Distributed Real-Time Pricing Control for Large-Scale Unidirectional

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TII.2016.2569584, IEEE Transactions on Industrial Informatics

1

Distributed Real-Time Pricing Control for Large Scale Unidirectional V2G with Multiple Energy Suppliers Navid Rahbari-Asr, Mo-Yuen Chow, Jiming Chen, and Ruilong Deng  Abstract— With the increasing trend in adoption of Plug-in Hybrid and Plug-in Electric Vehicles (PHEVs/PEVs), they will play a prominent role in the future electric energy market by acting as responsive loads to increase the grid stability and facilitate the integration of renewables. However, due to the large number of controllable devices in the future grid, central Vehicle to Grid (V2G) management would be challenging and vulnerable to single points of failure. This paper introduces a novel distributed approach for optimal management of unidirectional V2G considering multiple energy suppliers. Each charging station as well as each energy supplier is equipped with a local price regulator to control the price paid to the energy suppliers and the price paid by the vehicles through coordination with their neighbors. In response to the updated prices, the vehicles adjust their charging rates and energy suppliers adjust their production to maximize their benefit. The main advantages of the proposed approach are: 1) manages unidirectional V2G in a fully distributed way considering multiple energy suppliers and vehicles, 2) converges to the global optimum despite the greedy behavior of the individuals. Index Terms— Unidirectional V2G; Demand Response; Distributed Control; Distributed Optimization; Consensus Networks; KKT Conditions;

I.

INTRODUCTION

The predictions indicate that by 2023, there would be 1.8 -7 million Plug-in Hybrid/Plug-in Electric Vehicles (PHEVs/PEVs) on the road in United States [1]. The collective charging power of these vehicles can be a threat as well as an opportunity for the grid. On one hand, if the charging process is not controlled, the peak electricity demand would be increased which can overload the local infrastructures in certain areas. On the other hand due to the flexible nature of the charging load of the electric vehicles, they can be used for ancillary services (such as frequency regulation, smoothing the load curve, etc.) to the benefit of the grid [2]. This is mainly because the charging process has a high power rating as well as large ramp up/ramp down Manuscript received October 30, 2014; Revised on June 11, 2015; October 14, 2015; January 19, 2016; April 22, 2016; Accepted for publication on April 29, 2014. Navid Rahbari-Asr and Mo-Yuen Chow are with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27606, USA (e-mail: [email protected] and [email protected]). Ruilong Deng is with the Department of Electrical and Computer Engineering, University of Alberta, Canada (email: [email protected]).

rates. The role of PHEVs/PEVs as ancillary service providers becomes especially important as larger number of renewables are integrated to the grid increasing the intermittency and fluctuations in the power profile [3]. The enabling technology to mitigate the possible threats of PHEVs/PEVs to the grid, and to use them as potential service providers, is called the Vehicle to Grid (V2G) technology which allows interaction between the vehicle and the grid. V2G can be unidirectional, where the power flow is only from the grid to the PHEVs/PEVs and vehicles can regulate their charging power, or bidirectional where the connected PHEVs/PEVs can also discharge to the grid [4]. The focus of this paper is on optimal dispatch of PHEVs/PEVs using unidirectional V2G technology. Most of the existing scheduling algorithms for charging of PHEVs/PEVs in the literature are centralized. The charging stations/vehicles are required to transmit data to a control center called aggregator. The control center performs the necessary computations and determines the optimal power allocation for each unit and transmits it to the charging stations/vehicles [5], [6]. The centralized approach works well for small-scale problems. However, as the number of PHEVs/PEVs increases and they spread over a wide geographical area, the centralized approach would not scale well and becomes fragile to single points of failure. The situation is aggravated when multiple distributed energy suppliers are also added to the grid, and the control center has to get data and coordinate with them as well. For this reason, many of researchers have proposed decentralized/distributed approaches to do the charge optimization of PHEVs/PEVs to relieve the computational burden from the control center [7]–[13]. In [7]–[9] the energy provider sends a central price/event signal to all the charging stations to manage the aggregate charging load; In [10], [12] decentralized methods are introduced based on a multi-agent framework in which central aggregator broadcasts signals to lower level aggregators, which send pricing signals to individual vehicles. Although the aforementioned works in [10], [12] are decentralized, they are still vulnerable to single points of

Jiming Chen is with the Department of control science and engineering, of Industrial Control Technology and Institute of Industrial Process Control at Zhejiang University (email: [email protected]). Copyright © 2009 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]

1551-3203 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


2

Fig. 1. General Concept for Distributed Real-Time Pricing Control (DRPC)

failures, because higher level agents broadcast real-time pricing signals to lower level nodes. In [13], a fully distributed algorithm is introduced which does not require any control center, coordinator or aggregator, and therefore it is not only scalable but also robust against single point of failures. However, this work considers a single energy provider. Some approaches based on non-cooperative game theory which require minimum or no communication are also introduced in some papers such as [14]. Due to the noncooperative nature of these approaches, they converge to suboptimal (nearly optimal) solutions. Another game theory based approach for price based demand response is introduced in [15] in which the algorithm is proven to converge to a Nash equilibrium. However, Nash equilibriums are not necessarily the global optimum. In [16], a cooperative game theory approach is introduced for scheduling of residential loads which converges to the global optimum, but it only considers a single energy provider. In this paper, we propose a Distributed Real-time Pricing Control (DRPC) technology for energy management of large-scale electrical vehicles considering multiple energy providers and renewables. Fig. 1 shows the basic concept of the proposed approach: Each charging station, each individual energy supplier, and each demand is connected to a local price regulator with the communication capability. The price regulators control the price that each individual device is charged/paid for energy usage/production. There is one price regulator for each device. Price regulators coordinate with their neighbors and update the local energy price. In response to the updated prices, the vehicles adjust their charging rates to maximize their utility and energy suppliers adjust their production rate to maximize their own benefit. It is shown that by iteratively repeating this process, the entire system converges to the global optimum. The main advantages of this algorithm are: 1. While most of the existing distributed charge scheduling algorithms assume only a single energy provider, this technology considers multiple heterogeneous energy providers. This assumption is more suited to the future of the smart-grid. 2. This technology is fully distributed, meaning that it requires no central coordinator/leader/aggregator. 3. It respects the privacy of individual units. No device has to share its preferences, energy generation/consumption, etc. with other units.

