynamic SoC (Morstyn et al., Li etare al., same 2017). Reasons ofvalue balancing the SoCs varying load demands canfuel beand reduced or vanished, de- 2015; of as aspects in (Cai and creasing the Bragard usage of fossil electricity costdischarg(Suberu et al., al., 2014; 2014; et al., al., 2010). Charging and of EESSs EESSs areal.,summarized summarized as two two aspects in the (CaiSoCs and et Bragard et 2010). Charging and discharg2015; Li et 2017). Reasons of balancing of EESSs are summarized as two aspects in (Cai and creasing the usage of fossil fuel and electricity cost (Suberu Hu, 2016; Xu et al., 2017). Firstly, it protects EESSs et 2014;control Bragardofetsuch al., 2010). and dischargingal., power storageCharging units plays plays a crucial crucial Hu, 2016; Xu et al., 2017). Firstly, it protects EESSs ing power control ofetsuch such storage units of EESSs are two aspects in (Cai and 2016; Xu summarized et al., 2017).asFirstly, it aprotects EESSs et al., 2014; Bragard al., supporting 2010). Charging and dischargfrom overcharging or at higher ing power control of storage units plays aa crucial role in saving energy and load operation. So Hu, from 2016; overcharging or overdischarging overdischarging at aprotects higher system system role in saving energy and supporting load operation. So Hu, Xu et al., 2017). Firstly, it EESSs from overcharging or overdischarging at a higher system ing power control ofand such storage units a crucial Secondly, EESSs with SoCs have role in saving energy andcontrol supporting loadplays operation. So level. far, dispatch of storage device level. overcharging Secondly, the the EESSs with balancing balancing SoCssystem have far, energy energy dispatch and control of single single storage device from or EESSs overdischarging at the a higher Secondly,power the with balancing SoCs have role inbeen saving energyand and supporting load operation. So level. the maximum capacity. Without SoC balance far, energy dispatch control ofvia single storage device have extensively researched optimization and the maximum power capacity. Without the SoC balance have been extensively researched via optimization and level. Secondly, the EESSs with balancing SoCs have maximum some powerEESSs capacity. Without the SoC balance far, energy dispatchaiming andresearched control ofvia single storage device consideration, will stop when their have been extensively optimization and control techniques, to maximize maximize the renewable en- the consideration, some EESSs willWithout stop working working when their control techniques, aiming to the renewable enthe maximum power capacity. the SoC balance consideration, some EESSs will stop working when their have been extensively researched via optimization and SoCs reach the high or low limit. In this case, the power control techniques, aiming to maximize the renewable energy usage and simultaneously satisfy load demands and SoCs reach the high or low limit. In this case, the power ergy usage and simultaneously simultaneously satisfy load load demands and and consideration, stop working when their reach thesome high EESSs orEESSs low will limit. In this case, thea power control techniques, aiming to maximize the renewable en- SoCs capacity of may with worse ergy usage and satisfy demands constraints (Sichilalu et Tazvinga et capacity of the overall overall EESSs mayIndecrease decrease with a power worse constraints (Sichilalu et al., al., 2016; 2016; Tazvinga et al., al., 2015; 2015; SoCs reach high or low limit. this case, the capacity of the overall EESSs may decrease with a worse ergy usage and simultaneously satisfy load demands and system performance. Typical results incorporating the constraints (Sichilalu et al., Tazvinga et the al., devel2015; system performance. Typical results incorporating the two Zhu et et al., al., 2015; 2015; Sichilalu and2016; Xia, 2015). 2015). With two Zhu Sichilalu and Xia, With the develcapacity of theobtained overall EESSs may decrease with athe worse performance. Typical results incorporating two constraints (Sichilalu et al., 2016; Tazvinga et al., 2015; system balances are in (Lu et al., 2014, 2015; Yang Zhu et al., 2015; Sichilalu andand Xia,sensor 2015).network, With the development of battery technology multiple balances are obtained in (Lu et al., 2014, 2015; Yang opment of battery battery technology and sensor network, multiple system performance. Typical results incorporating two are Cai obtained in 2016; (Lu etXual., 2014, 2015;the Yang Zhu et al., 2015; Sichilalu andand Xia, 2015).network, With the devel- balances et and et al., In (Lu opment of technology sensor multiple package-level electrical energy storage systems (EESSs) et al., al., 2017; 2017; Cai and Hu, Hu, 2016; Xual., et 2014, al., 2017). 