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Projected Alternating Direction Method of Multipliers for Hybrid Systems
The International Federation of Congress Automatic Control Proceedings of the 20th World Proceedings of 20th World Congress Proceedings of the the 20th9-14, World Toulouse, France, July 2017 The International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International International Federation of of Automatic Automatic Control The Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 8423–8428 Projected Alternating Direction Method of Multipliers for Hybrid Systems Projected Alternating Direction Method of Multipliers for Hybrid Systems Projected Method of for Systems Projected Alternating Alternating Direction Direction Method of Multipliers Multipliers for Hybrid Hybrid Systems , Chengang Feng* **, Romain Bourdais*
Chengang Feng*,,, **, Romain Bourdais* Chengang Chengang Feng* Feng* **, **, Romain Romain Bourdais* Bourdais* * CentraleSupélec, 35510 Cesson-Sevigne, France (e-mail : [email protected], * CentraleSupélec, 35510 Cesson-Sevigne, France (e-mail :: [email protected], [email protected]). * 35510 Cesson-Sevigne, France (e-mail * CentraleSupé CentraleSupélec, lec,**Xi’an [email protected]). Cesson-Sevigne, France (e-mail : [email protected], [email protected], Jiaotong University, 710049, Xi’an, China [email protected]). [email protected]). **Xi’an Jiaotong University, 710049, Xi’an, China **Xi’an **Xi’an Jiaotong Jiaotong University, University, 710049, 710049, Xi’an, Xi’an, China China Abstract: In large-scale control optimization problems, a decentralized control structure can offer great Abstract: In control optimization problems, aa decentralized control structure can offer great scalability andlarge-scale rapidity advantages over a centralized implantation. Alternating Direction Method of Abstract: In large-scale control optimization problems, control structure can offer great Abstract: In large-scale control optimization problems, a decentralized decentralized control structure can Method offer great scalability and rapidity advantages over a centralized implantation. Alternating Direction of Multipliers (ADMM) is a decentralized optimization algorithm which has the important benefit of being scalability and rapidity advantages over aa centralized implantation. Alternating Direction Method of scalability and overoptimization implantation. Alternating Direction Method of Multipliers (ADMM) is aaadvantages decentralized algorithm which has the important benefit of being quite general in rapidity its scope and applicability incentralized continuous systems. In this paper, a new projected ADMM Multipliers (ADMM) is decentralized optimization algorithm which has the important benefit of being Multipliers (ADMM) is a decentralized optimization algorithm which has the important benefit of being quite general in its scope in continuous systems. In this projected ADMM algorithm is defined and itand canapplicability work in hybrid systems. The key point ispaper, to addaaanew convexification and a quite general in scope and applicability in continuous systems. In this paper, projected ADMM quite general in its its during scope and applicability inADMM continuous systems. Inhave thisisapplied paper, aaitnew new projected ADMM algorithm is defined and it can work in hybrid systems. The key point to add convexification and a projection process each iteration of algorithm. We to a charging control algorithm is defined and can work hybrid systems. The key point is to aa convexification and algorithm isprocess definedduring and it iteach can work in in results hybrid systems. The key point isapplied to add addcan convexification and aa projection iteration of ADMM algorithm. We have it to a charging control problem of electric vehicles. Simulation show that the proposed algorithm converge to a similar projection process process during during each each iteration iteration of of ADMM ADMM algorithm. algorithm. We We have have applied applied it it to to aa charging charging control control projection problem vehicles. Simulation show that the proposed can converge to aa similar result as of a electric centralized control within aresults limited iteration time. Due algorithm to its availability, simplicity and problem of electric vehicles. Simulation results show that the proposed algorithm can converge to similar problem of electric vehicles. Simulation results show that the proposed algorithm can converge to a similar result as a centralized control within a limited iteration time. Due to its availability, simplicity and scalability, the projected ADMM algorithm may be attractive in some practical engineering application. result as centralized control within aa limited iteration time. Due to simplicity and result as aa the centralized control within limited iteration time. Duepractical to its its availability, availability, simplicity and scalability, projected ADMM algorithm may be attractive in some engineering application. scalability, the projected ADMM algorithm may be attractive in some practical engineering application. © 2017, IFAC (International Federation of Automatic Control) Hosting Elsevierdesign Ltd. All reserved. scalability, the projected ADMM algorithm may be attractive in someby practical engineering application. Keywords: Projected Alternating Direction Method of Multipliers; Control forrights hybrid systems; Keywords: Projected Method of Control Decentralized control;Alternating Large scale Direction optimization problems; Electric vehicle Keywords: Projected Alternating Direction Method of Multipliers; Multipliers; Control design design for for hybrid hybrid systems; systems; Keywords: Projected Alternating Direction Method of Multipliers; Control design for hybrid systems; Decentralized control; Large scale optimization problems; Electric vehicle Decentralized control; Large scale optimization problems; Electric vehicle Decentralized control; Large scale optimization problems; Electric vehicle in a continuous system, no literature until now has integrated 1. INTRODUCTION theaa ADMM algorithm a hybriduntil system state is in continuous system, into no literature literature until nowwhose has integrated integrated continuous system, no now has 1. INTRODUCTION in a ADMM continuous system, no literature until nowvariables. has integrated A centralized optimal control can assure the satisfaction of in 1. defined by algorithm both continuous and discrete the algorithm into a hybrid hybrid system whose stateThe is 1. INTRODUCTION INTRODUCTION the ADMM into a system whose state is the ADMM algorithm into a hybrid system whose state is global constraints and has a real-time insight into operating A centralized optimal control can assure the satisfaction of objective of this paper is to propose a projected ADMM defined by both continuous and discrete variables. The A centralized optimal control can assure the of by both continuous and discrete variables. The A centralized optimal control can(Mets, assure the satisfaction satisfaction of defined defined by both continuous and discrete variables. The conditions at all points on network Verschueren, Haerick, global constraints and has a real-time insight into operating algorithm in which isaa suitable for aADMM hybrid objective of decentralized this paper paper is iscontrol to propose propose projected ADMM global constraints and has aa real-time real-time insight insight into into operating operating objective of this to projected global constraints and has objective of decentralized this paper iscontrol totopropose a suitable projected ADMM Develder, Turck, 2010). Nevertheless, it becomes conditions at&all allDe points on network (Mets, Verschueren, Haerick, system. The solution is easier implement in practice than a algorithm in which is for a hybrid conditions at points on network (Mets, Verschueren, Haerick, algorithm in decentralized control which is suitable for aa hybrid conditions at allDe points on network (Mets, Verschueren, Haerick, algorithm in decentralized control which is suitable for hybrid inefficient due to the huge dimensionality and thousands of Develder, & Turck, 2010). Nevertheless, it becomes purely continuous system. system. The solution is easier to implement in practice than aa Develder, & Turck, 2010). Nevertheless, Nevertheless, it it becomes becomes system. The solution is easier to implement in practice than Develder, & De De Turck, 2010). system. The solution is easier to implement in practice than a constraints when the number of nodes increases for large scaled inefficient due to the huge dimensionality and thousands of purely continuous system. inefficient due to the huge dimensionality and of continuous system. inefficient due tothe thenumber huge dimensionality andforthousands thousands of purely Increased use of electric vehicles has augmented power grid purely continuous system. problems. To address these issues, a decentralized control constraints when of nodes increases large scaled constraints when the of nodes increases for large scaled constraints when the number number nodes increases for many largecontrol scaled demand and many issueshas such as an incremental Increased use raised of electric electric vehicles has augmented power grid grid structure, where computation distributed across nodes Increased problems. To address theseofisissues, issues, a decentralized decentralized use of vehicles augmented power problems. To address these a control Increased use of electric vehicles has augmented power grid problems. To address these issues, a decentralized control investment for charging reinforcements, system overloads and demand and raised many issues such as an incremental and coordinated by a central instance, is often considered in a demand and raised many issues such as an incremental structure, where computation computation is distributed distributed across many nodes nodes structure, where is across many demand and raised many issues such as Fernandez