4.

Although each unit selfishly tries to choose the optimal local power consumption/generation to maximize its own benefit, the local price regulators adjust the prices such that the behavior of the overall system moves toward the global optimum. The paper is organized as follows. Section II formulates the unidirectional V2G management with multiple energy providers as a Social Welfare Maximization problem [17]. Section III introduces the DRPC algorithm as a cooperative distributed solution to solve the formulated problem. Section IV brings numeric simulations to validate the proposed approach and the concluding remarks come in Section V. II. PROBLEM FORMULATION Unidirectional V2G energy management with multiple energy suppliers can be formulated as a social welfare maximization problem, which is an optimization problem. It consists of an objective function, and a series of constraints. A. Objective Function Social welfare maximization at time t is defined as maximizing the summation of welfares of all the generation and consumer units [18]:   Wi , EV ( Pi , EV (t ), p (t ))   Wi , D ( Pi , D (t ), p (t ))     iEV i D Max  , P ( t ), p ( t ) ( ), ( )) ( ( ), ( )) W ( P t p t  W P t p t  i,R i, R    i ,G i ,G iR  iG 

(1)

where EV is the set of indices of electric vehicles, D is the set of indices of constant loads, G is the set of indices of dispatchable generation units, R is the set of indices of renewable generation units, Wi,.j is the welfare function of the device with index i of type j, Pi,j (t) is the power associated with the device with index i of type j, ( ) is the vector of powers of all generation and demand units, and p(t) is the price of energy at time t. 1) Welfare for Electric Vehicles The welfare for electric vehicles is the utility they get for being charged minus the cost they pay for charging: Wi, EV ( Pi , EV (t ), p(t ))  Ui , EV ( Pi, EV (t ))  p(t ) Pi , EV (t ) , (2) where for all i  EV , U i , EV ( Pi , EV (t )) is the utility of the vehicle with index i for getting a charging power of Pi , EV (t ) . We consider the utility function to be a differentiable concave function.



3 2) Welfare for Constant Loads Similar to electric vehicles, the utility for constant demands is utility minus cost. However, from the optimization point of view, at time t this utility is constant because the respective power cannot be changed by the optimization process: Wi , D ( Pi , D (t ), p (t ))  U i , D  Pi , D (t )   p (t ) Pi , D (t ), (3) where for all ∈ , , ( ) = , ( ) is the constant power associated with the demand with index i at time t, and , ( , ( )) is its respective utility. 3) Welfare for Dispatchable Generation Units The welfare for the dispatchable generation unit with index i can be defined as the net profit: Wi ,G ( Pi ,G (t ), p(t ))  p(t ) Pi ,G (t )  Ci ,G ( Pi ,G (t )), (4) where for all ∈ , , ( , ( )) is the cost of the dispatchable generation unit with index i for providing Pi,G (t). We consider the cost function to be a differentiable convex function. 4) Welfare for Renewable Generation Units The welfare for the renewable generation unit with index i is also the net profit, i.e. revenue minus cost. However, from the optimization point of view, at time t this cost is constant because the respective power cannot be changed by the optimization process: Wi , R ( Pi , R (t ), p (t ))  p (t ) Pi , R (t )  Ci , R  Pi , R (t )  (5) where for all ∈ , , ( ) = , ( ) is the constant power associated with the non-dispatchable generation unit with index i at time t, and , ( , ( )) is its respective cost. B. Constraints 1) Power Balance Constraint To ensure the stability of the power system, the summation of all the generated powers should equal the summation of all the consumed powers:  Pi,G (t )   Pi,R (t )   Pi, EV (t )   Pi,D (t ) . (6) iG

iR

iEV

iD

Note that in this paper, we do not consider the transmission losses. 2) Local Device Constraints Generators and vehicles have power rating and cannot produce or consume more power than a certain limit: i  EV  D  G  R, j   EV , D, G , R : , (7) 0  Pi , j (t )  Pi , j max (t ) where , ( ) is the maximum power that can be generated/consumed by the device with index i of type j at time t. Also, as we are considering unidirectional V2G the lower limit for charging power of electric vehicles is zero. III. DISTRIBUTED REAL TIME PRICING CONTROL In this section, the steps to derive the DRPC algorithm based on KKT multipliers and consensus networks are explained. A. Simplifying the Objective Function By replacing for welfare functions from (2)-(5) in (1), applying the power balance constraint, dropping the constant

terms (as they have no effect on the optimization), and representing the problem as a minimization, the optimization problem would be: 

Min

Pi , j ( t ):iEV G , j{ EV ,G} 

s.t.

