IEEE International Workshop on Smart Grid Communications and Networks
An Integer Linear Programming and Game Theory Based Optimization for Demand-side Management in Smart Grid Z. Zhu, J. Tang and S. Lambotharan
W. H. Chin and Z. Fan
Advanced Signal Processing Group, Loughborough University, Leicestershire, LE11 3TU, UK Email:{z.zhu2,j.tang,s.lambotharan}@lboro.ac.uk
Toshiba Research Europe, Telecommunications Research Laboratory 32 Queen Square, Bristol, BS1 4ND, UK Email:
[email protected],
[email protected]
Abstract—We propose a consumption scheduling mechanism for home and neighbourhood area load demand management in smart grid using integer linear programming (ILP) and game theory. The proposed mechanism is able to schedule both the optimal power and the optimal operation time for power-shiftable appliances and time-shiftable appliances respectively according to the power consumption patterns of all the individual appliances. Simulation results for both the centralised and the distributed management scenarios have been presented to demonstrate and verify the effectiveness of the proposed technique.
I. I NTRODUCTION The electric power industry is facing significant challenges of environmental issues, security and utility management. Experts believe that a fundamental evolution of the electric power system is needed. The transformation will be based on digitalised information and communications technologies in order to make the future power system green, reliable and intelligent. It is generally known as the Smart Grid. Countries around the world such as the US, Japan, China and several European countries have already started developing their smart grid and have already made some initial progress [1], [2]. Advanced electric technologies will improve the efficiency of the central bulk generation and transmission network. Medium/low voltage distributed generation (DG) system using various environment-friendly sources will be seamlessly integrated. Information and communication technologies (ICT) will be widely applied to the power grids to enhance its operation efficiency, reliability and flexibility. Indeed, one of the key developments for smart grid is flexible bi-directional demand response (DR) management. According to the US Department of Energy, demand response (DR) can be defined as the changes in electric usage by end-user from their normal consumption patterns in response to changes in the price of electricity over time [3]. Operators are able to effectively control the peak load and manage the changes in the electricity generation and supply by setting flexible price plans.
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Consumers can be proactive to manage their demand and make the best consumption scheduling plans. DR will be a core service provided by the smart grid since it directly connects the electricity consumers with the operators to form a transparent bi-directional utility management system. It has significant economical advantages and improves the flexibility of the grid. The primary goal of DR is to reduce the peak load and enhance the reliability of the grid. Currently, many smart grid researchers are working on designing demand-side management (DSM) approaches for DR by utilizing distributed algorithms and optimization techniques previously employed in communications research [4]–[6]. The focus of the work in this paper is on demand-side power consumption scheduling. As an analogy to the design of hierarchical internet routing, paper [4] designed a global optimization methodology to manage the demand of the whole network in a hierarchical decision tree structure. In [5], a powerful convex optimization technique was proposed to schedule the power of individual appliances. Both the centralised optimization based on linear programming and the decentralised game theoretic approach have been proposed. However, the proposed optimization framework might not be suitable for all appliances in practice. This is because some appliances have a fixed power consumption pattern, which means that once the appliance is turned on, it has to work according to its own power consumption pattern until the job is finished. In this case, we could optimize only the starting time, but the power consumption during the operation of the appliance is not under the control of the optimizer, hence it is not an optimization parameter. Inspired by [5], we propose a demand-side management method using integer linear programming technique and game theory. The proposed mechanism will be able to optimize the consumption schedule and satisfy both the user preference and specific requirements of all individual appliances. The mechanism can be applied to the home energy management
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Fig. 1.