2017). InYang (Lu package-level electrical energy storage systems (EESSs) balances are obtained in (Lu et 2015; et al., 2017; Cai and Hu, 2016; Xu et al., 2017). In (Lu opment of battery technology andstorage sensor network, multiple Yang et independent of package-level electrical energy systems (EESSs) also receive receive much attention. Every EESS is composed composed of et al., al., 2014, 2014, 2015; 2015; Yang et al., al., 2017), 2017), independent of also much attention. Every EESS is of al., andYang Hu, EESSs, 2016; et al.,independent 2017).adaptive In (Lu et al., 2017; 2014, Cai 2015; et al.,Xu 2017), of package-level electrical energy storage systems (EESSs) communications between decentralized also receive much attention. Every EESS is composed of series and/or parallel configured battery cells. communications between EESSs, decentralized adaptive series and/or parallel configured battery cells. et al., 2014, 2015; Yang et al., 2017), independent of betweenwere EESSs, decentralized adaptive also receive attention. Every EESScells. is composed of communications droop proposed to droop series and/ormuch parallel configured battery droop control control strategies strategies were proposed to tune tune the the droop communications between EESSs, decentralized adaptive For an an EESS, the configured charging/discharging power referrefer- droop control strategies were proposed to tune the droop series and/or parallel battery cells. coefficient according to SoC. In (Cai and Hu, 2016), for the For EESS, the charging/discharging power coefficient according to SoC. In (Cai and Hu, 2016), for the For EESS, thebycharging/discharging power droop control strategies weredesigned to tune the droop ence an is determined determined dispatch algorithm, algorithm, and and the refervolt- coefficient according to SoC. Inproposed (Cai and Hu, 2016), for the EESS with higher SoC, the distributed dispatch ence is by dispatch the voltEESS with higher SoC, the designed distributed dispatch For an EESS, the charging/discharging power reference is determined by dispatch algorithm, and the voltcoefficient according to SoC. In (Cai and Hu, 2016), for the age/current controller controller is is designed designed to to adjust adjust the the actual actual EESS withassigned higher SoC, the designed distributed dispatch algorithm more discharging (less charging) power age/current algorithm assigned morethe discharging (less charging) power ence istodetermined by dispatch algorithm, and theactual voltage/current controller is designed to adjust the EESS withassigned higher SoC, designed(less distributed dispatch power track the reference. There are two objectives in algorithm more discharging charging) power power to track track the reference. reference. There to areadjust two objectives objectives in age/current controller is designed the actual power to the There are two in algorithm assigned more discharging (less charging) power power to track the reference. There are two objectives in Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Federation of Automatic Control)
Proceedings, 2nd IFAC Conference on 229 Proceedings, 2nd IFAC Conference on 229 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference 229 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.283 Proceedings, 2nd IFAC Conference on 229 Systems Modelling, Identification and Control of Nonlinear Systems Modelling, and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018 Systems Identification
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than the estimated average value of desired total power, and for the one with lower SoC, assigned less discharging (more charging) power. In (Xu et al., 2017), a distributed power dispatch strategy was constructed by borrowing an economic dispatch algorithm. However, the determination of power reference in (Lu et al., 2014, 2015; Yang et al., 2017) had to rely on the information of all the EESSs and therefore were not distributed, constraints on power and SoC were not considered in (Cai and Hu, 2016), and in (Xu et al., 2017) the total charging/discharging power was only able to track the initial (not the real time) supply-demand mismatch. Based on these observations, energy dispatch of EESSs remains a hot topic of investigation. This paper concerns how to realize the energy dispatch task of EESSs with power and SoC constraints. The unified switching mode is firstly presented by using the sign function. The consensus control approach is applied to solve the dispatch problem. To guarantee the system stability under constraints, three algorithms are derived one by one. The first algorithm is designed in case of no constraints, then by additionally utilizing the dynamic saturation function, a second one is proposed in case of constrained power, and a final one is obtained in case of constrained power and SoC. Convergence of all the three algorithms is proved by Lyapunov method, under the assumption that the communication graph is undirected and connected. Simulation cases illustrate the effectiveness of the proposed third algorithm. The rest of this paper is organized as follows. Section 2 formulates the energy dispatch problem of multiple EESSs. Section 3 solves this problem using the consensus control approach. Simulation examples are provided in Section 4 to demonstrate the effectiveness of the proposed dispatch strategy. Finally, Section 5 concludes this work.