P

i ,G

iG

   U i , EV ( Pi , EV (t ))   Ci ,G ( Pi ,G (t ))  ,  iEV iG

(t )   Pi , R (t ) 

P

i , EV

iR

iEV

(t )   Pi , D (t ) , iD

i  EV  D  G  R, j   EV , D, G , R : 0  Pi , j (t )  Pi , j max

(8)

.

B. Augment the Objective Function with KKT Multipliers To solve (8), first, the objective function is augmented by the constraints and KKT multipliers. The Lagrangian would be: L    U i , EV ( Pi , EV (t ))   Ci ,G ( Pi ,G (t )) iEV

iG

,  (9)     Pi , EV (t )   Pi , D (t )   Pi ,G (t )   Pi , R (t )  iD iG iR   iEV is the KKT multiplier for the power balance where constraint. Note that we have not considered KKT multipliers for local device constraints, because those constraints can be handled by limiting the domain of each decision variable between its minimum and maximum. C. Dual Decomposition Rephrasing (9): L    U i , EV ( Pi , EV (t ))   Pi , EV (t )  iEV

    Pi ,G (t )  Ci ,G ( Pi ,G (t )) 

(10)

iG

      Pi , D (t )   Pi , R (t )  . iR   iD Using dual decomposition [19], an iterative approach that moves in the direction of minimizing L in terms of the primal vector P and maximizing L in terms of the dual variable λ can converge to the optimal point: i  EV : Pi ,kEV (t )  arg

min

 U

min

 

0  Pi , EV ( t )  Pi , EV max

i , EV

( Pi , EV (t ))   k Pi , EV (t ) 

, (11)

i  G : Pi ,kG (t )  arg

0  Pi ,G ( t )  Pi ,G max

k

Pi ,G (t )  Ci ,G ( Pi ,G (t )) 

 k 1   k  P k , where, P k 

P

i , EV

iEV

, (12) (13)

(t )   Pi , D (t )   Pi ,G (t )   Pi , R (t ) , (14) iD

iG

iR

and > 0 is the updating step size for dual variable . The convergence/stability of this iterative approach depends on the value of the updating step size. A. Distributed Observers to Estimate Global Information The iterative procedure represented by (11)-(13), consists of local optimizations that can be done at each device (eq.(11), and eq. (12) ). However, these local optimizations require knowledge of the global variable . Also, updating this global variable (eq. (13)) requires knowledge of generation and demand powers of all the devices at the k-th



4

Fig. 2. General Structure for Distributed Real-Time Pricing Control (DRPC)

iteration. Therefore, in this section with the help of consensus networks [20], a procedure is designed so that each node can estimate the value of the required global variables by collaboration with its neighbors. We introduce the following variables to serve as estimations of global variables: : Estimation of device with index of , Δ : Estimation of device with index i of average of Δ over all devices. The update rules for estimated values are as follows: i  G  R, j  {G, R}: w Pˆ k  Pˆ k  P k  P k 1 , (15) Pˆ k 1  Pˆ k  i

i

  ij

j

i

j N i



i, j

i, j

i  D  EV , j  {D, EV } : Pî k 1  Pî k 

 w  Pˆ  Pˆ   P ij

j

i

k 1 i, j

 Pi ,kj ,

(16)

j Ni

i  G  R  EV  D :

îk 1  îk   wij ˆ jk  îk  Pî k 1 , jN i





(17)

where Ni is the set of indices of neighbors of node i, > 0 is the updating step size for estimation of dual variable by node i, and wij=wji is the connectivity strength between node i and node j and is chosen such that 0 ≤ < max | | ,..,

to ensure the convergence [20].

Remark 1: The assumptions for the communications graph of the consensus network used for the distributed observers in (15)-(17) are as follows: 1. The graph is connected, 2. The graph is undirected i.e. any communication between any two nodes is bidirectional, 3. The graph is balanced i.e. ∀ , : = . Remark 2: In the scope of this paper, the communications among nodes is assumed to be perfect. However, in real world implementation the imperfections in communications such as delay and packet loss need to be considered. One approach to overcome the communications imperfections is to use gossip networks [21] instead of consensus networks as used in [22]. Developing a version of the algorithm that is more robust against communications imperfections would be considered in future publications. In (15) and (16), each unit updates its estimation of the average power imbalance based on its coordination with its neighbors and its local information. In (17), each unit updates its estimation of the dual variable through

coordination with neighbors and its estimate of the average power imbalance. Using the estimated values, each EV and each dispatchable generation unit will update their power consumption and generation as follows: min U i , EV ( Pi , EV )  îk Pi , EV , (18) i  EV : Pi ,kEV  arg



0  Pi , EV  Pi , EV max

i  G : Pi ,kG  arg

min

0 Pi ,G  Pi ,G max



 ˆ P k i

i ,G



 Ci ,G ( Pi ,G ) . (19)

If we compare (18), (19) with welfare definitions in section II-A, we find that can be interpreted as the price of energy offered to device i, at the k-th iteration. Remark 3: As can be interpreted as the price of energy offered to device i at the k-th iteration, we can see that the update equations (18), (19) are in accordance with the selfishness of electric vehicles and generators. Because update equation (18) means that, for the offered price the electric vehicle would decide for a consumption that maximizes its individual welfare i.e. , ( , ) , , and the generator would decide for a power that maximizes its individual net benefit, i.e. , , ( , ). B. Smart Transformers as Communication Nodes between Zones Equations (15)-(19), assume that the entire communication network among all generation and load devices forms a connected network. This can be realized among the devices inside each zone. However, it is not likely that devices located in different zones of the power network have the ability to exchange information with each other. One possible solution is to use smart transformers that beside their main functionality (stepping up/down the voltage), have the ability to exchange information among each other as well as with the devices inside their zone. It means in the communication network, the transformer nodes should have neighbors from their own zone, as well as neighbors from other transformers in the network such that the entire communication network forms a connected graph. Therefore, we add two more equations for transformers: i  T : Pî k 1  Pî k   wij Pˆjk  Pî k , (20)



jN i

 i  T : îk 1  îk 

 w  ˆ ij

j N i

k j





 îk   i  Pî k 1 ,

(21)

where T is the set of indices for transformers. Equations (15)(21) form the DRPC algorithm. The building blocks of this algorithm are shown in Fig. 2. Each energy device is