System components.
unit/gateway embed in the smart meters, which will probably work as a load control unit as well, in the home area network (HAN). It can also be applied to the distribution stations (central control node) in the neighbourhood/local areas to achieve centralised load management. The mechanism can be further applied to a group of interconnected consumers in the grid to achieve coordinated management. The rest of the paper is organised as follows. The proposed demand-side management mechanism is described in Section II. Section III presents system simulation results and conclusions are drawn in Section IV.
many smart meters are interconnected. The central control node will take the overall responsibility of scheduling the whole network and assigning individual meters their corresponding tasks. As in the content of distributed coordinative control, the system architecture is similar. However, it is more efficient and cost effective for the operators to apply distributed scheme since it does not require an additional infrastructure at the distribution node only for load control purpose. Individual users will be able to manage their demand in the area. The computation complexity will also be lower. However, in order to support larger and probably real-time information exchanges among different users, the requirements of communication infrastructure become higher. B. Centralised consumption scheduling The consumption scheduling mechanism can be described as a linear optimization problem which aims to minimise the hourly load, as shown below [5] min
L,xa,h ∈R
s.t.
Fig. 1 depicts the overall structure of the load management system. As shown in the figure, the smart meter is the key component in the system. It connects with the user interface to collect the user’s own power consumption plan and preference and display the scheduling information. On the other side, the smart meter connects with all the home appliances, not only to provide electricity for the appliances but also to determine the total requirements and power consumption patterns of all individual appliances. Based on all the collected information, the meter will globally optimize the hourly consumption and schedule all appliances. For non-shiftable appliances such as TV and fridge which have a fixed power requirement and operational period, the optimization will ensure continuous supply of power. The scheduling optimization will be carried out mainly for the shiftable appliances. For time-shiftable appliances, such as washing machine, the smart meter will be able to control the switch and provide sufficient electricity corresponding to the power pattern during the scheduled periods. For power-shiftable appliances, such as water boiler and electric vehicle chargers, the smart meter will schedule flexible power and ensure the total supply. Effective home area communication network is required for the system. However, as for non-real-time centralised scheduling, the information exchange may not require high quality communications. The system can be further extended to multiple users’ (neighbourhood/local area) scenario where
(1) xa,h ≤ L, ∀h ∈ H,
a∈A T
II. D EMAND - SIDE CONSUMPTION SCHEDULING A. System description
L
(2)
1 xa = la , ∀a ∈ A,
(3)
xa,h ≥ 0,
(4)
where 1 = [1, 1, . . . 1]T . We define a as an individual appliance in a set of appliances, A. xa = [xa,1 , xa,2 , . . . , xa,24 ]T denotes the schedule plan for appliance a and xa,h is the scheduling variable which represents the power consumption of appliance a in the particular hour h ∈ H, H = [1, 2, . . . , 24] of the day. The variable L represents the peak hourly load. The cost function of the above optimization problem is to minimize the hourly load L subject to the constraints in (2)-(4). The hourly load should be greater than or equal to the sum of the scheduled power for all appliances in that time. For each appliance, the total daily supply has to meet the requirement la . The power consumption xa,h must be non-negative value, i.e., xa,h ≥ 0. We use a time resolution of one hour for hourly scheduling in this paper, hence xa contains 24 elements. This resolution can be increased by increasing the length of xa . We formulate the requirements for all the individual appliances and the user preferences as constraints and add them into the optimization problem (1)-(4). For non-shiftable appliances a ∈ F ⊂ A, where F denotes the set of such appliances, the hourly power requirement is fixed at δa during its working period from has to haf (has , haf ∈ H). This constraint can be written as xa,h ≥ δa , ∀a ∈ F, ∀h ∈ [has , ha(s+1) , . . . , haf ].
(5)
Note that a similar constraint can also be applied to some of the shiftable appliances if the consumer has a specific requirement in terms of starting and finishing time for the
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appliance. For example, people usually control the electric heater using a programmable timer or a thermostat. This kind of appliance can be treated as a non-shiftable appliance.