2.1 Simplified structure of EESSs in a microgrid generations
PG1
Bus P1
G PG2
P2
G PGm
P3
1
2
EESS
EESS
Pn-1 3
EESS communication network
Pn
PL
n-1
n
EESS
EESS
Based on (2), the charging/discharging process of the ith EESS can be described by: Si (k + 1) = Si (k) − T ηi (k)Pi (k),
where k is the sampling time, T is the sampling interval, and 1 ηdi > 0, (discharging) ρi σi ζ Q = ρdiζ i Vi ηi (k) = Qi Vi i ci ηci > 0, (charging). Qi Vi Assume that all the EESSs have the same model, i.e., ηi (k) = η(k), thus Si (k + 1) = Si (k) − T η(k)Pi (k). (3)
Every EESS has three modes: charging, discharging and stop modes, and (3) is a switching system. Denote the global supply-demand mismatch by: (4) PM (k) = PL (k) − PG (k), where PL (k) and PG (k) are the total load demand and the total power generation respectively. If PM (k) > 0 (< 0), EESS i (i ∈ V) is in discharging or stop (charging or stop) operation with η(k) = ηd (ηc ) and Pi (k) ≥ 0 (≤ 0), where ηd = ηdi and ηc = ηci . If PM (k) = 0, the n EESSs stop working with Pi (k) = 0, which is not our consideration in this paper. Therefore, η(k) has the unified form: η(k) = 0.5sign(PM (k)) [sign(PM (k)) + 1] ηd
2. PROBLEM FORMULATION
G
ρi t (1) Ii (t)dt, i ∈ V, Si (t) = Si (0) − Qi 0 where V = {1, 2, · · · , n} is the index set of EESSs, Si (t) ∈ (0, 1) denotes the SoC of the ith EESS, and (ρi , Qi , Ii ) respectively represents the Coulombic efficiency, the capacity and the output current of the ith EESS. Coulombic efficiency ρi satisfies ρi = 1 in discharging mode and ρi ∈ (0, 1) in charging mode. Output current Ii is (Li et al., 2017): 1/ζdi , (discharging), Ii = σi Pi /Vi , σi = (charging), ζci , where Pi denotes the output power of the ith converter, Vi is the output voltage of the ith EESS, and σi is related to the converter efficiencies (ζdi , ζci ) of the ith converter during discharging and charging operations. It is assumed that Vi is time-invariant (Li et al., 2017). SoC equation (1) can be rewritten as: ρi σi t (2) Si (t) = Si (0) − Pi (t)dt, i ∈ V. Qi Vi 0
Loads
+ PM (k)
Fig. 1. Simplified structure of EESSs in microgrid. As shown in Fig. 1, renewable generations and EESSs are installed in the microgrid, in order to provide electrical energy to loads. The EESSs store the surplus power when the power generation is more than load demand, otherwise return the stored energy to loads. Suppose that there are n EESSs in the microgrid. 2.2 Unified model of charging/discharging process of EESSs The SoC of an EESS can be calculated by (Cai and Hu, 2016) 230
+ 0.5sign(PM (k)) [sign(PM (k)) − 1] ηc ,
(5)
− PM (k)
where sign(·) is the sign function. To make EESSs operate in appropriate modes, we propose the charging/discharging mode switching constraint: (6) PM (k)Pi (k) ≥ 0. Other two constraints for security of EESSs (Xu et al., 2015) are Pmin ≤ Pi (k) ≤ Pmax , (7) Smin ≤ Si (k) ≤ Smax ,
(8)
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where Pmax > 0, Pmin < 0, Pmax and Pmin are maximum power allowed to discharge and charge EESS i, and Smin and Smax are respectively the lower and the upper limits of SoC. Now, the unified switching model of EESSs is established as (3) and (5) with the constraints (6), (7) and (8). 