Optimal and Autonomous Incentive-based Energy Consumption Scheduling Algorithm for Smart Grid

Transcription

1 1 Optimal and Autonomous Incentive-based Energy Consumption Sceduling Algoritm for Smart Grid Amir-Hamed Mosenian-Rad, Vincent W.S. Wong, Juri Jatskevic and Robert Scober Department of Electrical and Computer Engineering Te University of Britis Columbia, Vancouver, Canada {amed, vincentw, jurij, Abstract In tis paper, we consider deployment of energy consumption sceduling () devices in smart meters for autonomous demand side management witin a neigborood, were several buildings sare an energy source. Te devices are assumed to be built inside smart meters and to be connected to not only te power grid, but also to a local area network wic is essential for andling two-way communications in a smart grid infrastructure. Tey interact automatically by running a distributed algoritm to find te optimal energy consumption scedule for eac subscriber, wit an aim at reducing te total energy cost as well as te peak-to-average-ratio (PAR) in load demand in te system. Incentives are also provided for te subscribers to actually use te devices via a novel pricing model, derived from a game-teoretic analysis. Simulation results confirm tat our proposed distributed algoritm significantly reduces te PAR and te total cost in te system. I. INTRODUCTION According to a report by te U.S. Department of Energy in 8 [1], 74% of te nation s electricity consumption occurs in buildings. Tis represents 39% of te total energy consumption among all sectors. Currently, te electricity consumption is not efficient in most buildings, leading to te waste of billions of dollars and a major amount of extra greenouse gas emissions. Tere are two general approaces for energy consumption management in buildings: reducing consumption and sifting consumption []. Te former can be done troug raising awareness among subscribers for more careful consumption patterns as well as constructing more energy efficient buildings, e.g., wit better eat isolations, less energy consuming ligting, etc. However, tere is also an important need for practical solutions to sift ig-load ouseold appliances to off-peak ours in order to reduce te peak-to-average ratio (PAR) in load demand. Appropriate load-sifting is foreseen to become even more crucial as plug-in ybrid electric veicles (PHEVs) become more popular. Most PHEVs need. -.3 kw of carging power for one mile of driving [3]. Tis will represent a significant new load on te existing distribution system. In particular, during te carging time, te PHEVs double te average ouseold load. Unbalanced conditions resulting from an increasing number of PHEVs will drastically exacerbate te already ig PAR of te load demand, leading to a degradation of te power quality, voltage problems, and even potential damage to utility and consumer equipment [3]. Load management, also known as demand side management [4] [6], as been practiced since te early 198s in different forms suc as direct load control and small-scale voluntary load management programs, wit varying degrees of success. However, tanks to te advancements in smart metering tecnologies [7] and te increasing interest in smart grid infrastructure (cf. [3], [8] [1]) wit two-way digital communication capability troug computer networking, we can pus a modernized load management system forward and introduce energy consumption sceduling () devices (e.g., as part of a smart meter) tat can optimally coordinate te timing of ouseold energy consumption in eac neigborood, a ig-rise building, or a large PHEV parking lot, troug communication among devices and also between te devices and te control and dispatc centers. Despite te importance of an efficient energy consumption sceduling system, suc large-scale sceduling plans cannot be implemented unless intelligent pricing scemes are used to provide incentives for te subscribers to follow tem. Te incentives can be in form of lower utility carges. In tis paper, we consider a scenario were a source of energy (e.g., a generator or a step-down substation transformer wic is connected to te grid) is sared by several subscribers, eac one equipped wit an device. Te devices are deployed inside te smart meters and are connected to not only te power line, but also to a communication network wic is essentially needed to andle two-way data communications. Te devices interact automatically by running a distributed algoritm to find te optimal energy consumption scedule for eac subscriber, wit an aim at reducing te PAR in te system. Interestingly, we can sow wit a game-teoretic analysis (cf. [11]) tat a simple pricing mecanism can provide te subscribers wit te incentives to cooperate in order to not only improve te system overall performance, but also to pay less individually. In oter words, troug an appropriate pricing sceme, te Nas equilibrium of an energy consumption game among te subscribers wo are saring a common energy source will be te exact global optimal solution of a system-wide optimization problem, making our design optimal and practical. Te rest of tis paper is organized as follows. We introduce te system model and notations in Section II. Tis includes an elaborate matematical formulation of te energy consumption sceduling problem as a convex optimization problem. We discuss te requirements of a valid billing sceme and also introduce te concept of an energy consumption game in Section III. Our distributed algoritm to be executed by te devices is presented in Section IV. Simulation results are given in Section V. Te paper is concluded in Section VI. All analytical proofs are provided in te Appendices.