Then, we rewrite the optimization problem in (1) by adding st as an integer optimization variable, together with other constraints as
For power-shiftable appliances a ∈ S ⊂ A with a standby power αa and a maximum working power βa , together with a possibly preferred working period, the consumption requirement can be written as
min L s.t. xa,h + xt,h ≤ L, ∀h ∈ H, L,xa,h ∈R,st ∈Z24×1
a∈A−T 1T xa = la ,
αa ≤ xa,h ≤ βa , ∀a ∈ S, ∀h ∈ [has , ha(s+1) , . . . , haf ]. (6) Now consider a time-shiftable appliance that consumes power la . If we impose only the constraint (3), the optimization will ensure supply of a total power of la over the 24 hours. For example, if a washing machine requires 1.5kWh for its operation, the constraint in (3) could ensure 1.5kWh power supply over 24 hours. This total power can be distributed in any possible ways. For example, the vector xa can have 0.25 for six hours and zeros for the rest. However, a washing machine cannot operate according to this power schedule. It might require fixed 1kW for the first hour of operation and another 0.5kW for the second hour of operation. In other words, this kind of appliances will need to operate according to their own power consumption pattern for correct operation. Therefore, only a constraint in the form of (3) is inadequate to ensure feasible operation of certain appliances. Considering this, we model this kind of requirement as follows. Suppose, an appliance a has a fixed consumption pattern pa = [pa,1 , . . . , pa,24 ]T , the schedule result xa has to be exactly the same as one of the cyclic shifts ca of that pattern. All the possible shifts for the vector pa and hence all the possible xa can be put in a matrix form as Pa = ca1 ca2 ... ca24 ⎡ ⎤ pa,1 pa,24 ... pa,3 pa,2 ⎢ pa,2 pa,1 ... pa,4 pa,3 ⎥ ⎢ ⎥ (7) =⎢ . .. . . .. .. ⎥ . ⎣ .. . . . . ⎦ pa,24 pa,23 ... pa,2 pa,1
(8)
Now the schedule plan can be written as xt = PTt st , ∀t ∈ T.
xa,h ≥ 0, xa,h ≥ δa , ∀a ∈ F, ∀h ∈ [has , . . . , haf ], αa ≤ xa,h ≤ βa , ∀a ∈ S, ∀h ∈ [has , . . . , haf ], 0 ≤ st ≤ 1, 1T st = 1, ∀t ∈ T, xt = PTt st , ∀t ∈ T.
(10)
Note that at this stage, the scheduling objective for the timeshiftable appliance is the switch vector sa , not the power xa,h . Therefore, we have to remove the corresponding appliance from the appliance set A. The set A − T now contains nonshiftable and power-shiftable appliances, whose power are still going to be scheduled by xa,h . The optimization problem in (10) is a mixed integer linear programming problem which contains both integer variables and non-integer variables. It can be solved using Branch and Bound method [7]. Note that if there are more than one optimal solution (with equal cost value) to the problem, the method will provide one of them. The above scheduling scheme is suitable for fixed price scenario where the unit price of the electricity consumption is fixed at any time. In this case, consumers have no way to reduce their cost by scheduling hence they have little incentive to shift their consumption. However, operators who want to reduce the peak load will be able to achieve direct load control without hurting the consumers preference by applying this centralised scheduling scheme. C. Distributed scheduling coordination
The above constraint cannot be directly formulated using linear programming. Hence, we apply integer linear programming (ILP) framework to formulate the constraint. Firstly, we define the schedule plan as xt = [xt,1 , xt,2 , . . . , xt,24 ]T ≥ 0, and the appliance set T ⊂ A for the time-shiftable appliance t ∈ T. Then, we define a binary integer vector st = [st,1 , st,2 , . . . , st,24 ]T as the switch control for the time-shiftable appliance. There is only one non-zero element in the vector st which is equal to one. This property is written as follows 1T st = 1, st ∈ {0, 1}24×1 , ∀t ∈ T.
t∈T
∀a ∈ A − T,
(9)
As for distributed demand management, the operator will be able to design incentive based approaches. For example, designing dynamic price plan where the rate changes according to the time of use (ToU) and the amount of consumption at particular time. In this case, users will have the incentive to shift the operation time and adjusting the operation power of appliances optimally to get the lowest price. Generally, operators will set higher rate at peak hours and lower rate at off-peak hours. As a result of such pricing and scheduling, there will be no peak load generated at peak hours. The goal of load control can be achieved. Since users would be proactive to schedule their consumption individually based on the knowledge of current demand condition and the provided price plan, centralised scheduling seems not necessary. Instead of that, there exists a potential
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coordination that all users can manage their usage together to achieve the optimal consumption plan for themselves. Referring to the framework of energy consumption game in [5], we apply the proposed integer linear programming model to design a distributed scheduling coordination scheme. We firstly define a daily ToU price plan as PP = [P P1 , P P2 , . . . , P P24 ] for all users and the hourly price P Ph is set to be a quadratic function of the load Lh (i.e. the unit rate increases with the total amount of load) at the particular time h ∈ H. P Ph = ωh L2h + φh , (11)
can be successfully formed. Recall the mixed integer linear programming model (10) and (12), we can formulate the optimization problem as min ( (ωh (Ln,h + L−n,h )2 + φh ), xn,a,h ∈R,st ∈Z24×1
s.t.