2.3 Communication network of EESSs To make the EESSs cooperatively charge and discharge for their group objective, we suppose that the EESSs compose a communication network based on wireless sensors (Sudevalayam and Kulkarni, 2011). In this network, they can share information (e.g., Si (k) and ηi (k)) with each other, as shown in Fig. 1. Model the information flow between them as a communication graph G(V, E, A), where V = {1, 2, · · · , n} is the index set of EESSs, E ⊂ V ×V is an edge set of ordered pairs of EESSs, and A = [aij ] ∈ Rn×n is the adjacency matrix with entries aij = 1 or aij = 0 (Olfati-Saber and Murray, 2004). If EESS i can receive information from EESS j, then (j, i) ∈ E, aij = 1 and EESS j is called the communication neighbor of EESS i, denoted by j ∈ Ni , where Ni = {j ∈ V|aij = 1}. If EESS i cannot have access to the information of EESS j, then (j, i) ∈ / E, aij = 0 and j ∈ Ni . Self-connection is not considered for G, i.e., aii = 0, ∀i ∈ V (Olfati-Saber and Murray, 2004). A graph G is undirected if aij = aji for any i, j ∈ V. The Laplacian matrix L = [lij ] ∈ Rn×n is defined as: if j = i; ij , −a n lij = aij , if j = i. j=1
For undirected connected graph, L = LT has the following properties (Olfati-Saber and Murray, 2004): − → (9) 0 = λ1 < λ2 ≤ λ3 ≤ · · · ≤ λn , L 1 = 0, − → where (λ1 , λ2 , · · · , λn ) are the eigenvalues of L, and 1 = [1, 1, · · · , 1]T ∈ Rn . To achieve distributed dispatch, we make the following Assumption. Assumption 1. The communication graph G(V, E, A) of EESSs is undirected and connected. 2.4 Energy dispatch objective
The energy dispatch objective of EESSs considered in this paper is stated as (Cai and Hu, 2016): under Assumption 1, design a distributed dispatch strategy Pi (i ∈ V) for EESS i only using the information of its communication neighbors and itself, such that under constraints (6)-(8), the SoCs of the n EESSs reach a balance Si (k) − Sj (k) → 0, ∀i, j ∈ V, (10) and the power tracking error n Pe (k) = Pi (k) − PM (k) (11) i=1
is minimized after (10) is achieved. Objective (10) ensures the SoC balancing between EESSs, and minimizing Pe (k) guarantees the supply-demand balance as well as reduces energy lost in charging and discharging operations. Reasons and physical meanings of the SoC balance can be found in (Cai and Hu, 2016). For the general case of
231
231
Smin Si Smax , Pi (k) can be adjusted such that Pe (k) → 0. For the special cases that Si (k) is close or equal to Smax /Smin in charging/discharging mode, |Pi (k)| must be small or zero such that Si (k+1) remains in [Smin , Smax ], and Pe (k) can only be minimized rather than Pe (k) → 0.
The energy dispatch objective is asymptotically achieved if for any initial SoCs satisfying Si ∈ [Smin , Smax ], the SoC balancing errors (Si (k) − Sj (k), i, j ∈ V) are asymptotically convergent to zero, the power tracking errors Pe (k) is minimized after Si (k) − Sj (k) → 0, and the constraints (6)-(8) are met. 3. DISTRIBUTED ENERGY DISPATCH In this section, we employ multiagent consensus control idea to develop the first dispatch scheme without considering constraints, then extend it to the case of constraints, solving the energy dispatch problem of multiple EESSs. 3.1 Algorithm design without constraints For sake of simplicity, we assume that the power and SoC variables are not subject to constraint. Based on the consensus control idea in (Olfati-Saber et al., 2007), we propose the distributed dispatch algorithm ε aij [Si (k) − Sj (k)] , (12) Pi (k) = Pave (k) + η(k) j∈Ni
where ε > 0, the term Pave (k) = PMn(k) is the nominal power component corresponding to thesupply-demand aij [Si (k) − balance, and the SoC balance error j∈Ni Sj (k)] is correction term. Lemma 1. If Assumption 1 holds and ε is a sufficiently small positive constant, then the dispatch strategy (12) guarantees the SoC balance errors Si (k) − Sj (k) (i, j ∈ V) globally asymptotically converge to zero and Pe (k) = 0. Proof. Define S(k) [S1 (k), S2 (k), · · · , Sn (k)]T ,
P (k) [P1 (k), P2 (k), · · · , Pn (k)]T − → = Pave (k) 1 + εη −1 (k)LS(k),
(13)
− → where 1 = [1, 1, · · · , 1]T ∈ Rn . As Assumption 1 holds, n − →T − → − → − → 1 L = 0, and Pi (k) = 1 T P (k) = 1 T [Pave (k) 1 + i=1
εη −1 (k)LS(k)] = PM (k), implying Pe (k) = 0.