5 connected to a local price regulator. Local price regulators use distributed observer equations (15), (16), and (20) to estimate the average imbalance of power. Based on the output of distributed observers and coordination with neighbors, the local price of energy offered to each device is updated. If the device is dispatchable resource or electric vehicle, it will respond to changes in the price by regulating its generation/consumption power in a greedy way (eq.(18), (19)). If the device is non-dispatchable resource or a constant load, it will not respond to the changes in price, but its generation/consumption information would be used as an input in the system. If the device is a transformer, its main role would be to propagate the pricing information to other zones (eq. (20), (21)). Remark 4: As shown in Fig. 2, the entities that exchange information among themselves are not customers/suppliers, but the local price regulators shown by red dotted lines. In the real application, they can be designed to be independently operated so that they have no incentive to report false data. However, it is possible that these independent operators are attacked to report false data in favor of certain individuals. The issue of attack in consensus based algorithms and how to mitigate its effects, has been extremely analyzed in multiple papers [23] [24]. The techniques introduced in these papers can be added to the proposed technology to make it secure against such attacks, and will be considered in the future work of the authors. Next, we provide a theorem that shows the fixed point of the DRPC algorithm satisfies KKT conditions of optimality. As the optimization problem is convex and all the constraints are affine, satisfaction of KKT conditions translates into global optimality. Theorem 1: If the optimization problem (8) is feasible and the cost/utility functions are continuously differentiable convex/concave functions, then the fixed points of the iterative algorithm presented by (15)-(21) starting from initially balanced states (i.e. for all i, Δ = Δ = 0) satisfy the KKT conditions of optimality for the optimization problem.■ Proof: The proof is provided in the supplementary material uploaded with the paper.■ Remark 5: Theorem 1 requires the algorithm to be triggered from an initially balanced state. In the power system, there are lower level controls (such as droop control [25] ), that try to balance the power generation and demand autonomously. Without intervention of any supervisory control, the system will be in a balanced state, but this state would not necessary be optimal. The technology proposed in this paper does not replace the local low level controllers that are already designed to maintain the balance of the power system. Rather, it provides a distributed way to change the set point of the low level controllers so that the system can operate optimally. Therefore, it is a reasonable assumption that the algorithm can be initiated from an initially balanced state.

The following theorem states that the convergence of the DRPC algorithm depends on the parameter η of the algorithm. Theorem 2: If the optimization problem (8) is feasible and the cost/utility functions for dispatchable generation units/EVs are continuously differentiable strictly convex/concave functions, there exist a positive value ε such that for all values of 0 < η < ε, the fixed points of the iterative algorithm presented by (15)-(21) is attractive. ■ Proof: The proof is provided in the supplementary material uploaded with the paper.■ Theorem 3: The electricity price at the convergence point of the iterative algorithm presented by (15)-(21) is ensured to be greater than or equal to zero if the following conditions hold: 1.

  0 : arg

min

C

 P

 U i , EV ( Pi , EV )   Pi , EV max

0 Pi ,G  Pi ,G max

i ,G

( Pi ,G )   Pi ,G   0 ,

   

2. 3.

arg

min

0  Pi , EV  Pi , EV max

i , EV

,

t :  Pi , EV max   Pi , D (t )   Pi , R (t ) . iEV

iD

iR

Proof: The proof is provided in the supplementary material uploaded with the paper. ■ Remark 6: The conditions in Theorem 3 have certain physical meanings. The first assumption asserts that if the generator is charged for producing power, then it should choose to not produce any amount of power. The second assumption asserts that if EVs are paid for using power, then they should choose to use maximum charging power. The third assumption asserts that the aggregate load of constant demands and electric vehicles with maximum charging rate should exceed the power produced by the renewables. Remark 7: The DRPC algorithm introduced in this paper has two basic pillars: Dual decomposition and consensus networks. Dual decomposition is used to distribute the computations (and thus reduce the computational complexity) and consensus algorithms are used to distribute the communications (and thus remove the need for global communications). Without the consensus algorithms, dual decomposition would need a central coordinator to gather information from all the units and broadcast the updated Lagrange multipliers. IV. NUMERICAL SIMULATIONS In this section, we validate the performance of the DRPC algorithm through numerical simulations and benchmarking it against centralized optimization. We test the algorithm on a 5 bus system (Garver Network). Fig. 3 shows the setup of the case study. It is a 5-bus system with multiple energy devices such as EVs, thermal generation units, renewables and constant loads. The energy devices are scattered in 5 different zones. The devices inside each zone have the ability to exchange information with each other as well as with the transformer that isolates the zone from the rest of the network. On a higher level, the



