Consensus-based Distributed Control for Economic Operation of Distribution Grid with Multiple Consumers and Prosumers Hajir Pourbabak, Student Member, IEEE Tao Chen, Student Member, IEEE Department of Electrical and Computer Engineering University of Michigan-Dearborn Dearborn, MI, USA

Abstract— This paper investigates the economic operation of distribution grid with multiple consumers and prosumers using consensus-based distributed control algorithms. Driven by the emerging Smart Grid technologies, a tremendous amount of electricity consumers and prosumers have the potential to actively participate into the retail electricity market. This paper formulates the utility and cost functions of a variety of consumers and prosumers and their local/global constraints. Case studies demonstrate the accuracy, robustness, effectiveness, and scalability of the proposed distributed control approaches for the economic operation of multiple consumers and prosumers under various operating conditions. The distributed control algorithms contained in this paper can apply to many other smart grid applications. Index Terms— Consensus Algorithm, Consumer, Prosumer, Economic Dispatch, and Smart Grid

I. INTRODUCTION The energy industry is undergoing through a reconstruction from a monopolistic electricity market to a more open and transactive one [1],[2]. The next-generation grid is a level playing field in terms of electricity transactions, where all customers have an equal opportunity [3],[4]. The new techniques, such as demand side management and demand response, make it possible for consumers to adapt their energy usage in response to dynamic pricing and other economic/environmental incentives. The emerging concepts of electricity prosumers are expected to have a significant impact on the retail electricity market. As a result, there is an urgent need to investigate the interactions among a number of consumers and prosumers.

Wencong Su, Member, IEEE Department of Electrical and Computer Engineering University of Michigan-Dearborn Dearborn, MI, USA [email protected]

Fig. 1. The schematic model of multi consumer and prosumer

The majority of existing literature focuses on the centralized control [5]. In most cases, the dedicated communication links are required to exchange data between the central controller (e.g., aggregator) and the local agents (e.g., consumers and prosumers). The centralized control approaches are suitable for relatively small-scale systems without reconstructing the existing communication and control networks. However, as the number of consumers and prosumers are increasing to hundreds of thousands, there are some technical barriers on the centralized control-based economic operations such as heavy computation burden and single point of failure.

Fig. 1 illustrates the proposed economic operation of distribution grid with multiple consumers and prosumers.

The decentralized control is an intermediate solution to address the abovementioned challenges. The overall objective is to maximize the benefits of local agents without coordinating with others. However, since there is no communication link between the different agents (i.e., consumers and prosumers), there is no guarantee that the decisions made by each local agent can contribute to the global optimal decision of the entire system as a whole. It is well known that consumers and prosumers have partial or complete conflicts. The strong interactions between different agents may prevent the entire system from achieving the global optimal operation.

It is challenging to achieve the economic operations of these heterogeneous market participants. The existing control approaches can be divided into three categories, namely, centralized control, decentralized control, and distributed control.

The distributed control has the potential to solve the economic operation problems of multiple consumers and prosumers. Local agents can share information through twoway communication links in order to find the global optimal decision.

The main contributions of this paper are summarized as follows: (1) Formulate the utility functions and cost functions of various consumers and prosumers; (2) Propose a global optimization problem to maximize the social welfare of all market participants subject to various global and local constraints; (3) Apply consensus-based algorithms to solve the proposed optimization problem in a fully distributed manner; and (4) Evaluate the performance of the proposed distributed algorithm in terms of accuracy, robustness, effectiveness, and scalability. The remaining sections of this paper are organized as follows. Section II formulates the utility functions and cost functions of consumers and prosumers and their local/global constraints. Section III presents the mathematical models of the proposed economic operation problem. Section IV introduces the consensus-based distributed algorithm. Section V discusses the simulation results. Section VI summarizes the major research findings of this paper. II. PROBLEM FORMULATION

Consumers

The envisioned electricity market consists of multiple consumers and prosumers, as shown in Fig. 2. We will detail the mathematical models of utility and cost functions for various consumers and prosumers. A number of local and global constraints will be also taken into account.

1

2

Prosumers

P2,1

3

..........

n

P2,i

P2,2

cost function for the -th prosumer with parameters and [8]. =

,

,

+

,

+

,

,

(2)

B) Constraints We will consider some generation and distribution constraints. Equation (3) represents the power balance for all consumers (i.e., load demand) and prosumers (i.e., generation output). ,

=

∀

(3)

Where is the load demand of consumer . The total generation of a prosumer should not exceed its maximum capacity. 0≤

,

≤

∀

(4)

Where , shows the power injection from prosumer to consumer . Since the proposed electricity market includes producers and users, we can extract a demand matrix shown in Equation (5). Notice that all elements of demand matrix are considered as the decision variables.