Under algorithm (12), the closed-loop form of EESSs is − → (14) S(k + 1) = −η(k)Pave (k) 1 + (In − εL)S(k),
where In ∈ Rn×n is the identity matrix. Define the Lyapunov function (Olfati-Saber and Murray, 2004): (15) V (k) = S T (k)LS(k) ≥ 0, where L is positive semi-definite. Compute ∇V (k) = V (k + 1) − V (k) T − → = −η(k)Pave (k) 1 + (In − εL)S(k) (16) − → × L −η(k)Pave (k) 1 + (In − εL)S(k) − V (k) = S T(k)LT (−2εIn + ε2 L)LS(k),
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− → where L = LT and L 1 = 0 are used. Since ε is sufficiently small, −2εIn + ε2 L is guaranteed negative definite. So, ∇V (k) ≤ 0. Based on Assumption 1 and the negative definite property of −2εIn + ε2 L, one has − → ∇V (k) = 0 ⇔ LS(k) = 0 ⇔ S(k) ∈ span{ 1 }, thus n the largest invariant set in {S(k) ∈ R |∇V (k) = 0} − → is {S(k) ∈ span{ 1 }}. According to the discrete time version of LaSalle invariance principle (Alberto et al., 2007), S(k) globally asymptotically convergent to {S(k) ∈ − → span{ 1 }}, and the SoC balance errors are therefore globally asymptotically vanished to zero. 3.2 Generation to constrained power case With power constraints (6) and (7), it is required to make the following Assumption. Assumption 2. The number of EESSs is determined such that the nominal power is within the permissible power range Pmin < Pave (k) < Pmax . Improve the dispatch scheme (12) as (17) Pi (k) = Pave (k) + η −1 (k)f [εLi S(k)], aij [Si (k) − Sj (k)] , Li is the ith where Li S(k) = j∈Ni
row of matrix L, and the bounded dynamic saturation function f (·) is given by χ(k), if χ(k) ∈ [fmin (k), fmax (k)], f [χ(k)] = fmax (k), if χ(k) > fmax (k), fmin (k), if χ(k) < fmin (k), + fmin (k) = PM (k) [−η(k)Pave (k)] (18) − + PM (k) [η(k)(Pmin − Pave (k))] , − fmax (k) = PM (k) [−η(k)Pave (k)] + + PM (k) [η(k)(Pmax − Pave (k))] .
Remark 1. Under Assumption 2, the function f (·) in (18) has the following features: χ(k)f [χ(k)] ≥ f 2 [χ(k)] ≥ 0, (19) f [χ(k)] = 0 ⇔ χ(k) = 0,
which are proved as follows. When PM (k) > 0, we have Pave (k) > 0 and fmin (k) = −η(k)Pave (k) < 0, fmax (k) = η(k)[Pmax − Pave (k)] > 0. When PM (k) < 0, we get Pave (k) < 0 and fmin (k) = η(k)[Pmin − Pave (k)] < 0, fmax (k) = −η(k)Pave (k) > 0.
In conclusion, fmin (k) < 0 < fmax (k), so f [χ(k)] = 0 ⇔ χ(k) = 0. By (18), it follows χ(k)f [χ(k)] = 2 if χ(k) ∈ [fmin (k), fmax (k)], χ (k) = f 2 [χ(k)], χ(k)fmax (k) > f 2 [χ(k)], if χ(k) > fmax (k), χ(k)fmin (k) > f 2 [χ(k)], if χ(k) < fmin (k). Lemma 2. Under Assumptions 1-2 and ε > 0 is sufficiently small, the distributed dispatch scheme (17) satisfies the constraints (6) and (7), and ensures the SoC balance errors and Pe (k) globally asymptotically convergent to zero. 232
Proof. (I) To verify that (6) and (7) are satisfied. The following three cases are taken into account. (a) For the case of εLi S(k) ∈ [fmin (k), fmax (k)], the algorithm (17) takes the form of Pi (k) = Pave (k) + η −1 (k)εLi S(k). As εLi S(k) ∈ [fmin (k), fmax (k)], it follows that Pi (k) ≤ Pave (k) + η −1 (k)fmax (k) Pmax , if PM (k) > 0 (discharging), = 0, if PM (k) < 0 (charging), Pi (k) ≥ Pave (k) + η −1 (k)fmin (k) 0, if PM (k) > 0 (discharging), = Pmin , if PM (k) < 0 (charging). Thismeans that 0 ≤ Pi (k) ≤ Pmax , if PM (k) > 0 (discharging), Pmin ≤ Pi (k) ≤ 0, if PM (k) < 0 (charging).
(b) For the case of εLi S(k) > fmax (k), (17) becomes Pi (k) = Pave (k) + η −1 (k)fmax (k) Pmax , if PM (k) > 0 (discharging), = 0, if PM (k) < 0 (charging). (c) For the case of εLi S(k) < fmin (k), (17) shows that Pi (k) = Pave (k) + η −1 (k)fmin (k) 0, if PM (k) > 0 (discharging), = Pmin , if PM (k) < 0 (charging).