6

TABLE I. DISPATCHABLE UNIT PARAMETERS CHOSEN FOR THE CASE STUDY

Node 3 4 9 10 14 16 17

7 12

PHEV/PEV Parameters Cap pmax Pmax SoC (kW) (kWh) (cents/kWh) 20 3.33 0.9 19.2 kW 0.3 19.2 kW 15 3.33 0.2 19.2 kW 15 2.00 0.7 19.2 kW 20 5.00 0.5 19.2 kW 30 3.33 0.5 19.2 kW 15 3.67 0.4 19.2 kW 25 2.50 Dispatchable Generation Unit Parameters b Pmax A (cents/kW2h) (cents/kWh) (kW) 0.00031 1.5 100 1.3 200 0.00051 TABLE II. GENERATED/CONSUMED POWER BY NONDISPATCHABLE UNITS

Node 2 6 14

18

10 5 0

where ai, bi, and ci are predetermined constants. For EVs we have used quadratic concave utility functions:

60

200

400 600 Iteration

Node = 7 Node = 12

40 20

800

1000

0

500 Iteration

(a)

cent/kWh

-0.5

0

the vehicle, and pi ,max (cents/kWh) is the maximum price the

Price Negotionation in Zone1

500 Iteration

1 2 3 4 5

1.5

Node = Node = Node = Node =

1

0.5

0

1000

0

500 Iteration

(c)

1000

ObjectiveValue 50 0

cents/h

100 kW

1 2 3 4

(d)

Power Imbalance ( Generation-  Demand) 150

50 0 -50

1000

(b)

Zone = Zone = Zone = Zone = Zone =

0.5 0

where Capi (kWh) is the energy capacity of the battery of the vehicle, SoCi (t ) is the State of Charge of the battery of vehicle with index i wills to pay for energy. Here, t (hours) is the length of the time step for optimization. The maximum power for electric vehicles is considered to be 19.2 kW which is roughly the maximum power that can be provided in level 2 AC charging. The generated power by the solar generation unit and the consumed power by the constant loads are as shown in TABLE II . The optimization problem is how to regulate the generated powers of the dispatchable generation units and charging powers of the PHEVs/PEVs so that the social welfare of the entire system is maximized.

80

0

0

1

(23)

3 4 9 10 14 16 17

Price Negotionation Among Zones 1.5

U i , EV ( Pi , EV (t )) 

,   pi ,max t 2  2   Pi , EV (t )   pi ,max 1  SoCi (t )  t  Pi , EV (t )  2Capi 

Node = Node = Node = Node = Node = Node = Node =

15

Power of Dispatchable Generation Units (kW) 100

kW

A. Case Study 1: Optimality of the convergence point The parameters for dispatchable devices (i.e. Thermal generation units and EVs) are shown in TABLE I. For thermal generation units, we have used quadratic convex cost functions [26]: Ci ,G ( Pi ,G (t ))  ai Pi ,G (t ) 2  bi Pi ,G (t )  ci , (22)

Charging Power for Electric Vehicles (kW) 20

cent/kWh

transformers have the ability to exchange information within themselves.

Applying the DRPC algorithm to do the price negotiation among zones and inside each zone, the plots of Fig. 4 are resulted. It can be seen that as the algorithm proceeds, the prices inside zones and among zones converge to a unique value, and the power imbalance goes to zero.

kW

Fig. 3. Five bus power network with multiple energy devices

Consumed Power by Constant Loads 15 kW 10 kW 20 kW Generated Power by Non-Dispatchable Generation Units 30 kW

Objective Value Optimal Objective Value

-50 -100 -150 -200

0

500 Iteration

1000

0

200

400 600 Iteration

800

1000

(e) (f) Fig. 4. DRPC algorithm for case study 1 a) Charging power for EVs, b) Generated power by dispatchable generation units, c) Evolution of prices among zones, d) Evolution of prices in zone 1, e) Power imbalance over the network, f) Objective value



7


( )/ where ( ) is the amount of time the , EV with index i at time t has before leaving the charging station (Time to Stay), and , and are two , positive constants. In this assignment, as time passes, ( ) would decrease and the vehicle would have a higher ( ), and thus for the same value of the price, it would , consume more power. In general, the problem can also be considered in a multiple time step framework, and the authors are working on the extended versions of this algorithm for future publications. Remark 9: Theorem 3 states that under the provided conditions, the price at the convergence point would be greater than or equal to zero. But would the price be large enough to cover the expenses of the energy suppliers? Actually the algorithm would take care of this by noting that each supplier of energy defines its cost function based on the expenses that are involved in producing the energy. If the price of energy is below a certain threshold such that any amount of production of energy has smaller revenue than cost, then that supplier will choose to produce zero amount of energy based on (19). If at this price, other producers of energy still have benefit in generating power, and the generated power is enough to cover the demands, then the price of energy would remain low. However, if the generated power by other producers, is not enough to cover the demand, then P k  0 , and according to (13), the price

cents/h

cents/h


-50

-100

-100

-150

-150 0

1000

2000 3000 Iteration

4000

0

5000

η = 1e-4

1000 Iteration

1500

2000

ObjectiveValue 50 0


-50 -100

cents/h

0

-150 -200

500

η = 5e-4

ObjectiveValue 50

-50


-100 -150

0

500

1000 Iteration

1500

-200

2000

0

η = 1.07e-3

500

1000 Iteration

1500

2000

η = 1.08e-3

ObjectiveValue

ObjectiveValue

200

200



100

cents/h

100

0

0

-100

-100

-200

Remark 8: The problem considered in this paper is a single time step problem. However, similar to [13] the effect of time could be implicitly included within the parameters of the utility function using heuristic relationships. For instance, the parameter , can be defined as a time varying ( )= , parameter and be assigned as 1+ ,

ObjectiveValue 0

-50

cents/h

Fig. 4 (a) shows that as the price evolves, EVs regulate their charging powers. The minimum charging power is allocated to the EV connected to node 3, which has the highest SoC. The maximum charging power is allocated to the EV connected to node 4, which has relatively small SoC, and is willing to pay more than the EV connected to node 9 which has smaller SoC than it. Fig. 4 (b) shows that on the generation side, dispatchable generation units also regulate their production in response to the evolution in the price. To validate that the algorithm converged to the optimal point, the same optimization problem was solved using centralized Quadratic Programming (QP). Table III (Case Study1) shows the results. Comparing Fig. 4 (a), (b) with the table, we can see that the powers of the electric vehicles and dispatchable generation units using the DRPC algorithm, converge to the same optimal values shown in Table III.