 P1,1 P  2,1     P4,1

P1,2 P2,2

 

 P4,2

 

P1,4   PUser1 P2,4   PUser2    P4,4   PUser n







PProsumer1

PProsumer2

PProsumeri

(5)

III. OPTIMIZATION PROBLEM

Pn,3 . . . . . . . . . .

Pn ,2

Pn ,1 1

2

Pn ,i n

3

Fig. 2. The schematic model of multi consumer and prosumer

A) Utility Funcations and Cost Functions In an electricity market, the consumers have their own load demand level based on different factors such as temperature changes, electricity price, and customer preference (comfort level). The deferent performance of users in a market can be modeled as a function of the satisfaction level of power consumption [6],[7]. The utility function ( , ) represents the satisfaction level of the -th consumer based on its load demand , that is demanded power from the prosumer by the consumer . Equation (1) models the quadratic utility function ( , ) for different consumers based on parameters and . , ,

=

−

,

,

<

2

(1)

, ≥ 4 2 The cost functions of prosumer are usually given by a quadratic function. The prosumers try to produce electricity in a cost-effective way. Equation (2) shows the quadratic

In this paper, we will consider the social welfare of all participating consumers and prosumers. The goal is to maximize the utility functions of consumers while minimizing the cost functions of consumers [6]. The overall objective function will be the maximization of the total social welfare defined in Equation (6). The objective function is the summation of utility functions minus cost functions. is defined as the cost of electricity sold by prosumer . ,

−

,

+

,

−

,

(6)

Equation (6) can be re-written as Equation (7). The objective function is also subject to constraints defined in Equations (3) and (4). ,

−

,

(7)

To protect the privacy of individual agents and keep the fairness the market clearing process [5], consumers prefer not to disclose their operating conditions to prosumers. Therefore, we will solve Equation (7) in a fully distributed

fashion, such that each agent only needs solve its own subproblem with a limited amount of local information from neighbors. In such a way, we will solve a set of sub-problems defined in Equation (8) collectively, in order to find the global optimal solution to Equation (7). −

,

−(

⎧ ⎪

(

− ) , + + ) , +

−

⎨ ⎪ ⎩

,

=

,

+

,

≤

,

,

+

∀

(8)

2

≥

2

Consumers

Fig. 3 illustrates a sample sub-problem for a single consumer and multiple prosumers according to Equation (8). A number of sub-problems can be solved in parallel.

1

2

Prosumers

P2,1

..........

3

2

n

3

Equation (9) represents Lagrange multiplier ( ∗ , ) for each sub-problem by taking derivative of Equation (8) with respect to , . The optimal decision variables (i.e., each element of the demand matrix) can be found by solving Equation (10). We will detail the proposed distributed algorithm for solving Equation (10) in Section IV.

∗ ,

=

(

)

=

∀

,

−2(

+

−2

,

)

,

+(

−

⎧( − ) − ⎪ 2( + ) = ⎨(− ) − ⎪ 2( ) ⎩

−

(

−

[ + 1] =

Fig. 3. A sub-problem

,

=

)

,

<

2

,

≥

2

,

<

2

,

≥

)

(11)

Where, is the entry of adjacency matrix. The discretetime consensus algorithm is described as

n

..........

∗

̇ =

P2,i

P2,2

1

and only if ( , ) ∈ [11]. Besides, the entry of adjacency matrix is a positive weight if ( , ) ∈ and = 0 if ( , ) ∉ . Moreover, the second smallest eigenvalue of its corresponding Laplace matrix is called algebraic connectivity. According to [12], the algebraic connectivity of interaction topology quantifies the speed of convergence of consensus algorithms. B) Consensus-based Distributed Algorithm In the consensus problem, all agents aim to reach a consensus. In other word, each agent’s state is driven toward the state of its neighbors [13]. Regarding each agents in a graph with singleintegrator dynamic ̇ = , where represent the state variable, in [11], a continuous-time consensus algorithm is given as

(9)

(10)

[ ]

(12)

Where [ + 1] is updated state of [ ] at iteration of [ + 1], [ ] are the local information discovered by agent at iteration , is the entry of adjacency matrix. Here, similar to [14], the elements of the adjacency matrix is defined as: 1 , ∉ (13) = | | 0, ∈ Where is a set of neighbors of vertex and |. | is its cardinality. The formulation of and of each sub-problem for distributed algorithm are mentioned in this part. Therefore, the internal incremental cost, output power and local power mismatch between generation and load of each sub-problem is calculated by Equations (14)-(16), respectively. is the iteration index and is a small positive constant that should be small enough for more stability of algorithm [8]. According to Equations (14)-(16), each agent (i.e., consumer and prosumer) needs to know information from its neighbors only.