anoverloads incremental structure, where computation is distributed across manyalready nodes reliability problems as shown in the works et al. investment for charging reinforcements, system large-scale system. Many decentralized algorithms and coordinated by a central instance, is often considered in a investment for for charging charging reinforcements, reinforcements, system system overloads overloads and and and coordinated by aa central instance, is often considered in investment and and coordinated by an central instance, is often considered in aa reliability (2011), Eising et alas (2014). For instance, with more et 20% problems shown in the works Fernandez al. exist to determine optimal control strategy, such as Nash large-scale system. Many decentralized algorithms already reliability problems as shown in the works Fernandez et al. large-scale system. Many decentralized algorithms algorithms already already reliability problems as shown in the works the Fernandez et20% al. large-scale system. Many decentralized penetration of EVs in the power system, peak power (2011), Eising et al (2014). For instance, with more certainty equivalence methodology (Ma, Callaway, & Hiskens, exist to determine an optimal control strategy, such as Nash (2011), Eising et al (2014). For instance, with more 20% exist to to determine determine an an optimal optimal control control strategy, strategy, such such as as Nash Nash (2011), et al in(2014). instance, withZhou, morepower 20% exist demand Eising willof be EVs increased by power upFor to 35.8% (Qian, Allan, penetration of EVs the power system, the peak 2010), distributed stochastic charging algorithm (Gan, Topcu, penetration certainty equivalence methodology (Ma, Callaway, & Hiskens, in the system, the peak power certainty equivalence methodology (Ma, Callaway, & Hiskens, penetration of EVs in the power system, the peak power certainty equivalence methodology (Ma, Callaway, &algorithm Hiskens, & Yuan,will 2011). Alternatively, a proper charging demand will be increased increased by up upwith to 35.8% 35.8% (Qian, Zhou,control, Allan, & Low, 2012), stochastic price-based two-layer control 2010), distributed charging algorithm (Gan, Topcu, be by to (Qian, Zhou, Allan, 2010), distributed stochastic charging algorithm (Gan, Topcu, demand demand will be of increased upwith to 35.8% (Qian, Zhou,control, Allan, 2010), distributed stochastic charging algorithm (Gan, Topcu, theYuan, penetration EVs canbynot only the power grid by & Yuan, 2011). Alternatively, abenefit proper charging (Gharesifard, Başar, & Domínguez-Garcí a, 2013) and & Low, 2012), price-based two-layer control algorithm & 2011). Alternatively, with a proper charging control, & Low, Low, 2012), 2012), price-based price-based two-layer two-layer control control algorithm algorithm & Yuan, 2011). Alternatively, with abenefit proper charging control, & flattening the overall load profile and but also optimize the the penetration of EVs can not only the power grid by integration of resource coordination and (Gharesifard, Başar, & a, Branch-and2013) and and the penetration of EVs can not only benefit the power grid by (Gharesifard, Başar, allocation & Domínguez-Garcí Domínguez-Garcí a, 2013) the penetration of EVs can not onlyand benefit the power grid the by (Gharesifard, Başar, & Domínguez-Garcí a, 2013) and charging price of individual users by acting as controllable flattening the overall load profile but also optimize Bound (Luo, Bourdais, van den Boom, & De Schutter, 2015). integration of of resource resource allocation allocation coordination coordination and the overall load profile and but also optimize the integration and Branch-andBranch-and- flattening flattening the overall load profile and but also optimize the integration of resource allocation coordination and Branch-andloads (Lopes, Soares, Almeida, & Da Silva, 2009). What’s charging price price of of individual individual users users by by acting acting as as controllable controllable However, these algorithms require convex2015). cost charging Bound (Luo, Bourdais, van den den Boom,a & &strictly De Schutter, Schutter, 2015). Bound (Luo, Bourdais, van Boom, De charging of users acting controllable Bound (Luo, Bourdais, van den Boom, &strictly De Schutter, 2015). more, (Lopes, if price EVs Soares, areindividual equipped with Vehicle Grid (V2G) loads (Lopes, Soares, Almeida, & by Da Silva,toas 2009). What’s function which is not available in all situations. Alternating However, these algorithms require a convex cost loads Almeida, & Da Silva, 2009). What’s However, these algorithms require aa strictly convex cost loads (Lopes, Soares, Almeida, & Da Silva, 2009). What’s However, these algorithms require strictly convex cost technology, which allows bidirectional energy exchange more, if EVs are equipped with Vehicle to Grid (V2G) Directionwhich Method algorithm can deal more, if EVs are equipped with Vehicle to Grid (V2G) function which is of notMultipliers available (ADMM) in all all situations. situations. Alternating function is not available in Alternating more, if EVs EVswhich aretheequipped withthey Vehicle to Grid (V2G) function which is of not available infunction all situations. Alternating between and power grid, could offer numerous technology, allows bidirectional energy exchange with a non-strictly convex cost and it maps onto Direction Method Multipliers (ADMM) algorithm can deal technology, which allows bidirectional energy exchange Direction Method of Multipliers (ADMM) algorithm can deal technology, which allows bidirectional energy exchange Direction Method of Multipliers (ADMM) algorithm can deal services to the power grid, such as active power support, between EVs EVs and and the the power power grid, grid, they they could could offer offer numerous numerous several distributed reasonably with non-strictly convex programming cost function function models and it it maps maps onto between with aaa standard non-strictly convex cost and onto between andpower the power they could offer numerous with non-strictly convex cost & function and it maps onto services reactive EVs compensation help power to solve the services topower the grid, grid, suchand as active active power support, well (Boyd, Parikh, Chu, Peleato, Eckstein, 2011). several standard distributed programming models reasonably to the power grid, such as support, several standard distributed programming models reasonably services to the power grid, such as active power support, several standard distributed programming models reasonably intermittency issue of the renewable energy resources (Tan, reactive power compensation and help to solve the well (Boyd, (Boyd, Parikh, Chu, Chu, Peleato, Peleato, & & Eckstein, Eckstein, 2011). 2011). power compensation and help to the well reactive power compensation and energy help to solve solve (Tan, the Therefore, number of Peleato, recent researches studied the reactive well (Boyd,aParikh, Parikh, Chu, & Eckstein,have 2011). Ramachandaramurthy, & Yong, 2016). We implemented the intermittency issue of the renewable resources intermittency issue of the renewable energy resources (Tan, intermittency issue of the renewable energy resources (Tan, ADMM algorithm in decentralized control. Tan, Yang, & Therefore, a number of recent researches have studied the projected ADMM algorithm to an EV charging control problem Ramachandaramurthy, & Yong, 2016). We implemented the Therefore, a number of recent researches have studied the Ramachandaramurthy, & Yong, 2016). We implemented the Therefore, a number ofdecentralized recent researches have studied the Ramachandaramurthy, & Yong, 2016). We implemented the Nehorai, (2014) applied ADMM in a distributed control ADMM algorithm in control. Tan, Yang, & with the objectives of price-based optimization. projected ADMM algorithm to an EV charging control problem ADMM in control. Tan, Yang, & ADMM algorithm to an EV charging control problem ADMM algorithm algorithm in decentralized decentralized control. Tan,reduce Yang, & projected projected ADMM algorithm to an EV charging control problem optimization to flatten the demand response and the Nehorai, (2014) applied ADMM in a distributed control with the the objectives objectives of of price-based price-based optimization. optimization. Nehorai, (2014) applied ADMM in a distributed control Nehorai, (2014) applied distributed control The of thisofpaper is organized as follows: the projected withreminder the objectives price-based optimization. electrical bill forflatten each individual userin in aa smart grid. Chen, & with optimization to flatten the ADMM demand response and reduce the optimization to the demand response and reduce the optimization to flatten the demand response and reduce the ADMM algorithm is defined in Section 2 and applied in the The reminder of this paper is organized as follows: follows: the projected projected Cheng, (2014) a distributed stochastic ADMM electrical bill for for proposed each individual individual user in in aa smart smart grid. Chen, Chen, & The reminder of this paper is organized as the electrical bill each user grid. & The reminder of this paper is organized as follows: the projected electrical bill for each individual user in a smart grid. Chen, & charging control of electric vehicles in Section 3. Section ADMM algorithm is defined in Section 2 and applied in the the4 algorithm to help frequency regulation in the power grid. The Cheng, (2014) (2014) proposed proposed aa distributed distributed stochastic stochastic ADMM ADMM ADMM algorithm is defined in Section 2 and applied in Cheng, ADMM algorithm iselectric defined in Section 2 andand applied in the4a Cheng, (2014) proposed aregulation distributed stochastic ADMM presents the numerical simulation study finally, charging control of vehicles in Section 3. Section paper (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, 2013) algorithm to help frequency in the power grid. The charging control of vehicles in Section 3. Section algorithm to frequency regulation in power grid. The charging ofinelectric electric vehicles instudy Section Section 44a algorithm toanhelp help frequency regulation in the the power grid.2013) The conclusioncontrol is given Section 5. presents the numerical simulation and3. finally, finally, introduced electric vehicle (EV) ADMM framework to presents paper (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, Jacobsen, the numerical simulation study and aa paper (Rivera, Wolfrum, Hirche, Goebel, & 2013) presents the numerical simulation study and finally, paper (Rivera, Wolfrum, Hirche, Goebel, & charging Jacobsen, 2013) conclusion is is given given in in Section Section 5. 