ObjectiveValue 0

cents/h

TABLE III. OPTIMAL GENERATED/CONSUMED POWERS BY DISPATCHABLE UNITS USING CENTRALIZED QUADRATIC PROGRAMMING (kW) Case Study 1 Thermal Generations Electric Vehicles 3 4 9 10 14 16 17 7 12 0 80.38 0 19.2 9.81 2.83 15.37 11.08 7.08 Case Study 3 Electric Vehicles Thermal Generations 3 4 9 10 14 16 17 7 12 0 19.2 8.48 2.13 13.78 10.36 5.31 0 109.26

-200

0

500

1000 Iteration

1500

2000

0

500

1000 Iteration

1500

2000

η = 1.10e-3 η = 1.50e-3 Fig. 5. Behavior of the DRPC algorithm for case study 1 under different choices of the parameter η

would increase until a point where the produced power equals the existing demand. For instance in case study 1, the price at the convergence point is around 1.4 cents/kWh, which is not beneficial for generator located at node 7. Nevertheless, this price is still beneficial for generator located at node 12. Therefore, generator located at node 7 does not produce any amount of energy, and the energy produced by the generator located at node 12 covers the existing demand. B. Case study 2: Effect of parameter on the convergence of the algorithm In Theorem 2, it was stated that the convergence of the algorithm depends on the choice of the parameter η. In this part, we show how different choices of η affects the behavior of the algorithm. Fig. 5 shows the development of the objective value for case study 1 for different choices of η. It can be seen that for the values of ≤ 1.07 3 the algorithm is stable and the objective value converges to the optimal value. However, as we increase by 1 5 beyond 1.07 3, the algorithm becomes unstable, and the objective value cannot converge. This behavior retains for larger values of as well. These observations are in agreement with the statement of theorem 2, that there exist a small positive threshold such that for all values of smaller than this threshold, the global optimum point is stable. Remark 10: Another observation from Fig. 5 is that the convergence speed of the algorithm is related to the value of the parameter . While for = 1 4 it takes around 2500



8 Charging Power for Electric Vehicles (kW)

Power of Dispatchable Generation Units (kW) 120

20

kW

10

5

0

3 4 9 10 14 16 17

100 80

kW

Node = Node = Node = Node = Node = Node = Node =

15

60

20 0

0

200

400

Node = 7 Node = 12

40

600 800 Iteration

1000

0

200

400

1200

(a)

600 800 Iteration

1000

1200

(b)

Price Negotionation Among Zones 1.5

Power Imbalance ( Generation-  Demand) 150 Zone = Zone = Zone = Zone = Zone =

0.5

1 2 3 4 5

0

-0.5

Fig. 7. Iterations required for convergence of the DRPC algorithm as a function of number of zones in the system

100

kW

cent/kWh

1

50

0

0

200

400

600 800 Iteration

1000

1200

-50

0

200

400

600 800 Iteration

1000

1200

(c) (d) Fig. 6. Response of DRPC algorithm to changes in the loading condition of the system a) EV response, b) Generation response, c) Price response, d) Power imbalance

iterations for the algorithm to converge, when = 1.07e 3 the algorithm converges in about 600 iterations. However, increasing beyond this value results in instability of the algorithm. Therefore, the proper choice of is extremely important to ensure the stability of the algorithm and its proper convergence speed. C. Case Study 3: Response of the algorithm to variations in renewable and load forecast information Now consider the same case study, with the difference that while the algorithm is still running, information regarding power production of the solar unit and some of the constant loads is updated. At iteration 800 of the algorithm, solar production changes from 30 kW to 20 kW, and the load connected to node 2, changes from 15 kW to 40 kW. The results are shown in Fig. 6 . It can be seen that in response to these changes (decrease in the renewable generation and increase in constant load), the price among zones increases and as a result the generators increase their production and EVs decrease their consumption and the system converges to a new equilibrium point. Similar to the previous case study, the optimization problem is also solved using centralized QP and the results are shown in Table III (Case Study2). Comparing Fig. 6 (a), (b) with the table, it can be seen that the DRPC convergence point is similar to the centralized QP solution. This shows that the algorithm has the capability of responding in realtime to changes in the system conditions to track the optimal operating point. D. Case Study 4: Scalability analysis Another important issue about the proposed algorithm is how its convergence rate changes by increasing the number of zones in the system. In this part, we provide a scalability analysis of the algorithm. To do this study, we increase the number of zones from 3 to 100. For each value of the number of zones, we have created 20 random scenarios and looked