2 ,

IV. SOLUTION ALGORITHM A) Graph Theory An undirected graph here is used to model the interaction topology of a network of agents [9],[10]. Denote = ( , ) as graph with a set of vertices = {1,2, … } and edges ⊆ × . An undirected edge ( , ) denotes that agents I and j can obtain information from each other. The set of neighbors of vertex is denoted by = { ∈ |( , ) ∈ }. Two vertices are called connected if there is a distinct path from agent to agent . An undirected graph is connected if there exist an undirected path between any pair of vertices. The adjacency matrix = ∈ × of an undirected graph is symmetric, which means ( , ) ∈ if

,

∆

( + 1) =

,

( + 1) = ( − ) − ( + 1) ⎧ ⎪ 2( + ) (− ) − ( + 1) ⎨ ⎪ 2( ) ⎩ ,

( ) + ℇ∆

,

,

≤ ≥

( )

,

2

(14)

∀

(15)

2

( + 1) = ∆

,

( )−

,

( + 1) −

,

( )

(16)

Table I: Parameters for Consumer/Prosumer Function in Case Study I Parameters for Consumers’ Utility Function ,

5.06 4.87 6.73 5.51

1 2 3 4

0.0935 0.0417 0.1007 0.0561

91.79 147.29 91.41 62.96 Total Load: 393.45

Parameters for Prosumers’ Cost Function 8.71 8.53 7.58 6.89

1 2 3 4

0.0031 0.0074 0.0066 0.0063

0 0 0 0

113.23 179.1 90.03 106.41

(cent/kWh)

In this section, we will provide numerical examples to evaluate the performance of the proposed distributed algorithm for the economic operation problems of multiple consumers and prosumers. The simulation results solved by the proposed distributed algorithms will be further compared with the centralized methods. The centralized methods will be implemented using YALMIP toolbox. For easy of illustration, we will consider a relatively small system with multiple consumers and prosumers. In case study I, we will take into account four different seller and users. However, it is worth to mention that the proposed solution algorithms can be applied to larger-scale problems because of its distributed nature; thus, we consider case study II with 40 agents including four seller and ten users. A) Case I Firstly, we scale up the test system to consider four consumers and four prosumers. They all have their own utility functions and cost functions. The parameters for consumers and prosumers are shown in Table I [15]. The generation output power of each prosumer is shown in Fig. 4. Two prosumer reach their generation limits: = , + , + , + , = 17.65 + 38.03 + 19.05 + 15.30 = 90.03 and = , + , + , + , = 20.06 + 44.75 + 21.54 + 20.06 = 106.41 The Lagrange multipliers (incremental costs) for subproblems are shown in Fig. 5. Each sub-problem has its own specific value of Lagrange multipliers. Lagrange multipliers of prosumers = 3 and 4 are fixed, because they cannot produce more than its limit. The total incremental cost derived from all sub-problem for whole load are indicated in Fig. 6. Fig. 7 shows that the summation of prosumers’ power is equal to total load. Table II compares the proposed distributed method with the centralized method. The results of distributed method are the same as that using centralized method.

Fig. 6. Total Incremental Cost

P(kW)

V. NUMERICAL RESULTS

Fig. 7. The summation of all prosumer’s generation Table II: Benchmarking Comparison with Distributed methods Optimal Incremental Cost (cent/kW) Prosumers 1 2 3 4

Distributed Method 9.3202 9.9725 8.7685 8.2310

Centralized Method 9.3247 9.9785 8.7684 8.2312

Optimal Output Power (kW) Fig. 4. Generation Output Power of Prosumers for Sub-problems in Case 9 8

1 2 3 4

99.4230 97.4708 90.0391 106.4299

99.1408 97.8692 90.0300 106.4100

7 6 5 4 3

sub-problem sub-problem sub-problem sub-problem

2 1 0

0

200

400

600

800

1000

Iteration

Fig. 5. Internal Lagrange Multipliers (Incremental Cost) for Subproblems in Case Study I

1 2 3 4 1200

B) Case II As mentioned before, 40 agents including 10 consumers and 4 prosumers are considered as third case study. Table III shows the parameters of all consumers. The prosumers parameters are as the previous case. The generation output power of each prosumer is shown in Fig. 8. In addition, Fig. 9 The Lagrange multipliers (incremental costs) for subproblems. The results of distributed method for this case study are benchmarked against centralized approach

provided by YALMIP (Table. IV). The results are the same as centralized method. Table III: Parameters for Consumer Function in Case Study II Parameters for Consumers’ Utility Function ,