5. achieve valley filling and minimal-cost goals. introduced an electric vehicle (EV) ADMM framework to conclusion introduced an electric vehicle (EV) framework to introduced electric vehicle (EV) ADMM ADMM framework to conclusion is given in Section 5. Although valley theanabove mentioned ADMM algorithm worksgoals. well achieve valley filling and minimal-cost minimal-cost charging goals. achieve filling and charging achieve valley filling and minimal-cost charging goals. Although the the above mentioned mentioned ADMM algorithm algorithm works well well Although Although the above above mentioned ADMM ADMM algorithm works works well Copyright © 2017 IFAC 8757 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2017 responsibility IFAC 8757Control. Peer review of International Federation of Automatic Copyright © © 2017 2017 IFAC IFAC 8757 Copyright 8757 10.1016/j.ifacol.2017.08.1571
Proceedings of the 20th IFAC World Congress 8424 Chengang Feng et al. / IFAC PapersOnLine 50-1 (2017) 8423–8428 Toulouse, France, July 9-14, 2017
In this paper, we consider a collection of 𝑁𝑁 independent agents in a hybrid system that share a common resource. Each of them has to minimize a local objective, while fulfilling various local constraints. Among these agents, the first 𝑀𝑀 of them are governed by discrete inputs and while the other (𝑁𝑁 − 𝑀𝑀) ones are governed by continuous inputs. The global optimization problem can be formulated as follows: 2. PROBLEM FORMULATION
min𝑥𝑥𝑖𝑖 ∑Ni=1 fi
subject to:
(xi )
(1)
∑Ni=1 Ai xi =0,
∀i=1…N, Ri ≤ Bi xi ≤ Ri ∀i=1…M, xi ∈Xdi
∀i=M+1…N, xi ∈Xc𝑖𝑖
In this optimization problem, the set Xid is a discrete set while Xic is a continuous convex set. The equality constraint ∑Ni=1 Ai xi =0 defines the resource to be shared and the inequations Ri ≤ Bi xi ≤ Ri represent the local constraints. To solve the problem, a well-known method, the so-called ADMM, can be a useful tool. However, its convergence cannot be guaranteed in the presence of discrete set. Consequently, the first step of our demarche is to relax these discrete constraints by convexification, which means transferring discrete sets into continuous sets. As an example, a discrete variable, xdi ∈{-1,0,1} can be transferred into xci ∈[-1,1] . With the translation of the discrete variables, the new optimization problem can be settled as follows: min𝑥𝑥𝑖𝑖 ∑Ni=1 fi (xi )
subject to:
(2)
∑Ni=1 Ai xi =0,
∀i=1…N, Ri ≤ Bi xi ≤ Ri ∀i=1…N,
xi ∈Xci
For the first 𝑀𝑀 optimization variables 𝑥𝑥𝑖𝑖 , the corresponding set
Xic is a convexification version of Xid . With this first manipulation, the ADMM method can be applied. It consists of an iterative mechanism (Boyd, Parikh, Chu, Peleato, & Eckstein, 2011), in which at each iteration, each agent has to solve its own optimization problem in parallel by: xi k+1 = arg min xi∈xc fi (xi )+ykT Ai xi + ‖ Ai xi -xi k +x̅k ‖2 (3) ρ
2
2
i
2
2
xi k+1 = arg min xi∈xc fi (xi )+ ‖Ai xi -xi k +x̅k +uk ‖2 2
ρ
2
i
uk+1 =uk +x̅k+1
(6)
In order to get a solution of the initial hybrid optimization problem, the third step of our demarche is to project the solution xi (∀i=1…M) found in (5) back into the discrete set Xdi before updating the average value 𝑥𝑥̅ 𝑘𝑘 and scaled dual variable u. This is what we call in this article the projected ADMM. The projection could be done based on either equal-probability distribution, normal distribution or even heuristic decision. There is no standard projection mechanism but should correspond to the characteristics of system constraints. In the example, 𝑥𝑥𝑖𝑖 lies in the interval [-1, 1] and can be projected based on equal-probability distribution mechanism:
xi =
1
-1,
when xi ϵ [-1, - )
0,
when xi ϵ [- , ]
1
3 1
3 1 3
(7)
1, when xi ϵ ( , 1] { 3 However, a simple projection can sometimes make a suboptimal solution infeasible (i.e. local constraints are not satisfied). Therefore, some conditions should be added in the projection to ensure the feasibility. The idea is to add some extra conditions in the projection rules corresponding to each constraint of system. For example, if Bi xi > Ri , then xi will be 1 projected to 0 even it lies in the interval ( , 1]. In this way, the 3 local constraints could be satisfied and the solution is feasible in a non-convex problem. As for the convergence criterion, Boyd, Parikh, Chu, Peleato, & Eckstein, (2011) suggest a reasonable stopping criterion, which is that the primal 𝑟𝑟 𝑘𝑘 and dual 𝑠𝑠 𝑘𝑘 residuals must be relatively small, i.e. Norm( rk ) ≤ εPri
(8)
Norm(sk ) ≤ εdual
(9)
where εPri > 0 and εdual > 0 are feasibility tolerances for the primal and dual feasibility conditions. The primal feasibility and dual feasibility can be written as: (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, 2013)
sk =[s1 k ,s2 k …,sN+1 k ]
(4)
In these two equations, 𝑘𝑘 is the current iteration number, ρ > 0 is the penalty parameter, 𝑦𝑦 is the dual variable and 1 x̅k = ∑Ni=1 x𝑘𝑘i is the average profile of all agents. If the N optimization problem (2) is strongly convex, this iterative mechanism will be guaranteed to converge to the optimal solution.
(5)
Also, the coordinator has to update the scaled dual variable as follows:
rk = x̅k
And a coordinator has to update a quantity, which we call the Lagrangian multiplier: yk+1 =yk +ρ x̅k+1
‖ Ai xi -xi k +x̅k ‖ and replacing the dual variable 𝑦𝑦 by the 2 scaled dual variable u = y/ρ. Then, the iteration (3) becomes:
ρ
(10) T
with si k =-ρN (xki -xk-1 ̅k-1 -x̅k ). i +x
(11)
In summary, the projected ADMM algorithm is presented as follows:
In the work of Boyd, Parikh, Chu, Peleato, & Eckstein, (2011), we can also find that (3) can be written in a slightly different form, which is often more convenient, by combining yk xi and 8758
Projected ADMM algorithm Variables: the total number of agents: N, the number of agents who are governed by discrete inputs: M, parameters of constraints: Ai , Bi , Ri , Ri , feasible set: Xdi , control horizon: 𝑇𝑇
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Chengang Feng et al. / IFAC PapersOnLine 50-1 (2017) 8423–8428
8425
Optimization variables: xi Select a sufficiently large total iteration number K. Initialize xi 0 (t), u0i (t), t∈ T, i∈ N such that they satisfy all constrains. For 𝑘𝑘 = 0, 1 … 𝐾𝐾:
1) Convexification: for each 𝑖𝑖 ∈ (1…M), transfer the discrete feasible input set into a continuous set 2) Calculation: Each agent 𝑖𝑖 solves the problem in (5) to get xi k+1 (t) at each time slot t
3) Projection: project each continuous xi k+1 (𝑖𝑖 ∈ (1…M)) into discrete value with several extra conditions 4) Update: calculate uk+1 by (6) with the new x̅k+1
Figure 1. System architecture
5) Termination: Stop the iteration if the convergence criteria hold for all agents or if the maximum number of iterations K is reached; otherwise return to step 1 This algorithm offers a suitable mechanism to find a suboptimal solution of the original problem, as long as we can include a projection that ensures the feasibility of each local constraints. 3. APPLICATION: CHARGING CONTROL OF ELECTRIC VEHICLES As for controlling the charging of EV, it is much easier to control just the discrete charging decision of each EV other than the continuous charging power in reality. But the power supply from the grid to charge electric vehicle is a continuous variable. Thus, the charging control system can be seen as a hybrid system. By assuming the profile of the electricity price, the number of EVs needed to be charged and their arrival and departure times given, we need to decide an optimal charging control decision of a fleet of EVs at a charging station. The goal is to charge all the EVs up to the required levels within the determined time under the constrained grid condition while the total cost on charging the EVs is minimized.
Each EV is able to predict the amount of charging energy needed and is always in constant power charging option (Constant power – Constant Voltage charging option) (Marra, Yang, Træholt, Larsen, Rasmussen, & You, 2012). Thus, the EV 𝑖𝑖’s charging rate Mi is a constant. Upon departing, EV 𝑖𝑖 demands its battery level to be between [ Di , Di ]. Assume EV 𝑖𝑖’s initial battery level when it arrives is B0i and its battery capacity is B𝑐𝑐i , we can have: Di ≤ B0i + ∑Tt=1 Mi ∆t xi (t) /Bci ≤ Di
(12) can be reformulated as
Ri =Di Bci - B0i Bci ≤ ∑Tt=1 Mi ∆t xi (t) ≤ Ri =Di Bci - B0i Bci (13)
𝑅𝑅𝑖𝑖 , Ri are minimal and maximum charging power required for EV i, respectively.
In addition, the battery level in an EV should be limited in a reasonable interval (Marra, Yang, Træholt, Larsen, Rasmussen, & You, 2012). The choice for such an interval relates mainly to battery lifetime aspects: charging or discharging EV beyond the interval leads to a quicker battery degradation. Let [ Si , Si ] denote the interval of EV 𝑖𝑖, then we have: c Si ≤ B0i + ∑T' t=1 Mi ∆t xi (t) /Bi ≤ Si
3.1 System model As shown in Fig.1, we consider a system consisting of an aggregator, EVs, no-EV load (residential habitants for instance), renewable energy source and a power grid. The aggregator is an energy administrator which obtains energy form the power grid and renewable energy source and supply it to energy consumers. The aggregator negotiates with 𝑁𝑁 EVs (i∈(1,…,N)) at a charging station for a daily charging schedule over 𝑇𝑇 time slots (t∈( 1,…,T)). V2G service is fully employed whenever necessary and each EV obeys the guide of aggregator.