The above three cases imply that (6) and (7) are fulfilled. (II) To verify that the SoC balance errors and Pe (k) are tending to zero. − → By (17), we have P (k) = Pave (k) 1 + η −1 (k)f (εLS(k)), where f (εLS(k)) = [f (εL1 S(k)), · · · , f (εLn S(k))]T ∈ Rn . Compute ∇V (k) = V (k + 1) − V (k) as T
∇V (k) = [S(k) − f (εLS(k))] L[S(k) − f (εLS(k))] − V (k)
≤ f T (·)(L − 2ε−1 In )f (·), where f (·) = f (εLS(k)), and the first inequality in (19) is utilized. As ε > 0 is sufficiently small, L − 2ε−1 In is negative definite and ∇V (k) ≤ 0. Under Assumptions 1-2 and ε > 0, Remark 1 shows that f (εLS(k)) = 0 ⇔ S(k) ∈ − → span{ 1 }. So, S(k) globally asymptotically converges to − → {S(k) ∈ span{ 1 }} which is the largest invariant set in n {S(k) ∈ R |∇V (k) = 0} (Alberto et al., 2007). This implies that the SoC balance errors and Pe (k) globally asymptotically tend to zero. 3.3 Generation to the case of constrained power and SoC
Considering both the power and the SoC constraints (6)(8), we modify (17) as if Sˆi (k + 1) ∈ [Smin , Smax ], Pˆi (k), −1 Pi (k) = η (k)[Si (k) − Smax ], if Sˆi (k + 1) > Smax , −1 η (k)[Si (k) − Smin ], if Sˆi (k + 1) < Smin , (20) where Pˆi (k) = Pave (k) + η −1 (k)f [εLi S(k)], (21) Sˆi (k + 1) = Si (k) − η(k)Pˆi (k).
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Theorem 1. Suppose that Assumptions 1-2 hold and ε > 0 is sufficiently small, then for any initial SoCs satisfying Si (0) ∈ [Smin , Smax ], the distributed algorithm (20) guarantees that the constraints (6)-(8) are satisfied, the SoC balance errors of EESSs Si (k) − Sj (k) (i, j ∈ V) are asymptotically decayed to zero, and the power tracking error Pe (k) is minimized after Si (k) − Sj (k) → 0, i, j ∈ V. Proof. (I) To prove (8). The closed-loop system is Si (k + 1) = Si (k) − η(k)Pi (k) Sˆi (k + 1) ∈ [Smin , Smax ], if Sˆi (k + 1) ∈ [Smin , Smax ], = Smax , if Sˆi (k + 1) > Smax , Smin , if Sˆi (k + 1) < Smin . (II) To prove (6)-(7).
If Sˆi (k + 1) ∈ [Smin , Smax ], it follows from Lemma 2 that Pi (k) = Pˆi (k) satisfies the conditions (6)-(7). If Sˆi (k + 1) > Smax , then EESS i is in charging/stop mode, and 0 ≥ Pi (k) = η −1 (k)[Si (k) − Smax ] = η −1 (k)[Sˆi (k + 1) − Smax ] + Pˆi (k) > Pˆi (k) ≥ Pmin ,
where Pˆi (k) ≥ Pmin has been proven by the proof of Lemma 2. If Sˆi (k+1) < Smax , then EESS i is in discharging mode, and 0 ≤ Pi (k) = η −1 (k)[Si (k) − Smin ] = η −1 (k)[Sˆi (k + 1) − Smin ] + Pˆi (k) < Pˆi (k) ≤ Pmax . Consequently, the power constraints (6) and (7) are satisfied. (III) To prove the convergence of SoC balance errors.
i=1 j∈Ni
ˆ + 1) = SˆT (k + 1)LS(k
By (22) and (23), it follows n {aij [Si (k + 1) − Sj (k + 1)]2 } V (k + 1) = i=1 j∈Ni
≤
n
i=1 j∈Ni
{aij [Sˆi (k + 1) − Sˆj (k + 1)]2 }
≤ V (k) + f T (·)(L − 2ε−1 In )f (·), which, together the proof of Lemma 2, shows that the SoC balance errors are asymptotically vanished to zero. (IV) To prove that Pe (k) is minimized after Si (k) − Sj (k) → 0.
Therefore, Si (k + 1) ∈ [Smin , Smax ].
According to the proof of Lemma 2, we have n {aij [Sˆi (k + 1) − Sˆj (k + 1)]2 }
233
(22)
≤ V (k) + f T (εLS(k))(L − 2ε−1 In )f (εLS(k)), ˆ + 1) = [Sˆ1 (k + 1), · · · , Sˆn (k + 1)]T . where S(k
If Sˆi (k + 1) and Sˆj (k + 1) are both in [Smin , Smax ], then [Si (k + 1) − Sj (k + 1)]2 = [Sˆi (k + 1) − Sˆj (k + 1)]2 .