5.06 4.87 6.73 5.51 6.07 5.12 4.88 5.02 5.63 4.90

1 2 3 4 5 6 7 8 9 10

0.0935 0.0417 0.1007 0.0561 0.0549 0.0461 0.0517 0.0830 0.0620 0.0400

69.79 45.29 49.41 60.96 32.61 42.15 53.50 47.10 38.00 49.10 Total Load: 487.91

VI. CONCLUSION In this paper, we model the interactions among multiple consumers and prosumers considering their utility functions and cost functions. We proposed a consensus-based distributed algorithm to find the global optimal solutions for the economic operations of multiple consumers and prosumers subject to a number of local/global constraints. The proposed distributed algorithm completely reduces our reliance on the central controller. The case studies demonstrate the accuracy, robustness, effectiveness, and scalability of the proposed distributed algorithm. In future, we will fully investigate the computational/communication overhead of the proposed distributed algorithm under various scenarios. References [1]

25

20

[2]

15

[3] 10

5

[4] 0 0

100

200

300

400

500

600

700

800

900

1000

Iteration

Fig. 8. Generation Output Power of Prosumers for Sub-problems in Case

[5]

[6]

[7]

[8]

Fig. 9. Internal Lagrange Multipliers (Incremental Cost) for Sub-problems Table IV: Comparison of Distributed Methods and Centralized Methods

[9]

[10]

Optimal Incremental Cost (cent/kW) Prosumers 1 2 3 4 5 6 7 8 9 10 1 2 3 4

Distributed Method Centralized Method 6.3384 6.3349 4.1439 4.1444 3.8339 3.8350 4.3127 4.3137 2.8558 2.8568 3.9112 3.9145 4.5947 4.5958 5.0049 5.0124 3.5934 3.5935 4.1631 4.1630 Optimal Generation Output Power (kW) 63.4130 63.4130 73.4694 73.4694 148.200 148.201 202.8269 202.8282

[11]

[12]

[13] [14]

[15]

A. Q. Huang, M. L. Crow, G. T. Heydt, J. P. Zheng, and S. J. Dale, “The Future Renewable Electric Energy Delivery and Management (FREEDM) System: The Energy Internet,” Proc. IEEE, vol. 99, no. 1, pp. 133–148, Jan. 2011. W. Su, “The Role of Customers in the U.S. Electricity Market: Past, Present and Future,” Electr. J., vol. 27, no. 7, pp. 112–125, Aug. 2014. N. Zhang, Y. Yan, and W. Su, “A game-theoretic economic operation of residential distribution system with high participation of distributed electricity prosumers,” Appl. Energy, vol. 154, pp. 471–479, Sep. 2015. “GridWise Architecture Council, January 2015, ‘GridWise Transactive Energy Framework version 1.0’, [Online] Available: www.gridwiseac.org/pdfs/te_framework_report_pnnl22946.pdf.” W. Su, J. Wang, and D. Ton, “Smart Grid Impact on Operation and Planning of Electric Energy Systems,” in Handbook of Clean Energy Systems, J. Yan, Ed. Chichester, UK: John Wiley & Sons, Ltd, 2015, pp. 1–13. R. Deng, Z. Yang, F. Hou, M.-Y. Chow, and J. Chen, “Distributed Real-Time Demand Response in Multiseller– Multibuyer Smart Distribution Grid,” IEEE Trans. Power Syst., vol. 30, no. 5, pp. 2364–2374, 2015. P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V. W. S. Wong, and J. Jatskevich, “Optimal Real-Time Pricing Algorithm Based on Utility Maximization for Smart Grid,” 2010 First IEEE Int. Conf. Smart Grid Commun., pp. 415–420, 2010. H. Xing, S. Member, Y. Mou, M. Fu, Z. Lin, and S. Member, “Distributed Bisection Method for Economic Power Dispatch in Smart Grid,” pp. 1–12, 2014. S. Yang, S. Tan, and J. Xu, “Consensus Based Approach for Economic Dispatch Problem in a Smart Grid,” vol. 28, no. 4, pp. 4416–4426, 2013. J. Cao, “Consensus-based distributed control for economic dispatch problem with comprehensive constraints in a smart grid,” Electronic Theses, Treatises and Dissertations,Florida State University, 2014. A. Olshevsky and J. N. Tsitsiklis, “Convergence Speed in Distributed Consensus and Averaging,” SIAM J. Control Optim., Feb. 2009. R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks. London: Springer London, 2011. G. Binetti, A. Davoudi, F. L. Lewis, D. Naso, and B. Turchiano, “Distributed Consensus-Based Economic Dispatch With Transmission Losses,” IEEE Trans. Power Syst., vol. 29, no. 4, pp. 1711–1720, 2014. N. Rahbari-Asr, U. Ojha, Z. Zhang, and M. Chow, “Incremental Welfare Consensus Algorithm for Cooperative Distributed Generation/Demand Response in Smart Grid,” IEEE Trans. Smart Grid, vol. 5, no. 6, pp. 2836–2845, Nov. 2014.