2 Energy Source C (.) S l 1 l l N-1 S S 1 N-1 S N l N LAN Power Line E n,a α n,a γn,a min γn,a max TABLE I SYSTEM PARAMETERS TO BE SET FOR EACH APPLIANCE a A n BY EACH SUBSCRIBER n N. Total energy to be sceduled. Beginning of te time interval tat consumption can be sceduled. End of te time interval tat consumption can be sceduled. Minimum sceduled power level. Maximum sceduled power level. Fig. 1. A sample smart grid system wit N load subscribers. A. Power System II. SYSTEM MODEL Consider a smart power system wit several load subscribers and one source of energy, e.g., a generator or a step-down substation transformer connected to te electric grid. An example of suc a system is illustrated in Fig. 1. We assume tat eac subscriber is equipped wit an device in its smart meter for sceduling te ouseold energy consumption. Te subscribers are all connected to te power line (solid line) coming from te energy source. Te devices are also connected to eac oter and also to te energy source troug a local area network (LAN) (dased line). Let N denote te set of subscribers, were N N. For eac subscriber n N, let A n denote te set of appliances: waser/dryer, refrigerator, PHEVs, etc. For eac appliance a A n, we define energy consumption sceduling vector x n,a [x 1 n,a,..., x H n,a], (1) were H = 4 ours. For eac our of te day H {1,..., H}, real-valued scalar x n,a denotes te corresponding one-our energy consumption tat is sceduled for appliance a from subscriber n. In tis case, we can define te total ourly energy consumption for eac subscriber n N as l n a A n x n,a, H. () We also define E n,a H x n,a (3) as te total daily energy consumption for appliance a from subscriber n. Here, we assume tat E n,a is pre-determined and set by te load subscriber according to er needs. For example, E n,a = 16 kw for a plug-in ybrid electric sedan for a 4- mile daily driving range [3]. In fact, our designed sceduler aims not to cange te amount of energy consumption, but instead to systematically manage and sift it, e.g., in order to reduce te PAR. In tis regard, te subscriber also needs to select te beginning α n,a H and te end H of a time interval tat te energy consumption for appliance a is valid to be sceduled 1. Clearly, α n,a <. For example, a subscriber may select α n,a = 1 and = 8 for er PHEV suc tat te battery carging finises before 8: AM wen 1 Te model ere can be easily extended to te case wen a particular appliance is needed to be sceduled multiple times during te day. However, ere we focus on single-time sceduling for te ease of exposition. se needs to use te veicle. Tis imposes certain constraints on vector x n,a. In fact, it is required tat and =α n,a x n,a = E n,a, (4) x n,a =, {1,..., α n,a 1} { +1,..., H}. (5) Te time range set by te subscriber needs to be larger tan or equal to te time interval needed to finis te carging. For example, for a single-pase PHEV te normal carging time is 3 ours [3]; terefore, it is required tat α n,a 3. We note tat certain appliances may ave very strict sceduling requirements, for example, a refrigerator, may require operation all te time. In tat case, α n,a = 1 and = 4. Many ome appliances may ave some maximum power levels γn,a max, for eac a A n. For example, a PHEV may be carged only up to γn,a max = 3.3 kw per our [3]. Tis imposes te following upper-bound constraints on te coice of energy consumption sceduling vector x n,a for eac appliance a: x n,a γ max n,a, {α n,a,..., }. (6) Some appliances also ave minimum stand-by power levels γ min n,a, for eac a N. In tat case, it is furter required tat x n,a γ min n,a, {α n,a,..., }. (7) In certain cases, we may require sceduling discrete power levels. Tis can be fulfilled by eiter rounding te continuous value of x n,a to te required discrete power levels, or redefining x n,a as a discrete variable wit desired discrete levels. Discrete-level energy consumption sceduling is beyond te scope of tis paper. Te information needed to be set by subscriber n for appliance a is summarized in Table I. B. Energy Cost Consider te total load at eac our of te day H: L l n. (8) We define a cost function C (L ) indicating te cost of generating or providing energy by te energy source at eac our H. We first notice tat in general C 1 (L) C (L), 1, H, 1. (9) In oter words, te cost of te same load can be different at different times of te day. In particular, te cost can be less