If they do not belong to [Smin , Smax ], then [Si (k + 1) − Sj (k + 1)]2 = 0 ≤ [Sˆi (k + 1) − Sˆj (k + 1)]2 ,
where Si (k + 1) = Sj (k + 1) = Smax or Smin due to the fact that all the EESSs are in the same charging/stop or discharging/stop mode. If one of them (assumed as Sˆi (k + 1)) belongs to [Smin , Smax ] and the other one does not, then [Si (k + 1) − Sj (k + 1)]2 = [Smax − Sˆj (k + 1)]2 < [Sˆi (k + 1) − Sˆj (k + 1)]2 , (charging), [Si (k + 1) − Sj (k + 1)]2 = [Smin − Sˆj (k + 1)]2
< [Sˆi (k + 1) − Sˆj (k + 1)]2 , (discharging). Summarizing the above analysis yields: [Si (k + 1) − Sj (k + 1)]2 ≤ [Sˆi (k + 1) − Sˆj (k + 1)]2 . (23) 233
After Si (k) − Sj (k) → 0 is achieved, denote Si (k) = S ∗ (k), ∀i ∈ V, and all the EESSs have the same power if Sˆi (k + 1) ∈ [Smin , Smax ], Pave (k), −1 ∗ Pi (k) = η (k)[S (k) − Smax ], if Sˆi (k + 1) > Smax , −1 η (k)[S ∗ (k) − Smin ], if Sˆi (k + 1) < Smin . For the case of Sˆi (k + 1) ∈ [Smin , Smax ], one gets Pe (k) = 0. For the case of Sˆi (k + 1) > Smax , only Pi (k) ∈ [η −1 (k)(S ∗ (k)−Smin ), 0] can ensure Pi (k) and Si (k+1) fulfill (6)-(8). Note that Pave (k) < η −1 (k)(S ∗ (k) − Smin ) and n nPave (k) = PM (k), so under (6)-(8), Pe (k) = | Pi (k) − i=1
PM (k)| is minimized at Pi (k) = η −1 (k)(S ∗ (k) − Smin ). Analogously, for the case of Sˆi (k + 1) < Smin ), Pe (k) is minimized at Pi (k) = η −1 (k)[S ∗ (k) − Smin ]). 4. SIMULATION RESULTS
In this section, numerical simulations are implemented to demonstrate the effectiveness of the third proposed algorithm. Suppose that there are ten EESSs connected by the communication graph depicted in Fig. 2. The model parameters of EESSs are (Cai and Hu, 2016) Qi = 120Ah, Vi = 220V, ζdi = 0.92, ζci = 0.95, i = 1 ∼ 10. The SoC range is [Smin , Smax ] = [0.1, 0.9], and the power range is [Pmin , Pmax ] = [−5kW, 5kW]. Initial SoCs are chosen as S(0) = 0.01 × [28, 31, 34, 37, 40, 44, 46, 49, 52, 55]T , and ε = 0.2. The sampling interval is selected as T = 1 hour, and the operation period is two days. In the two days, the global supply-demand mismatch is periodic with period one day, shown in Tab. I. 1
2
3
4
5
10
9
8
7
6
Fig. 2. Communication graph. Carry out the third algorithm for two days. Fig. 3 shows the SoC and power trajectories of every EESS. It can be observed that the SoC variables are balanced to a common dynamic value, and SoC and power variables satisfy the constraints (6)-(8). Fig. 4 presents the power tracking error Pe (k), implying that Pe (k) is convergent to zero. 5. CONCLUSION In this paper, three energy dispatch algorithms are proposed to determine the charging/discharging power refer-
2018 IFAC MICNON 234 Guadalajara, Mexico, June 20-22, 2018
Wenjing Xie et al. / IFAC PapersOnLine 51-13 (2018) 229–234
Table 1. Global supply-demand mismatch PM (k) (kW) time 00 : 00 01 : 00 02 : 00 03 : 00 04 : 00 05 : 00 06 : 00 07 : 00
mismatch 7.56 7.56 7.56 7.56 7.56 8.40 12.08 1.904
time 08 : 00 09 : 00 10 : 00 11 : 00 12 : 00 13 : 00 14 : 00 15 : 00
mismatch −12.88 −24.32 −37.76 −35.616 −35.04 −25.28 −29.28 −25.28
time 16 : 00 17 : 00 18 : 00 19 : 00 20 : 00 21 : 00 22 : 00 23 : 00
mismatch −11.84 −0.32 12.656 16.80 16.80 16.80 14.00 11.20
SoC Si (k) (%)
100 80 60
EESS 2
20
EESS 3
0
Power Pi (k) (Kw)
EESS 1
40
EESS 4 0
5
10
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20 25 time (h)
30
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45
EESS 5 EESS 6
6
EESS 7
4
EESS 8 EESS 9
2
EESS 10
0 −2 −4
0
5
10
15
20 25 time (h)
30
35
40
45
Power tracking error Pe (k) (Kw)
Fig. 3. SoC and power trajectories under the third algorithm. 1 0.8 0.6 0.4 0.2 0
0
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25 time (h)
30
35
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45