(12)
T' ∈ T
(14)
3.2 Projected ADMM algorithm for charging control of EVs As defined in the work Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, (2013), the EV charging control problem can be formulated as a standard exchange optimization problem using ADMM algorithm. In the case of price-based optimization which aims to minimize the total charging cost, using the variables definition displayed in Table 1, we can define the cost function of the aggregator: min
f(xa )=PT ∙xa = ∑Tt=1 P(t)∙xa (t)
subject to
∑Ni=1 xi ∙Mi ∙∆t - xa +Pload - Pnew =0
0 ≤ xa ≤ xa , xi (t)∈{-1,0,1} 8759
(15)
Proceedings of the 20th IFAC World Congress 8426 Chengang Feng et al. / IFAC PapersOnLine 50-1 (2017) 8423–8428 Toulouse, France, July 9-14, 2017
For each EV 𝑖𝑖 , xi (t)=1 means the EV i is charging, consuming energy provided by the aggregator at time slot t. xi (t)=-1 means that the EV i is discharging, using V2G service to feed energy back to the system at time slot t and when xi (t)=0, neither charging nor discharging will happen. In the contrast, the aggregator cannot give energy to the outside power grid and it also has a limitation of power bought from the outside power grid 𝑥𝑥𝑎𝑎 due to physical power transfer line limitation. The energy balance equation means that the demand and supply of energy in a system should be equal. Table 1. LIST OF SYMBOLS USED IN THIS PAPER Description
Type
Variable 𝑃𝑃
Profile of the electricity price
𝑥𝑥𝑎𝑎
Power bought from the power grid in the aggregator
𝑥𝑥i
Discrete charging sequence of EV i
𝑀𝑀𝑖𝑖 ∆𝑡𝑡
𝑃𝑃𝑛𝑛𝑛𝑛𝑛𝑛
𝑁𝑁
𝑥𝑥𝑎𝑎 𝑅𝑅𝑖𝑖 𝑅𝑅𝑖𝑖 𝑆𝑆𝑖𝑖 𝑆𝑆𝑖𝑖 𝐵𝐵𝑖𝑖
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
Constant
Time step duration
Scalar
Power consumed apart from EV in the system Number of EVs
Table 2. PROJECTION RULES FOR EV CHARGING ADMM ALGORITHM 𝒙𝒙𝒄𝒄𝒊𝒊 (𝐭𝐭)
𝒙𝒙𝒅𝒅𝒊𝒊 (𝐭𝐭)
Conditions
1
[-Mi ∆t, - Mi ∆t] 3
c
∑Tt=1 Mi ∆t xi (t) ≥ Ri (initially projected to 0)
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
(
1 1 - Mi ∆t, Mi ∆t) 3 3
∑Tt=1 Mi ∆t xi (t) ≤ Ri (initially projected to-1)
∑Tt=1 Mi ∆t xi (t) ≥ Ri (initially projected to 1)
0
Constant Vector ∈ 𝑅𝑅 𝑇𝑇 Scalar
Maximum charging power required for EV i
Scalar
Minimal battery level required in EV i Minimal battery level required in EV i Matrix to calculate the stored energy in EV i
Bi (T' )xi (T' )(t)=Si (t), T' =1,..,t ∈T
-1
Minimal charging power required for EV i
[
2
(16)
As each EV fully does Vehicle to Grid service, in order to achieve significant reduction in the charging cost, the cost function for each EV is set to be 0 (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, 2013). Then, for each EV 𝑖𝑖, the EV charging problem in ADMM form is: 2 k ̅k +uk ‖2 x𝑖𝑖 2 ‖xi -xi +x
subject to Ri ≤ ∑𝑇𝑇𝑡𝑡=1 Mi ∆t xi (𝑡𝑡) ≤ Ri xi ∈ Xic : continuous set
1
( ∑Ni=1 xk+1 +xk+1 a ) i
uk+1 =uk +x̅k+1
subject to -xa +Pload - Pnew ≤ xa ≤ Pload - Pnew
ρ
∑Tt=1 Mi ∆t xi (t) ≤ Ri (initially projected to 0)
̅xk+1 =
Lower triangular matrix ∈ 𝑅𝑅 𝑇𝑇∗𝑇𝑇
2
Bi (T' )xi (T' )(t)=Si (t), T' =1,..,t ∈T
For the average profile of all agents 𝑥𝑥̅ and scaled dual variable u update:
Constant Vector ∈ 𝑅𝑅 𝑇𝑇
ρ
∆t, Mi ∆t] c
Constant Vector ∈ 𝑅𝑅 𝑇𝑇
xa k+1 = arg min xa -PT xa + ‖xa -xa k +x̅k +uk ‖2
1 M 3 i
1
(15) can be reformulated in the ADMM form as:
S i ≤ B i xi ≤ S i ,
Before 𝑥𝑥̅ update and u update, the continuous set 𝑋𝑋𝑖𝑖 above needs to be projected into a discrete set 𝑋𝑋𝑖𝑖𝑑𝑑 . The three discrete value representing the charging decisions have almost the same chance to appear, so the equal-probability distribution methodology can be applied in the projection. Some conditions need to be added based on the existing local constraints. The projection rules are summarized in Table 2.
Scalar
Maximum power can be bought from the power grid
xi k+1 = arg min
Ri defines a bound on the power needed to charge EV 𝑖𝑖 up to the required level before departing from the charging station. The other set of inequality constraints Si ≤ Bi xi ≤ Si make sure that the (dis)charging power does not violate the maximal and minimal energy that the battery can support when EV 𝑖𝑖 is connected. 𝑋𝑋𝑖𝑖𝑐𝑐 is a continuous set transferred by a discrete set.
Discrete Vector ∈ 𝑅𝑅 𝑇𝑇
Charging rate of EV i in Constant power charging option
Available renewable energy source
𝑃𝑃𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
The first set of inequality constraints Ri ≤ ∑𝑇𝑇𝑡𝑡=1 Mi ∆t xi (𝑡𝑡) ≤
(17)
N+1
(18) (19)
The processes of projected ADMM in charging control of EVs is as follows: the aggregator first sends the average profile of all agents 𝑥𝑥̅ and scaled price 𝑢𝑢 to all EVs. Based on this information, the aggregator and the EVs solve their individual optimization problem defined in (16) and (17). Each EV i needs to project the obtained value 𝑥𝑥𝑖𝑖 into a discrete set, do some recalculations and then send their solutions to the aggregator in order to obtain the new 𝑥𝑥̅ . With this new value, the aggregator then updates the scaled price u. This sequence is repeated until the convergence criterion is met. 4 NUMERICAL SIMULATION STUDY In this section, simulations are conducted to verify the performance of the proposed algorithm. We considered a case where 30 EVs need to be charged in the daytime assuming these 30 vehicles arrive in the charging station and depart at the same time. The control horizon T was assumed to be 12 hours and the time slot was set to be 30 minutes, thus T =24. The battery capacity of EV was supposed to be identical, namely C. The initial battery level of EV i was
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generated randomly form [0.1C, 0.7C]. The required charging level at departing time and charging rate of all vehicles were assumed to be the same. Furthermore, we assumed that charging was not interrupted before departure of each EV and the renewable energy generated was always below to the demand in the system. The energy price profile P was obtained from the France RTE website shown in Fig. 2. For comparison, the uncontrolled charging scenario is also tested, that is, beginning to charge as soon as the EV arrives at the charging station and stopping charging as soon as the demand is satisfied. The simulation environment was MATLAB and Quadratic Programming (quadprog) is used for the optimizations.
Figure 3. Cost evolution with iteration time of ADMM algorithm, projected ADMM algorithm, centralized control algorithm (Branch-and-Bound) in a hybrid system, centralized control algorithm in a continuous system
Figure 2. Electricity price profile for simulation
Two kinds of charging control systems were simulated. In the first case, the charging power of each EV is controlled. Thus, the system is a continuous system. In the second case, the control variable is the charging decision of each EV, it is therefore a hybrid system. The total cost evolutions with iteration of different control strategies are illustrated in Fig. 3. Both ADMM algorithm and projected ADMM algorithm are finally converged to some values around the results of the centralized control strategies within an iteration time smaller than 2000. This demonstrates that the proposed ADMM algorithm can converge to a similar result as a centralized control algorithm.
Table 3 shows the relative cost of the controlled cases and no control case where we define the cost in no control system is a unite cost. It may be seen that that the total cost of the aggregator can be reduced by about 57% with a proper control. Additionally, the centralized control always outperforms the decentralized control. It can also be noticed that the control in a continuous system leads to a smaller cost than that in a hybrid system. However, the maximum difference between diverse control strategies is around 0.1% of the total cost, which is much smaller than the aforementioned reduction. In this way, the result obtained by the proposed algorithm is acceptable. Table 3. COMPARISON BETWEEN THE CONTROLLED CASES System Type Control strategy Normalized cost
Continuous system
Hybrid system
No control
100.00
Centralized control
42.79
ADMM control
42.82
Centralized control
42.85
Projected ADMM control
42.90
Fig. 4 illustrates the battery level evolution of an electric vehicle during charging. It can be seen that the vehicle charges when the electricity price is low and discharges when the price is high. The battery level at departing time falls into the required interval and it does not violate the maximal and minimal energy that the battery can support. This shows that the proposed algorithm can satisfy the constraints. In Fig. 5, we show the impact of the proposed ADMM algorithm on the power consumed for charging electric vehicles. It can be noticed that with the proposed ADMM algorithm, the vehicles consume much more energy during low electricity price period and consume less, even negative energy during high electricity price period, which means they send their stored energy back to the grid. It shows the idea of minimizing the total cost on charging the EVs: the vehicles are controlled to charge when the price is low and to discharge when the price is high. The simulation result validates the effectiveness of the proposed ADMM algorithm.
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conditions in the projection can guarantee the feasibility of the optimal solutions. This algorithm is effective and can lead to a similar result as a centralized control which makes it attractive. The features of the proposed algorithm are validated in a case study of EV charging control with the intention of minimizing the total cost for charging the EVs. REFERENCES
Figure 4. Example of the battery level evolution of an EV during charging process.
Figure 5. Power consumed evolution for charging electric vehicles with proposed ADMM control strategy
In reality, a hybrid system is easier to implement than a continuous system. For example, in an EV charging system, it is easier to control just the charging sequence composed of -1, 0, and 1 instead of the charging power. It is also beneficial to the cost reduction of control infrastructures. Moreover, the decentralized control structure can offer great scalability advantages over a centralized implantation. Because of all these aspects, the projected ADMM algorithm seems more practical in real environment. 5 CONCLUSIONS In this paper, a projected ADMM algorithm suitable for hybrid systems was introduced. The idea of the proposed algorithm is to add a convexification and a projection at the beginning and the end of each iteration of ADMM algorithm. The extra
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Mets, K., Verschueren, T., Haerick, W., Develder, C., & De Turck, F. (2010). Optimizing smart energy control strategies for plug-in hybrid electric vehicle charging. In Network Operations and Management Symposium Workshops (NOMS Wksps)(pp. 293-299). Ma, Z., Callaway, D., & Hiskens, I. (2010). Decentralized charging control for large populations of plug-in electric vehicles. In 49th IEEE conference on decision and control (CDC) (pp. 206-212). IEEE. Gan, L., Topcu, U., & Low, S. H. (2012). Stochastic distributed protocol for electric vehicle charging with discrete charging rate. In 2012 IEEE Power and Energy Society General Meeting (pp. 1-8). IEEE. Gharesifard, B., Başar, T., & Domínguez-García, A. D. (2013). Price-based distributed control for networked plug-in electric vehicles. In 2013 American Control Conference (pp. 5086-5091). IEEE. Luo, R., Bourdais, R., van den Boom, T. J., & De Schutter, B. (2015). Integration of resource allocation coordination and branch-and-bound. In 2015 54th IEEE Conference on Decision and Control (CDC) (pp. 4272-4277). IEEE. Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine Learning, 3(1), 1122. Tan, Z., Yang, P., & Nehorai, A. (2014). An optimal and distributed demand response strategy with electric vehicles in the smart grid. IEEE Transactions on Smart Grid, 5(2), 861-869. Chen, S., & Cheng, R. S. (2014). Distributed Charging Control of Plug-In Electric Vehicles via the Stochastic Alternating Direction Method of Multipliers. In 2014 IEEE 80th Vehicular Technology Conference (VTC2014-Fall) (pp. 1-5). IEEE. Rivera, J., Wolfrum, P., Hirche, S., Goebel, C., & Jacobsen, H. A. (2013). Alternating direction method of multipliers for decentralized electric vehicle charging control. In 52nd IEEE Conference on Decision and Control (pp. 6960-6965). IEEE. Fernandez, L. P., San Roman, T. G., Cossent, R., Domingo, C. M., & Frías, P. (2011). Assessment of the impact of plug-in electric vehicles on distribution networks. IEEE Transactions on Power Systems, 26(1), 206-213. Eising, J. W., van Onna, T., & Alkemade, F. (2014). Towards smart grids: Identifying the risks that arise from the integration of energy and transport supply chains. Applied Energy, 123, 448-455. Qian, K., Zhou, C., Allan, M., & Yuan, Y. (2011). Modeling of load demand due to EV battery charging in distribution systems. IEEE Transactions on Power Systems, 26(2), 802-810. Lopes, J. P., Soares, F. J., Almeida, P. M., & Da Silva, M. M. (2009). Smart charging strategies for electric vehicles: Enhancing grid performance and maximizing the use of variable renewable energy resources. In EVS24 Intenational Battery, Hybrid and Fuell Cell Electric Vehicle Symposium, Stavanger, Norveška. Tan, K. M., Ramachandaramurthy, V. K., & Yong, J. Y. (2016). Integration of electric vehicles in smart grid: A review on vehicle to grid technologies and optimization techniques. Renewable and Sustainable Energy Reviews, 53, 720-732. Marra, F., Yang, G. Y., Træholt, C., Larsen, E., Rasmussen, C. N., & You, S. (2012). Demand profile study of battery electric vehicle under different charging options. In 2012 IEEE Power and Energy Society General Meeting (pp. 1-7).