C=
h∈H
(ωh (
Ln,h )2 + φh ).
(12)
n
We assume that the average daily unit price is the same for all the users, that is, user’s total cost is only proportional to his/her total amount of consumption. This implies that all users are equally treated in the coordination. More importantly, it suggested that any user is able to find an optimal schedule and cost by optimizing the cost of the whole group. This relationship can be used to design the coordination game. Referring to [5], we define the coordinative scheduling game as follows: the players of the game are the N individual users connected in the neighbourhood/local area network. The strategies are the daily consumption schedules L that each player chooses. The payoffs of every player is the ‘minus’ group cost since the maximum payoff is achieved by minimizing the cost C. According to the concepts of game theory [8], at any particular time, given the knowledge of other players’ current consumption schedules (their strategies) L−n = [L1 , L2 , . . . , Ln−1 , Ln+1 , . . . , LN ] and the price plan P, player n is able to play a best response which can maximize his payoff and the best response will be the result of the following problem min C|L−n . (13) Ln,h ∈R
The game will be played repeatedly by every player in the group. Players broadcast their current strategies to the whole network after each play. If every player plays the best response to each other, the game has a potential to reach its Nash Equilibrium where no players will be able to play better and achieve a lower cost. Hence no player has the incentive to change their schedule again. The game is claimed to have a ‘solution’ at that point and hence the scheduling coordination
h∈H
xn,a,h +
a∈A−T 1T xn,a
−n
xn,t,h ≤ Ln,h , ∀h ∈ H,
t∈T
= ln,a , ∀a ∈ A − T,
xn,a,h ≥ 0, xn,a,h ≥ δa , ∀a ∈ F, ∀h ∈ [has , . . . , haf ], αa ≤ xn,a,h ≤ βa , ∀a ∈ S, ∀h ∈ [has , . . . , haf ],
T
where ωh ∈ ω = [ω1 , ω2 , . . . , ω24 ] represents the basic ToU price rate which can be hourly different and φh is the additional fee charged by the operators. We further define user n’s scheduled daily consumption results as Ln = [Ln,1 , Ln,2 , . . . , Ln,24 ]T , where Ln,h is the scheduled hourly load for user n. The daily cost for the group of users can be calculated as
0 ≤ st ≤ 1, 1T st = 1, ∀t ∈ T, xt = PTt st , ∀t ∈ T.
(14)
The problem is similar to (10) and it can also be solved using Branch and Bound method. However, the objective here becomes the cost not the load itself. In addition, since L−n,h is gathered from other smart meters’ scheduled results, the optimization variables will be only local variables xn,a,h and st . This optimization process is to be conducted by the individual user n locally, the computation complexity is lower than that of the centralised scheme. It is worth pointing out that although the cost function (12) is a convex function and xn,a,h is a continuous variable, after including integer variable st , the game should be treated as a non-continuous game. The existence and uniqueness of the equilibrium points of the game cannot be guaranteed as in some of the continuous games. There may be none or multiple equilibrium points in the game. However in this paper, we demonstrate the effectiveness of the coordination game using simulation results. III. S IMULATION RESULTS A set of system simulations has been carried out using Matlab to implement the proposed scheduling mechanism in both the centralised and the distributed coordination scenarios. In the simulations, we first define a set of home appliances and their individual requirements as listed in Table I. According to the settings, we are able to formulate the total power requirement and the individual consumption patterns to the corresponding constraint formats. For example for the hob and oven, we can use the expression in (5) and make the constraint as x1,h ≥ 1, ∀h ∈ [19, 20]. Similarly for the power-shiftable electric vehicle battery charger, we can write 0.1 ≤ x5,h ≤ 3, ∀h ∈ [1, 8] ∪ [20, 24]. For the time-shiftable washing machine, we defined the power pattern as p6 = [1, 0.5, 0, · · · , 0]T and the switch vector s6 to formulate the constraint as x6 = PT6 s6 . After incorporating all the constraints together, we can solve the ILP problem in (10) and achieve the optimal scheduling.