into the statistics of the required iterations by the algorithm to reach the vicinity of the global optimal. For each experiment, we stop the algorithm at iteration k once ‖ ‖ ≤ 0.1 where is the vector of power decisions at the k-th iteration of the algorithm, and is the vector of the optimal power decisions. Fig. 7 shows the maximum and average number of the required iterations as a function of number of zones. We can see that the there is no specific increasing pattern (such as exponential increase) in the number of the required iterations for convergence. As the number of the required computations on each device is directly related to the number of the iterations, this result suggests that the proposed technology is scalable with increasing the size of the system. V. CONCLUSION A novel distributed real-time pricing control for optimal management of unidirectional V2G was proposed which considers multiple energy suppliers. In this approach, each charging station as well as each individual energy supplier is equipped with a local price regulator which controls the local price of energy by coordinating with its neighbors and considering the balance of supply and demand. In response to the updated prices, the vehicles adjust their charging rates to maximize their utility while energy suppliers adjust their production rate to maximize their benefit. The main advantages of the proposed approach are: 1) solves V2G management problem in a fully distributed way considering the vehicle side and supplier side simultaneously, 2) converges to the global optimum despite the greedy behavior of the individuals.

[1] [2]

[3]

[4]

REFERENCES D. Block and J. Harrison, “Electric Vehicle Sales and Future Projections,” 2014. P. Denholm and W. Short, “An Evaluation of Utility System Impacts and Benefits of Optimally Dispatched Plug-In Hybrid Electric Vehicles,” National Renewable Energy Laboratory, 2006. D. Dallinger, S. Gerda, and M. Wietschel, “Integration of intermittent renewable power supply using grid-connected vehicles – A 2030 case study for California and Germany,” Applied Energy, vol. 104, pp. 666–682, Apr. 2013. H. Turton and F. Moura, “Vehicle-to-grid systems for sustainable development: An integrated energy analysis,” Technological Forecasting and Social Change, vol. 75, no. 8, pp. 1091–1108, Oct. 2008.



9 [5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15] [16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

C. Jin, J. Tang, and P. Ghosh, “Optimizing Electric Vehicle Charging With Energy Storage in the Electricity Market,” IEEE Transactions on Smart Grid, vol. 4, no. 1, pp. 311–320, 2013. D. Wu, D. C. Aliprantis, and L. Ying, “Load Scheduling and Dispatch for Aggregators of Plug-In Electric Vehicles,” IEEE Transactions on Smart Grid, vol. 3, no. 1, pp. 368–376, 2012. S. Stüdli, E. Crisostomi, R. Middleton, and R. Shorten, “A flexible distributed framework for realising electric and plug-in hybrid vehicle charging policies,” International Journal of Control, no. June 2012, pp. 37–41, 2012. Z. Fan, “A Distributed Demand Response Algorithm and Its Application to PHEV Charging in Smart Grids,” IEEE Transaction on Smart Grid, vol. 3, no. 3, pp. 1280–1290, 2012. N. Rotering and M. Illic, “Optimal charge control of plug-in hybrid electric vehicles in deregulated electricity markets,” IEEE Transactions on Power Systems, vol. 26, no. 3, pp. 1021–1029, 2010. C.-K. Wen, J.-C. Chen, J.-H. Teng, and P. Ting, “Decentralized Plug-in Electric Vehicle Charging Selection Algorithm in Power Systems,” IEEE Transaction on Smart Grid, vol. 3, no. 4, pp. 1779–1789, 2012. L. Gan, U. Topcu, and S. H. Low, “Optimal Decentralized Protocol for Electric Vehicle Charging,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 940–951, 2013. E. L. Karfopoulos and N. D. Hatziargyriou, “A Multi-Agent System for Controlled Charging of a Large Population of Electric Vehicles,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 1196–1204, 2013. N. Rahbari-Asr and M.-Y. Chow, “Cooperative Distributed Demand Management for Community Charging of PHEV/PEVs Based on KKT Conditions and Consensus Networks,” IEEE Transactions on Industrial Informatics, vol. 10, no. 3, pp. 1907– 1916, 2014. Z. Ma, D. Callaway, and I. Hiskens, “Decentralized Charging Control for Large Populations of Plug-in Electric Vehicles,” 49th IEEE Conference on Decision and Control (CDC), pp. 206–212, Dec. 2010. M. Ye and G. Hu, “Game Design for Price based Demand Response,” IEEE Transactions on Cybernetics, pp. 1–11, 2015. R. Deng, Z. Yang, J. Chen, N. Rahbari-Asr, and M.-Y. Chow, “Residential Energy Consumption Scheduling: A CoupledConstraint Game Approach,” IEEE Transactions on Smart Grid, vol. 5, no. 3, pp. 1340 – 1350, 2013. N. Rahbari-Asr, U. Ojha, Z. Zhang, and M.-Y. Chow, “Incremental Welfare Consensus Algorithm for Cooperative Distributed Generation/Demand Response in Smart Grid,” IEEE Transactions on Smart Grid, vol. 5, no. 6, pp. 2836–2845, 2014. P. Samadi, H. Mohsenian-rad, R. Schober, and V. W. S. Wong, “Advanced Demand Side Management for the Future Smart Grid Using Mechanism Design,” IEEE Transactions on Smart Grid, vol. 3, no. 3, pp. 1170–1180, 2012. S. Boyd, P. Neal, E. Chu, B. Paleato, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations and Trends® in Machine Learning, vol. 3, no. 1, pp. 1–122, 2010. R. Olfati-Saber and J. Fax, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the IEEE, vol. 95, no. 1, pp. 215–233, 2007. S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Randomized Gossip Algorithms,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2508–2530, 2006. Y. Zhang, N. Rahbari-asr, and M.-Y. Chow, “A robust distributed system incremental cost estimation algorithm for smart grid economic dispatch with communications information losses,” Journal of Network and Computer Applications, 2015. W. Zeng and M.-Y. Chow, “Resilient distributed control in the presence of misbehaving agents in networked control systems,” IEEE Transactions on Cybernetics, vol. 44, no. 11, pp. 2038– 2049, 2014. F. Pasqualetti, A. Bicchi, and F. Bullo, “Consensus computation in unreliable networks: A system theoretic approach,” IEEE Transactions on Automatic Control, vol. 57, no. 1, pp. 90–104, Jan. 2012. K. De Brabandere, B. Bolsens, J. Van den Keybus, A. Woyte, J. Driesen, and R. Belmans, “A Voltage and Frequency Droop