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 8423–8428 Projected Alternating Direction Method of Multipliers for Hybrid Systems Projected Alternating Direction Method of Multipliers for Hybrid Systems Projected Method of for Systems Projected Alternating Alternating Direction Direction Method of Multipliers Multipliers for Hybrid Hybrid Systems , Chengang Feng* **, Romain Bourdais*
Chengang Feng*,,, **, Romain Bourdais* Chengang Chengang Feng* Feng* **, **, Romain Romain Bourdais* Bourdais* * CentraleSupélec, 35510 Cesson-Sevigne, France (e-mail : [email protected], * CentraleSupélec, 35510 Cesson-Sevigne, France (e-mail :: [email protected], [email protected]). * 35510 Cesson-Sevigne, France (e-mail * CentraleSupé CentraleSupélec, lec,**Xi’an [email protected]). Cesson-Sevigne, France (e-mail : [email protected], [email protected], Jiaotong University, 710049, Xi’an, China [email protected]). [email protected]). **Xi’an Jiaotong University, 710049, Xi’an, China **Xi’an **Xi’an Jiaotong Jiaotong University, University, 710049, 710049, Xi’an, Xi’an, China China Abstract: In large-scale control optimization problems, a decentralized control structure can offer great Abstract: In control optimization problems, aa decentralized control structure can offer great scalability andlarge-scale rapidity advantages over a centralized implantation. Alternating Direction Method of Abstract: In large-scale control optimization problems, control structure can offer great Abstract: In large-scale control optimization problems, a decentralized decentralized control structure can Method offer great scalability and rapidity advantages over a centralized implantation. Alternating Direction of Multipliers (ADMM) is a decentralized optimization algorithm which has the important benefit of being scalability and rapidity advantages over aa centralized implantation. Alternating Direction Method of scalability and overoptimization implantation. Alternating Direction Method of Multipliers (ADMM) is aaadvantages decentralized algorithm which has the important benefit of being quite general in rapidity its scope and applicability incentralized continuous systems. In this paper, a new projected ADMM Multipliers (ADMM) is decentralized optimization algorithm which has the important benefit of being Multipliers (ADMM) is a decentralized optimization algorithm which has the important benefit of being quite general in its scope in continuous systems. In this projected ADMM algorithm is defined and itand canapplicability work in hybrid systems. The key point ispaper, to addaaanew convexification and a quite general in scope and applicability in continuous systems. In this paper, projected ADMM quite general in its its during scope and applicability inADMM continuous systems. Inhave thisisapplied paper, aaitnew new projected ADMM algorithm is defined and it can work in hybrid systems. The key point to add convexification and a projection process each iteration of algorithm. We to a charging control algorithm is defined and can work hybrid systems. The key point is to aa convexification and algorithm isprocess definedduring and it iteach can work in in results hybrid systems. The key point isapplied to add addcan convexification and aa projection iteration of ADMM algorithm. We have it to a charging control problem of electric vehicles. Simulation show that the proposed algorithm converge to a similar projection process process during during each each iteration iteration of of ADMM ADMM algorithm. algorithm. We We have have applied applied it it to to aa charging charging control control projection problem vehicles. Simulation show that the proposed can converge to aa similar result as of a electric centralized control within aresults limited iteration time. Due algorithm to its availability, simplicity and problem of electric vehicles. Simulation results show that the proposed algorithm can converge to similar problem of electric vehicles. Simulation results show that the proposed algorithm can converge to a similar result as a centralized control within a limited iteration time. Due to its availability, simplicity and scalability, the projected ADMM algorithm may be attractive in some practical engineering application. result as centralized control within aa limited iteration time. Due to simplicity and result as aa the centralized control within limited iteration time. Duepractical to its its availability, availability, simplicity and scalability, projected ADMM algorithm may be attractive in some engineering application. scalability, the projected ADMM algorithm may be attractive in some practical engineering application. © 2017, IFAC (International Federation of Automatic Control) Hosting Elsevierdesign Ltd. All reserved. scalability, the projected ADMM algorithm may be attractive in someby practical engineering application. Keywords: Projected Alternating Direction Method of Multipliers; Control forrights hybrid systems; Keywords: Projected Method of Control Decentralized control;Alternating Large scale Direction optimization problems; Electric vehicle Keywords: Projected Alternating Direction Method of Multipliers; Multipliers; Control design design for for hybrid hybrid systems; systems; Keywords: Projected Alternating Direction Method of Multipliers; Control design for hybrid systems; Decentralized control; Large scale optimization problems; Electric vehicle Decentralized control; Large scale optimization problems; Electric vehicle Decentralized control; Large scale optimization problems; Electric vehicle in a continuous system, no literature until now has integrated 1. INTRODUCTION theaa ADMM algorithm a hybriduntil system state is in continuous system, into no literature literature until nowwhose has integrated integrated continuous system, no now has 1. INTRODUCTION in a ADMM continuous system, no literature until nowvariables. has integrated A centralized optimal control can assure the satisfaction of in 1. defined by algorithm both continuous and discrete the algorithm into a hybrid hybrid system whose stateThe is 1. INTRODUCTION INTRODUCTION the ADMM into a system whose state is the ADMM algorithm into a hybrid system whose state is global constraints and has a real-time insight into operating A centralized optimal control can assure the satisfaction of objective of this paper is to propose a projected ADMM defined by both continuous and discrete variables. The A centralized optimal control can assure the of by both continuous and discrete variables. The A centralized optimal control can(Mets, assure the satisfaction satisfaction of defined defined by both continuous and discrete variables. The conditions at all points on network Verschueren, Haerick, global constraints and has a real-time insight into operating algorithm in which isaa suitable for aADMM hybrid objective of decentralized this paper paper is iscontrol to propose propose projected ADMM global constraints and has aa real-time real-time insight insight into into operating operating objective of this to projected global constraints and has objective of decentralized this paper iscontrol totopropose a suitable projected ADMM Develder, Turck, 2010). Nevertheless, it becomes conditions at&all allDe points on network (Mets, Verschueren, Haerick, system. The solution is easier implement in practice than a algorithm in which is for a hybrid conditions at points on network (Mets, Verschueren, Haerick, algorithm in decentralized control which is suitable for aa hybrid conditions at allDe points on network (Mets, Verschueren, Haerick, algorithm in decentralized control which is suitable for hybrid inefficient due to the huge dimensionality and thousands of Develder, & Turck, 2010). Nevertheless, it becomes purely continuous system. system. The solution is easier to implement in practice than aa Develder, & Turck, 2010). Nevertheless, Nevertheless, it it becomes becomes system. The solution is easier to implement in practice than Develder, & De De Turck, 2010). system. The solution is easier to implement in practice than a constraints when the number of nodes increases for large scaled inefficient due to the huge dimensionality and thousands of purely continuous system. inefficient due to the huge dimensionality and of continuous system. inefficient due tothe thenumber huge dimensionality andforthousands thousands of purely Increased use of electric vehicles has augmented power grid purely continuous system. problems. To address these issues, a decentralized control constraints when of nodes increases large scaled constraints when the of nodes increases for large scaled constraints when the number number nodes increases for many largecontrol scaled demand and many issueshas such as an incremental Increased use raised of electric electric vehicles has augmented power grid grid structure, where computation distributed across nodes Increased problems. To address theseofisissues, issues, a decentralized decentralized use of vehicles augmented power problems. To address these a control Increased use of electric vehicles has augmented power grid problems. To address these issues, a decentralized control investment for charging reinforcements, system overloads and demand and raised many issues such as an incremental and coordinated by a central instance, is often considered in a demand and raised many issues such as an incremental structure, where computation computation is distributed distributed across many nodes nodes structure, where is across many demand and raised many issues such as Fernandez anoverloads incremental structure, where computation is distributed across manyalready nodes reliability problems as shown in the works et al. investment for charging reinforcements, system large-scale system. Many decentralized algorithms and coordinated by a central instance, is