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Type
1.Hob and oven 2.heater
Nonshiftable
3.Fridge and freezer 4.Water boiler
Nonshiftable Powershiftable Powershiftable
5.Electric vehicle (15 miles daily driving ) 6.Washing machine
Timeshiftable
7.Dish washer
Timeshiftable
1.5
User preference and power requirement Operating period: 7pm-8pm(hob/oven) 9pm-10pm, 3am-5am(heater) hourly consumption: 1kWh Operating 24hrs hourly consumption: 0.12kWh Hourly consumption: 0-1.5kWh daily requirement: 3kWh Preferred charging period: 8pm-8am charging power: 0.1kW-3kW daily requirement: 5kWh [9] Operating 2hrs, once per day 1kWh for the 1st hr 0.5kWh for the 2nd hr power: 0.8kWh for 1 hour daily requirement: 0.8kWh
app1
Power consumption (kwh)
Name
app3
app4
app5
app6
app7
1
0.5
0
0
2
4
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10
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14
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Time (hour)
Fig. 3.
Optimal schedule for individual appliances.
2.5
Scheduled load
TABLE I A PPLIANCES AND POWER CONSUMPTION PATTERNS Power consumption (kwh)
2
1.5 Scheduled load
Power consumption (kwh)
app2
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1.5
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0.5
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0
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Time (hour)
Fig. 4.
0
1
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5
Fig. 2.
6
7
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9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (hour)
Hourly power consumption schedule.
Fig. 2 depicts the scheduled hourly load. It can be seen that the maximum hourly load of 1.22kWh appears during the hours of 3-5, 14 and 19-22. From the appliances consumption requirement, we can see that when the heater is turned on between 21 and 22, and considering the power of the fridge and the charging of electric vehicle battery, the required load is 1 + 0.12 + 0.1 = 1.22 (kWh). For any time of the day, the scheduled load is no more than 1.22kWh. Hence we claim that the optimal hourly scheduling with a minimum possible peak is achieved. Fig. 3 depicts the scheduling result for the individual appliances. Clearly we can see that during 9-10pm, the heater (app2) is consuming fixed 1kWh and the fridge (app3) is consuming 0.12kWh. The water boiler (app4) has been totally shut down and only 0.1kWh which is the minimum charging requirement for the electric vehicle (app5) is provided. Timeshiftable appliances were not scheduled to operate during the period. Indeed, the washing machine (app6) and the dish washer (app7) are both scheduled in the day time when there is lower demand from other appliances. Now consider a small neighbourhood area network scenario
Hourly power consumption schedule for multiple households.
with 4 households and a total of 43 units daily demand. Each household has similar appliances as in the previous case but with different load requirements, power patterns and consumer preferences. Now a centralised scheduling which will meet all the requirements is achievable. Fig. 4 depicts the scheduling result. As seen, the optimized hourly peak load is 2.14kWh and the overall load allocation is more balanced compared to the previous case. The peak to average ratio is 2.14/(43/24) = 1.19, which means the peak load is just 19% higher than the daily average. It can be observed that the mechanism is able to improve the reliability of the grid due to reducing the peak to average load. In the next step, we implement the scheduling coordination game in a distributed control scenario with 10 connected individual users. The daily demand of the group is 260 units. A basic ToU price plan was set. The plan has higher rate during the daytime and include two peak periods at 9-10am in the morning and 5-6pm in the evening. Users randomize their schedules as initial strategies and play the coordination game by solving problem (14) at random times. Fig. 5 depicts the convergence of the proposed game and shows the effectiveness of the distributed coordination. We can observe that after 17 iterations, the optimal cost of the whole group has converged to £36.69. The game indeed has an equilibrium and is able to converge at a reasonably fast speed.