Fig. 4. Power tracking error Pe (k) under the third algorithm. ence for every EESS with the objectives of SoC and supplydemand balances. Consensus control approach, Lyapunov method and graph theory are employed in designing these algorithms. The third one asymptotically realizes the energy dispatch task under SoC and power constraints. Future work may lie on the distributed energy dispatch problem of EESSs with different model parameters. REFERENCES Alberto, L.F., Calliero, T.R., and Martins, A.C. (2007). An invariance principle for nonlinear discrete autonomous dynamical systems. IEEE Transactions on Automatic Control, 52(4), 692–697. Bragard, M., Soltau, N., Thomas, S., and De Doncker, R.W. (2010). The balance of renewable sources and user demands in grids: Power electronics for modular battery energy storage systems. IEEE Transactions on Power Electronics, 25(12), 3049–3056. Cai, H. and Hu, G. (2016). Distributed control scheme for package-level state-of-charge balancing of gridconnected battery energy storage system. IEEE Transactions on Industrial Informatics, 12(5), 1919–1929. 234
Li, C., Coelho, E.A.A., Dragicevic, T., Guerrero, J.M., and Vasquez, J.C. (2017). Multiagent-based distributed state of charge balancing control for distributed energy storage units in AC microgrids. IEEE Transactions on Industry Applications, 53(3), 2369–2381. Lu, X., Sun, K., Guerrero, J.M., Vasquez, J.C., and Huang, L. (2015). Double-quadrant state-of-charge-based droop control method for distributed energy storage systems in autonomous DC microgrids. IEEE Transactions on Smart Grid, 6(1), 147–157. Lu, X., Sun, K., Guerrero, J.M., Vasquez, J.C., and Huang, L. (2014). State-of-charge balance using adaptive droop control for distributed energy storage systems in DC microgrid applications. IEEE Transactions on Industrial Electronics, 61(6), 2804–2815. Morstyn, T., Hredzak, B., and Agelidis, V.G. (2015). Distributed cooperative control of microgrid storage. IEEE Transactions on Power Systems, 30(5), 2780– 2789. Olfati-Saber, R., Fax, J.A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215–233. Olfati-Saber, R. and Murray, R.M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Sichilalu, S., Tazvinga, H., and Xia, X. (2016). Optimal control of a fuel cell/wind/pv/grid hybrid system with thermal heat pump load. Solar Energy, 135, 59–69. Sichilalu, S.M. and Xia, X. (2015). Optimal power dispatch of a grid tied-battery-photovoltaic system supplying heat pump water heaters. Energy Conversion and Management, 102, 81–91. Suberu, M.Y., Mustafa, M.W., and Bashir, N. (2014). Energy storage systems for renewable energy power sector integration and mitigation of intermittency. Renewable and Sustainable Energy Reviews, 35, 499–514. Sudevalayam, S. and Kulkarni, P. (2011). Energy harvesting sensor nodes: Survey and implications. IEEE Communications Surveys & Tutorials, 13(3), 443–461. Tazvinga, H., Zhu, B., and Xia, X. (2015). Optimal power flow management for distributed energy resources with batteries. Energy Conversion and Management, 102, 104–110. Xu, Y., Li, Z., Zhao, J., and Zhang, J. (2017). Distributed robust control strategy of grid-connected inverters for energy storage system’s state-of-charge balancing. IEEE Transactions on Smart Grid, published online. Xu, Y., Zhang, W., Hug, G., Kar, S., and Li, Z. (2015). Cooperative control of distributed energy storage systems in a microgrid. IEEE Transactions on Smart Grid, 6(1), 238–248. Yang, H., Qiu, Y., Li, Q., and Chen, W. (2017). A selfconvergence droop control of no communication based on double-quadrant state of charge in dc microgrid applications. Journal of Renewable and Sustainable Energy, 9(3), 034102. Zhu, B., Tazvinga, H., and Xia, X. (2015). Switched model predictive control for energy dispatching of a photovoltaic-diesel-battery hybrid power system. IEEE Transactions on Control Systems Technology, 23(3), 1229–1236.