often considered in a investment for for charging charging reinforcements, reinforcements, system system overloads overloads and and and coordinated by aa central instance, is often considered in investment and and coordinated by an central instance, is often considered in aa reliability (2011), Eising et alas (2014). For instance, with more et 20% problems shown in the works Fernandez al. exist to determine optimal control strategy, such as Nash large-scale system. Many decentralized algorithms already reliability problems as shown in the works Fernandez et al. large-scale system. Many decentralized algorithms algorithms already already reliability problems as shown in the works the Fernandez et20% al. large-scale system. Many decentralized penetration of EVs in the power system, peak power (2011), Eising et al (2014). For instance, with more certainty equivalence methodology (Ma, Callaway, & Hiskens, exist to determine an optimal control strategy, such as Nash (2011), Eising et al (2014). For instance, with more 20% exist to to determine determine an an optimal optimal control control strategy, strategy, such such as as Nash Nash (2011), et al in(2014). instance, withZhou, morepower 20% exist demand Eising willof be EVs increased by power upFor to 35.8% (Qian, Allan, penetration of EVs the power system, the peak 2010), distributed stochastic charging algorithm (Gan, Topcu, penetration certainty equivalence methodology (Ma, Callaway, & Hiskens, in the system, the peak power certainty equivalence methodology (Ma, Callaway, & Hiskens, penetration of EVs in the power system, the peak power certainty equivalence methodology (Ma, Callaway, &algorithm Hiskens, & Yuan,will 2011). Alternatively, a proper charging demand will be increased increased by up upwith to 35.8% 35.8% (Qian, Zhou,control, Allan, & Low, 2012), stochastic price-based two-layer control 2010), distributed charging algorithm (Gan, Topcu, be by to (Qian, Zhou, Allan, 2010), distributed stochastic charging algorithm (Gan, Topcu, demand demand will be of increased upwith to 35.8% (Qian, Zhou,control, Allan, 2010), distributed stochastic charging algorithm (Gan, Topcu, theYuan, penetration EVs canbynot only the power grid by & Yuan, 2011). Alternatively, abenefit proper charging (Gharesifard, Başar, & Domínguez-Garcí a, 2013) and & Low, 2012), price-based two-layer control algorithm & 2011). Alternatively, with a proper charging control, & Low, Low, 2012), 2012), price-based price-based two-layer two-layer control control algorithm algorithm & Yuan, 2011). Alternatively, with abenefit proper charging control, & flattening the overall load profile and but also optimize the the penetration of EVs can not only the power grid by integration of resource coordination and (Gharesifard, Başar, & a, Branch-and2013) and and the penetration of EVs can not only benefit the power grid by (Gharesifard, Başar, allocation & Domínguez-Garcí Domínguez-Garcí a, 2013) the penetration of EVs can not onlyand benefit the power grid the by (Gharesifard, Başar, & Domínguez-Garcí a, 2013) and charging price of individual users by acting as controllable flattening the overall load profile but also optimize Bound (Luo, Bourdais, van den Boom, & De Schutter, 2015). integration of of resource resource allocation allocation coordination coordination and the overall load profile and but also optimize the integration and Branch-andBranch-and- flattening flattening the overall load profile and but also optimize the integration of resource allocation coordination and Branch-andloads (Lopes, Soares, Almeida, & Da Silva, 2009). What’s charging price price of of individual individual users users by by acting acting as as controllable controllable However, these algorithms require convex2015). cost charging Bound (Luo, Bourdais, van den den Boom,a & &strictly De Schutter, Schutter, 2015). Bound (Luo, Bourdais, van Boom, De charging of users acting controllable Bound (Luo, Bourdais, van den Boom, &strictly De Schutter, 2015). more, (Lopes, if price EVs Soares, areindividual equipped with Vehicle Grid (V2G) loads (Lopes, Soares, Almeida, & by Da Silva,toas 2009). What’s function which is not available in all situations. Alternating However, these algorithms require a convex cost loads Almeida, & Da Silva, 2009). What’s However, these algorithms require aa strictly convex cost loads (Lopes, Soares, Almeida, & Da Silva, 2009). What’s However, these algorithms require strictly convex cost technology, which allows bidirectional energy exchange more, if EVs are equipped with Vehicle to Grid (V2G) Directionwhich Method algorithm can deal more, if EVs are equipped with Vehicle to Grid (V2G) function which is of notMultipliers available (ADMM) in all all situations. situations. Alternating function is not available in Alternating more, if EVs EVswhich aretheequipped withthey Vehicle to Grid (V2G) function which is of not available infunction all situations. Alternating between and power grid, could offer numerous technology, allows bidirectional energy exchange with a non-strictly convex cost and it maps onto Direction Method Multipliers (ADMM) algorithm can deal technology, which allows bidirectional energy exchange Direction Method of Multipliers (ADMM) algorithm can deal technology, which allows bidirectional energy exchange Direction Method of Multipliers (ADMM) algorithm can deal services to the power grid, such as active power support, between EVs EVs and and the the power power grid, grid, they they could could offer offer numerous numerous several distributed reasonably with non-strictly convex programming cost function function models and it it maps maps onto between with aaa standard non-strictly convex cost and onto between andpower the power they could offer numerous with non-strictly convex cost & function and it maps onto services reactive EVs compensation help power to solve the services topower the grid, grid, suchand as active active power support, well (Boyd, Parikh, Chu, Peleato, Eckstein, 2011). several standard distributed programming models reasonably to the power grid, such as support, several standard distributed programming models reasonably services to the power grid, such as active power support, several standard distributed programming models reasonably intermittency issue of the renewable energy resources (Tan, reactive power compensation and help to solve the well (Boyd, (Boyd, Parikh, Chu, Chu, Peleato, Peleato, & & Eckstein, Eckstein, 2011). 2011). power compensation and help to the well reactive power compensation and energy help to solve solve (Tan, the Therefore, number of Peleato, recent researches studied the reactive well (Boyd,aParikh, Parikh, Chu, & Eckstein,have 2011). Ramachandaramurthy, & Yong, 2016). We implemented the intermittency issue of the renewable resources intermittency issue of the renewable energy resources (Tan, intermittency issue of the renewable energy resources (Tan, ADMM algorithm in decentralized control. Tan, Yang, & Therefore, a number of recent researches have studied the projected ADMM algorithm to an EV charging control problem Ramachandaramurthy, & Yong, 2016). We implemented the Therefore, a number of recent researches have studied the Ramachandaramurthy, & Yong, 2016). We implemented the Therefore, a number ofdecentralized recent researches have studied the Ramachandaramurthy, & Yong, 2016). We implemented the Nehorai, (2014) applied ADMM in a distributed control ADMM algorithm in control. Tan, Yang, & with the objectives of price-based optimization. projected ADMM algorithm to an EV charging control problem ADMM in control. Tan, Yang, & ADMM algorithm to an EV charging control problem ADMM algorithm algorithm in decentralized decentralized control. Tan,reduce Yang, & projected projected ADMM algorithm to an EV charging control problem optimization to flatten the demand response and the Nehorai, (2014) applied ADMM in a distributed control with the the objectives objectives of of price-based price-based optimization. optimization. Nehorai, (2014) applied ADMM in a distributed control Nehorai, (2014) applied distributed control The of thisofpaper is organized as follows: the projected withreminder the objectives price-based optimization. electrical bill forflatten each individual userin in aa smart grid. Chen, & with optimization to flatten the ADMM demand response and reduce the optimization to the demand response and reduce the optimization to flatten the demand response and reduce the ADMM algorithm is defined in Section 2 and applied in the The reminder of this paper is organized as follows: follows: the projected projected Cheng, (2014) a distributed stochastic ADMM electrical bill for for proposed each individual individual user in in aa smart smart grid. Chen, Chen, & The reminder of this paper is organized as the electrical bill each user grid. & The reminder of this paper is organized as follows: the projected electrical bill for each individual user in a smart grid. Chen, & charging control of electric vehicles in Section 3. Section ADMM algorithm is defined in Section 2 and applied in the the4 algorithm to help frequency regulation in the power grid. The Cheng, (2014) (2014) proposed proposed aa distributed distributed stochastic stochastic ADMM ADMM ADMM algorithm is defined in Section 2 and applied in Cheng, ADMM algorithm iselectric defined in Section 2 andand applied in the4a Cheng, (2014) proposed aregulation distributed stochastic ADMM presents the numerical simulation study finally, charging control of vehicles in Section 3. Section paper (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, 2013) algorithm to help frequency in the power grid. The charging control of vehicles in Section 3. Section algorithm to frequency regulation in power grid. The charging ofinelectric electric vehicles instudy Section Section 44a algorithm toanhelp help frequency regulation in the the power grid.2013) The conclusioncontrol is given Section 5. presents the numerical simulation and3. finally, finally, introduced electric vehicle (EV) ADMM framework to presents paper (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, Jacobsen, the numerical simulation study and aa paper (Rivera, Wolfrum, Hirche, Goebel, & 2013) presents the numerical simulation study and finally, paper (Rivera, Wolfrum, Hirche, Goebel, & charging Jacobsen, 2013) conclusion is is given given in in Section Section 5. 5. achieve valley filling and minimal-cost goals. introduced an electric vehicle (EV) ADMM framework to conclusion introduced an electric vehicle (EV) framework to introduced electric vehicle (EV) ADMM ADMM framework to conclusion is given in Section 5. Although valley theanabove mentioned ADMM algorithm worksgoals. well achieve valley filling and minimal-cost minimal-cost charging goals. achieve filling and charging achieve valley filling and minimal-cost charging goals. Although the the above mentioned mentioned ADMM algorithm algorithm works well well Although Although the above above mentioned ADMM ADMM algorithm works works well Copyright © 2017 IFAC 8757 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2017 responsibility IFAC 8757Control. Peer review of International Federation of Automatic Copyright © © 2017 2017 IFAC IFAC 8757 Copyright 8757 10.1016/j.ifacol.2017.08.1571