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aims at finding the optimal schedule to comply with the price plan which was designed mainly for load control purpose. If the operator considers the peak at night is harmful for the grid, it will probably change the price plan according to the response. Besides, we can also see in Fig. 5, if the users quit the game (after 20 iterations) and tried to schedule separately instead, the optimal cost is £38.20. They would not able to obtain lower cost due to the loss of information of the others. The scheduled load is shown in Fig. 7. As seen, more loads are scheduled in the peak time. Therefore, it is better off for all the users to join the coordination game. At this stage, the goal of load control can be achieved. Furthermore, from the simulations we note that there exist different schedule results to achieve the same optimal cost even if the game has converged. We hence considered that this game has multiple equilibrium points. As discussed in the previous section, this might be caused by the non-continuous nature of the game. The current work is focused on theoretical analyzing the behaviour of this game in terms of uniqueness of the Nash Equilibrium.
65 coordinative scheduling individual scheduling 60
Cost (GBP)
55
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Fig. 5.
15
20 Iteration
25
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Optimal cost of the coordination game.
25 scheduled load
Power consumption (kWh)
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IV. C ONCLUSION 5
0
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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (hour)
Fig. 6.
Optimal schedule of the coordination game.
In this scenario, in order to form a coordinative scheduling, only an average of 2 rounds of play per user is required. Fig. 6 depicts the coordinative schedule result of the whole group. As seen, the minimum loads are scheduled during the high price periods and more demand are scheduled during the night time when unit rate is lower. However, comparing with the centralised scheduling result as in Fig. 4, the peak load in Fig. 6 is 20.40kWh and the peak to average ratio is 1.88 which is much higher. It seems the coordination approach did not perform better in terms of peak load reduction. This is because the scheduling method no longer focuses on balancing the load over the day to directly reduce the peak. Instead, it 20 scheduled load 18 16
Power consumption (kWh)
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Fig. 7.
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We have proposed an ILP and game theory based optimization mechanism for the home demand-side management in a smart grid. The proposed mechanism is able to schedule both the optimal power and the optimal operation time for powershiftable appliances and time-shiftable appliances respectively according to user preference and the power consumption patterns of individual appliances. The mechanism can be applied to both centralised control and distributed coordinative management schemes. Simulation results demonstrate the convergence of the game theoretical algorithm. R EFERENCES [1] F. Li, et al., “Smart transmission grid: Vision and framework,” IEEE transactions on Smart Grid, vol. 1, no. 2, pp. 168–177, Sep.2010. [2] US DoE, “Smart Grid Research & Development: Multi-Year Program Plan(MYPP) 2010-2014,” Available http://www.oe.energy.gov. [3] US DoE, “Benefits of demand response in electricity markets and recommendations for achieving them,” Report to the US Congress., Feb. 2006. Available http://eetd.idi.gov. [4] V. Bakker, M. Bosman, A. Molderink, J. Hurink, and G. Smit, “Demand side load management using a three step optimization methodology,” in 1st IEEE International Conference on Smart Grid Communications, p. 431436, Gaithersburg, Maryland, USA, Oct. 2010. [5] A. Mohsenian-Rad, V. Wong, J. Jatskevich, R. Schober, and A. LeonGarcia, “Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid,” IEEE transactions on Smart Grid, vol. 1, no. 3, pp. 320–331, Dec. 2010. [6] J. Medina, N. Muller, and I. Roytelman, “Demand response and distribution grid operations: Opportunities and challenges,” IEEE Transactions on Smart Grid, vol. 1, no. 2, pp. 193 –198, Sep. 2010. [7] G. L. Nemhauser and L. A. Wolsey, Integer and combinatorial optimization. John Wiley and Sons, New York, 1998. [8] M. J. Osborne, An introduction to game theory. Oxford University Press, 2004. [9] Renault. Fluence Z.E. Specifications, http://www.renault-ze.com/.
Non-coordinative scheduling.
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