[26]

Control Scheme for Parallel Inverters,” IEEE Transactions on Industrial Electronics, vol. 22, no. 1, pp. 1107–1115, 2007. J. J. Grainger, Power System Analysis. McGraw-Hill Education, 2003.

Navid Rahbari-Asr (S’09, M’16) received his B.S. in Electrical Engineering from University of Tabriz, Iran, in 2008. In 2011, he received his M.Sc. in Control Engineering from Tarbiat Modares University, Tehran, Iran, and his Ph.D. degree from the Advanced Diagnosis, Automation, and Control Laboratory at North Carolina State University, Raleigh in 2015. His research interests include distributed control, distributed optimization, big data, and computational intelligence with applications to smart grids. Mo-Yuen Chow (S’81, M’82, SM’93, F’07) earned his degree in Electrical and Computer Engineering from the University of Wisconsin-Madison (B.S., 1982); and Cornell University (M. Eng., 1983; Ph.D., 1987). Dr. Chow is a Professor in the Department of Electrical and Computer Engineering at North Carolina State University. Dr. Chow was a Changjiang Scholar and a Qiushi Professor at Zhejiang University. Dr. Chow’s recent research focuses on distributed control and management on smart grids, batteries, and robotic systems. Dr. Chow has established the Advanced Diagnosis, Automation, and Control Laboratory. He is an IEEE Fellow, a Co-Editor-in-Chief of IEEE Trans. on Industrial Informatics, Editor-in-Chief of IEEE Transactions on Industrial Electronics 2010-2012. He has received the IEEE Region-3 Joseph M. Biedenbach Outstanding Engineering Educator Award, the IEEE ENCS Outstanding Engineering Educator Award, the IEEE ENCS Service Award, the IEEE Industrial Electronics Society Anthony J Hornfeck Service Award. He is a Distinguished Lecturer of IEEE IES. Jiming Chen (M'08, SM'11) received B.Sc degree and Ph.D degree both in Control Science and Engineering from Zhejiang University in 2000 and 2005, respectively. He was a visiting researcher at University of Waterloo from 2008 to 2010. Currently, he is a full professor with Department of control science and engineering, and vice director of the State Key laboratory of Industrial Control Technology and Institute of Industrial Process Control at Zhejiang University, China. He serves/served associate editors for several international Journals including IEEE Transactions on Parallel and Distributed System, IEEE Network, IEEE Transactions on Control of Network Systems, etc. He was a guest editor of IEEE Transactions on Automatic Control, etc. His research interests include cyber security, sensor networks, smart-grid, networked control. Ruilong Deng (S'11, M'14) received the B.Sc. and Ph.D. degrees both in Control Science and Engineering from Zhejiang University, China, in 2009 and 2014, respectively. He was a Visiting Scholar at Simula Research Laboratory, Norway, in 2011, and the University of Waterloo, Canada, from 2012 to 2013. He was a Research Fellow at Nanyang Technological University, Singapore, from 2014 to 2015. Currently, he is an AITF Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Alberta, Canada. His research interests include smart grid, cognitive radio, and wireless sensor network. Dr. Deng currently serves as an Editor for IEEE/KICS Journal of Communications and Networks, and a Guest Editor for IEEE Transactions on Emerging Topics in Computing and Journal of Computer Networks and Communications. He also serves/served as a Technical Program Committee Member for IEEE Globecom, IEEE ICC, IEEE Smart Grid Comm, EAI SGSC, etc.


Distributed Real-Time Pricing Control for Large-Scale Unidirectional

Distributed Real-Time Pricing Control for Large-Scale Unidirectional

Suggest Documents

Distributed algorithms for Unidirectional networks - Semantic Scholar

Distributed Realtime Interaction and ... - Jan-Michael Frahm

Distributed Optimization for Control

Distributed QoS Guarantees for Realtime Traffic in Ad Hoc Networks

Realtime Monitoring for Parallel Distributed Processing System based ...

Realtime Monitoring for Parallel Distributed Processing System based

A LargeScale Gene-Trap Screen for Insertional

Realtime

A New Space-vector-modulated Control For A Unidirectional Three ...

Distributed Optimistic Concurrency Control for

Distributed Optimistic Concurrency Control for

Realtime

Distributed Computation for Cooperative Control

Distributed Window Concurrency Control for

DISTRIBUTED CONTROL NETWORK FOR ... - MIcroLab

Unidirectional Encoder Rate Control Scheme for Transform Domain

Realtime Control for Motion Creation of 3D Avatars | SpringerLink

Distributed Storage Schemes over Unidirectional Ring Networks - arXiv

Framework for Distributed Engine Control System for

Framework for Distributed Engine Control System for

HAFNI-enabled largescale platform for neuroimaging informatics ...

Unidirectional panel

Distributed Control Loop Patterns for Managing Distributed Applications

Transmission Pricing of Distributed Multilateral Energy ... - CiteSeerX