Proceedings of the 20th IFAC World Congress 8424 Chengang Feng et al. / IFAC PapersOnLine 50-1 (2017) 8423–8428 Toulouse, France, July 9-14, 2017
In this paper, we consider a collection of 𝑁𝑁 independent agents in a hybrid system that share a common resource. Each of them has to minimize a local objective, while fulfilling various local constraints. Among these agents, the first 𝑀𝑀 of them are governed by discrete inputs and while the other (𝑁𝑁 − 𝑀𝑀) ones are governed by continuous inputs. The global optimization problem can be formulated as follows: 2. PROBLEM FORMULATION
min𝑥𝑥𝑖𝑖 ∑Ni=1 fi
subject to:
(xi )
(1)
∑Ni=1 Ai xi =0,
∀i=1…N, Ri ≤ Bi xi ≤ Ri ∀i=1…M, xi ∈Xdi
∀i=M+1…N, xi ∈Xc𝑖𝑖
In this optimization problem, the set Xid is a discrete set while Xic is a continuous convex set. The equality constraint ∑Ni=1 Ai xi =0 defines the resource to be shared and the inequations Ri ≤ Bi xi ≤ Ri represent the local constraints. To solve the problem, a well-known method, the so-called ADMM, can be a useful tool. However, its convergence cannot be guaranteed in the presence of discrete set. Consequently, the first step of our demarche is to relax these discrete constraints by convexification, which means transferring discrete sets into continuous sets. As an example, a discrete variable, xdi ∈{-1,0,1} can be transferred into xci ∈[-1,1] . With the translation of the discrete variables, the new optimization problem can be settled as follows: min𝑥𝑥𝑖𝑖 ∑Ni=1 fi (xi )
subject to:
(2)
∑Ni=1 Ai xi =0,
∀i=1…N, Ri ≤ Bi xi ≤ Ri ∀i=1…N,
xi ∈Xci
For the first 𝑀𝑀 optimization variables 𝑥𝑥𝑖𝑖 , the corresponding set
Xic is a convexification version of Xid . With this first manipulation, the ADMM method can be applied. It consists of an iterative mechanism (Boyd, Parikh, Chu, Peleato, & Eckstein, 2011), in which at each iteration, each agent has to solve its own optimization problem in parallel by: xi k+1 = arg min xi∈xc fi (xi )+ykT Ai xi + ‖ Ai xi -xi k +x̅k ‖2 (3) ρ
2
2
i
2
2
xi k+1 = arg min xi∈xc fi (xi )+ ‖Ai xi -xi k +x̅k +uk ‖2 2
ρ
2
i
uk+1 =uk +x̅k+1
(6)
In order to get a solution of the initial hybrid optimization problem, the third step of our demarche is to project the solution xi (∀i=1…M) found in (5) back into the discrete set Xdi before updating the average value 𝑥𝑥̅ 𝑘𝑘 and scaled dual variable u. This is what we call in this article the projected ADMM. The projection could be done based on either equal-probability distribution, normal distribution or even heuristic decision. There is no standard projection mechanism but should correspond to the characteristics of system constraints. In the example, 𝑥𝑥𝑖𝑖 lies in the interval [-1, 1] and can be projected based on equal-probability distribution mechanism:
xi =
1
-1,
when xi ϵ [-1, - )
0,
when xi ϵ [- , ]
1
3 1
3 1 3
(7)
1, when xi ϵ ( , 1] { 3 However, a simple projection can sometimes make a suboptimal solution infeasible (i.e. local constraints are not satisfied). Therefore, some conditions should be added in the projection to ensure the feasibility. The idea is to add some extra conditions in the projection rules corresponding to each constraint of system. For example, if Bi xi > Ri , then xi will be 1 projected to 0 even it lies in the interval ( , 1]. In this way, the 3 local constraints could be satisfied and the solution is feasible in a non-convex problem. As for the convergence criterion, Boyd, Parikh, Chu, Peleato, & Eckstein, (2011) suggest a reasonable stopping criterion, which is that the primal 𝑟𝑟 𝑘𝑘 and dual 𝑠𝑠 𝑘𝑘 residuals must be relatively small, i.e. Norm( rk ) ≤ εPri
(8)
Norm(sk ) ≤ εdual
(9)
where εPri > 0 and εdual > 0 are feasibility tolerances for the primal and dual feasibility conditions. The primal feasibility and dual feasibility can be written as: (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, 2013)
sk =[s1 k ,s2 k …,sN+1 k ]
(4)
In these two equations, 𝑘𝑘 is the current iteration number, ρ > 0 is the penalty parameter, 𝑦𝑦 is the dual variable and 1 x̅k = ∑Ni=1 x𝑘𝑘i is the average profile of all agents. If the N optimization problem (2) is strongly convex, this iterative mechanism will be guaranteed to converge to the optimal solution.
(5)
Also, the coordinator has to update the scaled dual variable as follows:
rk = x̅k
And a coordinator has to update a quantity, which we call the Lagrangian multiplier: yk+1 =yk +ρ x̅k+1
‖ Ai xi -xi k +x̅k ‖ and replacing the dual variable 𝑦𝑦 by the 2 scaled dual variable u = y/ρ. Then, the iteration (3) becomes:
ρ
(10) T
with si k =-ρN (xki -xk-1 ̅k-1 -x̅k ). i +x
(11)
In summary, the projected ADMM algorithm is presented as follows:
In the work of Boyd, Parikh, Chu, Peleato, & Eckstein, (2011), we can also find that (3) can be written in a slightly different form, which is often more convenient, by combining yk xi and 8758
Projected ADMM algorithm Variables: the total number of agents: N, the number of agents who are governed by discrete inputs: M, parameters of constraints: Ai , Bi , Ri , Ri , feasible set: Xdi , control horizon: 𝑇𝑇
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Optimization variables: xi Select a sufficiently large total iteration number K. Initialize xi 0 (t), u0i (t), t∈ T, i∈ N such that they satisfy all constrains. For 𝑘𝑘 = 0, 1 … 𝐾𝐾:
1) Convexification: for each 𝑖𝑖 ∈ (1…M), transfer the discrete feasible input set into a continuous set 2) Calculation: Each agent 𝑖𝑖 solves the problem in (5) to get xi k+1 (t) at each time slot t
3) Projection: project each continuous xi k+1 (𝑖𝑖 ∈ (1…M)) into discrete value with several extra conditions 4) Update: calculate uk+1 by (6) with the new x̅k+1
Figure 1. System architecture
5) Termination: Stop the iteration if the convergence criteria hold for all agents or if the maximum number of iterations K is reached; otherwise return to step 1 This algorithm offers a suitable mechanism to find a suboptimal solution of the original problem, as long as we can include a projection that ensures the feasibility of each local constraints. 3. APPLICATION: CHARGING CONTROL OF ELECTRIC VEHICLES As for controlling the charging of EV, it is much easier to control just the discrete charging decision of each EV other than the continuous charging power in reality. But the power supply from the grid to charge electric vehicle is a continuous variable. Thus, the charging control system can be seen as a hybrid system. By assuming the profile of the electricity price, the number of EVs needed to be charged and their arrival and departure times given, we need to decide an optimal charging control decision of a fleet of EVs at a charging station. The goal is to charge all the EVs up to the required levels within the determined time under the constrained grid condition while the total cost on charging the EVs is minimized.
Each EV is able to predict the amount of charging energy needed and is always in constant power charging option (Constant power – Constant Voltage charging option) (Marra, Yang, Træholt, Larsen, Rasmussen, & You, 2012). Thus, the EV 𝑖𝑖’s charging rate Mi is a constant. Upon departing, EV 𝑖𝑖 demands its battery level to be between [ Di , Di ]. Assume EV 𝑖𝑖’s initial battery level when it arrives is B0i and its battery capacity is B𝑐𝑐i , we can have: Di ≤ B0i + ∑Tt=1 Mi ∆t xi (t) /Bci ≤ Di
(12) can be reformulated as
Ri =Di Bci - B0i Bci ≤ ∑Tt=1 Mi ∆t xi (t) ≤ Ri =Di Bci - B0i Bci (13)
𝑅𝑅𝑖𝑖 , Ri are minimal and maximum charging power required for EV i, respectively.
In addition, the battery level in an EV should be limited in a reasonable interval (Marra, Yang, Træholt, Larsen, Rasmussen, & You, 2012). The choice for such an interval relates mainly to battery lifetime aspects: charging or discharging EV beyond the interval leads to a quicker battery degradation. Let [ Si , Si ] denote the interval of EV 𝑖𝑖, then we have: c Si ≤ B0i + ∑T' t=1 Mi ∆t xi (t) /Bi ≤ Si
3.1 System model As shown in Fig.1, we consider a system consisting of an aggregator, EVs, no-EV load (residential habitants for instance), renewable energy source and a power grid. The aggregator is an energy administrator which obtains energy form the power grid and renewable energy source and supply it to energy consumers. The aggregator negotiates with 𝑁𝑁 EVs (i∈(1,…,N)) at a charging station for a daily charging schedule over 𝑇𝑇 time slots (t∈( 1,…,T)). V2G service is fully employed whenever necessary and each EV obeys the guide of aggregator.
(12)
T' ∈ T
(14)
3.2 Projected ADMM algorithm for charging control of EVs As defined in the work Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, (2013), the EV charging control problem can be formulated as a standard exchange optimization problem using ADMM algorithm. In the case of price-based optimization which aims to minimize the total charging cost, using the variables definition displayed in Table 1, we can define the cost function of the aggregator: min
f(xa )=PT ∙xa = ∑Tt=1 P(t)∙xa (t)
subject to
∑Ni=1 xi ∙Mi ∙∆t - xa +Pload - Pnew =0
0 ≤ xa ≤ xa , xi (t)∈{-1,0,1} 8759
(15)
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For each EV 𝑖𝑖 , xi (t)=1 means the EV i is charging, consuming energy provided by the aggregator at time slot t. xi (t)=-1 means that the EV i is discharging, using V2G service to feed energy back to the system at time slot t and when xi (t)=0, neither charging nor discharging will happen. In the contrast, the aggregator cannot give energy to the outside power grid and it also has a limitation of power bought from the outside power grid 𝑥𝑥𝑎𝑎 due to physical power transfer line limitation. The energy balance equation means that the demand and supply of energy in a system should be equal. Table 1. LIST OF SYMBOLS USED IN THIS PAPER Description
Type
Variable 𝑃𝑃
Profile of the electricity price
𝑥𝑥𝑎𝑎
Power bought from the power grid in the aggregator
𝑥𝑥i
Discrete charging sequence of EV i
𝑀𝑀𝑖𝑖 ∆𝑡𝑡
𝑃𝑃𝑛𝑛𝑛𝑛𝑛𝑛
𝑁𝑁
𝑥𝑥𝑎𝑎 𝑅𝑅𝑖𝑖 𝑅𝑅𝑖𝑖 𝑆𝑆𝑖𝑖 𝑆𝑆𝑖𝑖 𝐵𝐵𝑖𝑖
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
Constant
Time step duration
Scalar
Power consumed apart from EV in the system Number of EVs
Table 2. PROJECTION RULES FOR EV CHARGING ADMM ALGORITHM 𝒙𝒙𝒄𝒄𝒊𝒊 (𝐭𝐭)
𝒙𝒙𝒅𝒅𝒊𝒊 (𝐭𝐭)
Conditions
1
[-Mi ∆t, - Mi ∆t] 3
c
∑Tt=1 Mi ∆t xi (t) ≥ Ri (initially projected to 0)
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
(
1 1 - Mi ∆t, Mi ∆t) 3 3
∑Tt=1 Mi ∆t xi (t) ≤ Ri (initially projected to-1)
∑Tt=1 Mi ∆t xi (t) ≥ Ri (initially projected to 1)
0
Constant Vector ∈ 𝑅𝑅 𝑇𝑇 Scalar
Maximum charging power required for EV i
Scalar
Minimal battery level required in EV i Minimal battery level required in EV i Matrix to calculate the stored energy in EV i
Bi (T' )xi (T' )(t)=Si (t), T' =1,..,t ∈T
-1
Minimal charging power required for EV i
[
2
(16)
As each EV fully does Vehicle to Grid service, in order to achieve significant reduction in the charging cost, the cost function for each EV is set to be 0 (Rivera, Wolfrum, Hirche, Goebel, & Jacobsen, 2013). Then, for each EV 𝑖𝑖, the EV charging problem in ADMM form is: 2 k ̅k +uk ‖2 x𝑖𝑖 2 ‖xi -xi +x
subject to Ri ≤ ∑𝑇𝑇𝑡𝑡=1 Mi ∆t xi (𝑡𝑡) ≤ Ri xi ∈ Xic : continuous set
1
( ∑Ni=1 xk+1 +xk+1 a ) i
uk+1 =uk +x̅k+1
subject to -xa +Pload - Pnew ≤ xa ≤ Pload - Pnew
ρ
∑Tt=1 Mi ∆t xi (t) ≤ Ri (initially projected to 0)
̅xk+1 =
Lower triangular matrix ∈ 𝑅𝑅 𝑇𝑇∗𝑇𝑇
2
Bi (T' )xi (T' )(t)=Si (t), T' =1,..,t ∈T
For the average profile of all agents 𝑥𝑥̅ and scaled dual variable u update:
Constant Vector ∈ 𝑅𝑅 𝑇𝑇
ρ
∆t, Mi ∆t] c
Constant Vector ∈ 𝑅𝑅 𝑇𝑇
xa k+1 = arg min xa -PT xa + ‖xa -xa k +x̅k +uk ‖2
1 M 3 i
1
(15) can be reformulated in the ADMM form as:
S i ≤ B i xi ≤ S i ,
Before 𝑥𝑥̅ update and u update, the continuous set 𝑋𝑋𝑖𝑖 above needs to be projected into a discrete set 𝑋𝑋𝑖𝑖𝑑𝑑 . The three discrete value representing the charging decisions have almost the same chance to appear, so the equal-probability distribution methodology can be applied in the projection. Some conditions need to be added based on the existing local constraints. The projection rules are summarized in Table 2.
Scalar
Maximum power can be bought from the power grid
xi k+1 = arg min
Ri defines a bound on the power needed to charge EV 𝑖𝑖 up to the required level before departing from the charging station. The other set of inequality constraints Si ≤ Bi xi ≤ Si make sure that the (dis)charging power does not violate the maximal and minimal energy that the battery can support when EV 𝑖𝑖 is connected. 𝑋𝑋𝑖𝑖𝑐𝑐 is a continuous set transferred by a discrete set.
Discrete Vector ∈ 𝑅𝑅 𝑇𝑇
Charging rate of EV i in Constant power charging option
Available renewable energy source
𝑃𝑃𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
Continuous Vector ∈ 𝑅𝑅 𝑇𝑇
The first set of inequality constraints Ri ≤ ∑𝑇𝑇𝑡𝑡=1 Mi ∆t xi (𝑡𝑡) ≤
(17)
N+1
(18) (19)
The processes of projected ADMM in charging control of EVs is as follows: the aggregator first sends the average profile of all agents 𝑥𝑥̅ and scaled price 𝑢𝑢 to all EVs. Based on this information, the aggregator and the EVs solve their individual optimization problem defined in (16) and (17). Each EV i needs to project the obtained value 𝑥𝑥𝑖𝑖 into a discrete set, do some recalculations and then send their solutions to the aggregator in order to obtain the new 𝑥𝑥̅ . With this new value, the aggregator then updates the scaled price u. This sequence is repeated until the convergence criterion is met. 4 NUMERICAL SIMULATION STUDY In this section, simulations are conducted to verify the performance of the proposed algorithm. We considered a case where 30 EVs need to be charged in the daytime assuming these 30 vehicles arrive in the charging station and depart at the same time. The control horizon T was assumed to be 12 hours and the time slot was set to be 30 minutes, thus T =24. The battery capacity of EV was supposed to be identical, namely C. The initial battery level of EV i was
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generated randomly form [0.1C, 0.7C]. The required charging level at departing time and charging rate of all vehicles were assumed to be the same. Furthermore, we assumed that charging was not interrupted before departure of each EV and the renewable energy generated was always below to the demand in the system. The energy price profile P was obtained from the France RTE website shown in Fig. 2. For comparison, the uncontrolled charging scenario is also tested, that is, beginning to charge as soon as the EV arrives at the charging station and stopping charging as soon as the demand is satisfied. The simulation environment was MATLAB and Quadratic Programming (quadprog) is used for the optimizations.
Figure 3. Cost evolution with iteration time of ADMM algorithm, projected ADMM algorithm, centralized control algorithm (Branch-and-Bound) in a hybrid system, centralized control algorithm in a continuous system
Figure 2. Electricity price profile for simulation
Two kinds of charging control systems were simulated. In the first case, the charging power of each EV is controlled. Thus, the system is a continuous system. In the second case, the control variable is the charging decision of each EV, it is therefore a hybrid system. The total cost evolutions with iteration of different control strategies are illustrated in Fig. 3. Both ADMM algorithm and projected ADMM algorithm are finally converged to some values around the results of the centralized control strategies within an iteration time smaller than 2000. This demonstrates that the proposed ADMM algorithm can converge to a similar result as a centralized control algorithm.
Table 3 shows the relative cost of the controlled cases and no control case where we define the cost in no control system is a unite cost. It may be seen that that the total cost of the aggregator can be reduced by about 57% with a proper control. Additionally, the centralized control always outperforms the decentralized control. It can also be noticed that the control in a continuous system leads to a smaller cost than that in a hybrid system. However, the maximum difference between diverse control strategies is around 0.1% of the total cost, which is much smaller than the aforementioned reduction. In this way, the result obtained by the proposed algorithm is acceptable. Table 3. COMPARISON BETWEEN THE CONTROLLED CASES System Type Control strategy Normalized cost
Continuous system
Hybrid system
No control
100.00
Centralized control
42.79
ADMM control
42.82
Centralized control
42.85
Projected ADMM control
42.90
Fig. 4 illustrates the battery level evolution of an electric vehicle during charging. It can be seen that the vehicle charges when the electricity price is low and discharges when the price is high. The battery level at departing time falls into the required interval and it does not violate the maximal and minimal energy that the battery can support. This shows that the proposed algorithm can satisfy the constraints. In Fig. 5, we show the impact of the proposed ADMM algorithm on the power consumed for charging electric vehicles. It can be noticed that with the proposed ADMM algorithm, the vehicles consume much more energy during low electricity price period and consume less, even negative energy during high electricity price period, which means they send their stored energy back to the grid. It shows the idea of minimizing the total cost on charging the EVs: the vehicles are controlled to charge when the price is low and to discharge when the price is high. The simulation result validates the effectiveness of the proposed ADMM algorithm.
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conditions in the projection can guarantee the feasibility of the optimal solutions. This algorithm is effective and can lead to a similar result as a centralized control which makes it attractive. The features of the proposed algorithm are validated in a case study of EV charging control with the intention of minimizing the total cost for charging the EVs. REFERENCES
Figure 4. Example of the battery level evolution of an EV during charging process.
Figure 5. Power consumed evolution for charging electric vehicles with proposed ADMM control strategy
In reality, a hybrid system is easier to implement than a continuous system. For example, in an EV charging system, it is easier to control just the charging sequence composed of -1, 0, and 1 instead of the charging power. It is also beneficial to the cost reduction of control infrastructures. Moreover, the decentralized control structure can offer great scalability advantages over a centralized implantation. Because of all these aspects, the projected ADMM algorithm seems more practical in real environment. 5 CONCLUSIONS In this paper, a projected ADMM algorithm suitable for hybrid systems was introduced. The idea of the proposed algorithm is to add a convexification and a projection at the beginning and the end of each iteration of ADMM algorithm. The extra
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