3 Total Daily Cost (Dollars) (a) Step (Slope: 8.3 c/kw) Step 1 (Slope: 5.9 c/kw) Energy Consumption (kw) Total Daily Cost (Dollars) (b) Energy Consumption (kw) Fig.. Two sample convex and increasing cost functions: (a) Two-step conservation rate model used by BC Hydro [13]; (b) A quadratic cost function. during te nigt compared to te day time. Furtermore, we make te following assumptions: Assumption 1: Te cost functions are increasing in total per-our load. Tat is, for eac H, we ave C (ˆL ) C ( L ), ˆL L. (1) Te inequality in (1) simply implies tat te energy cost will always increase wen te total load increases. Assumption : Te cost functions are strictly convex. Tat is, for eac H, and any ˆL, L, we ave [1] C (θ ˆL + (1 θ) L ) θc (ˆL ) + (1 θ)c ( L ), (11) were θ 1. Examples of convex cost functions are sown in Fig.. A convex function can be a piece-wise linear function as in te two-step conservation rate used by Britis Columbia (BC) Hydro [13], as in Fig. (a); or a smoot differentiable quadratic function as in Fig. (b). Te cost function we assume is general and can represent eiter te actual energy cost or simply a cost model used by a utility company in order to impose a proper load-sifting. C. Optimization Problem Given complete knowledge about subscribers needs and a centralized control of te system in Fig. 1, an efficient energy consumption sceduling to be implemented by devices can be caracterized as te solution of te following problem: ) min x 1,...,x N s.t. H C ( a A n x n,a =α n,a x n,a =E n,a, a A n, n N, γ min n,a x n,a =, x n,a γ max n,a, H n,a, a A n, n N, H\H n,a, a A n, n N. (1) Here, for eac n N and any a A n, we ave H n,a {α n,a,..., }. (13) For eac subscriber n N, tensor x n is formed by stacking up energy consumption sceduling vectors x n,a. Te optimization problem in (1) is convex and can be solved using convex programming tecniques suc as te interior point metod (IPM) [1] in a centralized fasion. However, we are interested in solving problem (1) distributively at te devices wit minimum amount of information excanges among te devices and te energy source. In particular, we would like eac device be able to scedule te energy consumption at te ouseold according to te individual needs of te subscribers. It is also important to make sure tat te subscribers ave te incentive to actually use te devices and follow te scedules tey determine. A. Pricing and Billing III. ENERGY CONSUMPTION GAME For eac registered subscriber n N, let denote te daily amount in dollars to be carged to subscriber n by te utility company wic owns te energy source. In oter words, is te amount appears on te load subscriber n s bill at te end of te day. In general, we expect tat te following two key properties old for any billing model: 1) Property I: Clearly, we need to ave ) H, (14) C ( were te left and side denotes te total daily carge to te subscribers and te rigt and side denotes te total daily cost. In tis regard, we can define κ H C ( ) 1. (15) l n If κ = 1, ten te billing system is budget balanced and te utility company carges te subscribers only wit te same amount tat generating/providing energy costs for te utility. ) Property II: It is expected tat te carges for eac subscriber to be proportional to te er total daily load. Tat is, we ave H ln, n N. (16) In oter words, a fair billing leads to te following equality: b m = H l n H l m l n, n, m N. (17) After summing up (17) across all subscribers m N, for eac n N, we ave b m = ( ) H H b l m n H = b l m n H. l n l n (18)

4 Finally, from (), (4), (15), and (18), we ave ( ) H = l n H b m l m κ ( H H ( )) = l n H C l l m m κ ( a A = n E H ( n,a C a A m E m,a a A m x m,a )). (19) In oter words, te only billing model tat satisfies te axiomatic requirements in (14) and (17) is te model in (19). B. Energy Consumption Game From (19), te carge on eac subscriber would depend on ow se and oter subscribers scedule teir consumptions. Tis leads to te following game among subscribers: Game 1 (Energy Consumption Game Among Subscribers): Players: Registered subscribers in set N. Strategies: Energy consumption sceduling vectors x n for all subscribers and appliances. Payoffs: P n (x n ; x n ) for eac subscriber, were P n (x n ; x n ) = κ a A = n E n,a a A m E m,a ( H C ( a A m x m,a )) Here, x n denotes te energy consumption sceduling vectors for all subscribers oter tan subscriber n. In Game 1, te subscribers try to select teir energy consumption scedule to minimize teir payments to te utility. Teorem 1: Suppose te cost functions C ( ) are increasing and strictly convex for eac H. Te Nas equilibrium of Game 1 always exists and is unique. Te proof of Teorem 1 is given in Appendix A. Note tat Nas equilibrium is a solution concept in game teory tat caracterizes ow te players play a game [11]. Te energy consumption sceduling variables (x n, n N ) form a Nas equilibrium for Game 1 if and only if for eac n N, P n (x n; x n) P n (x n ; x n), x n. () If te energy consumption game is at its unique Nas equilibrium, ten none of te subscribers would try to deviate from scedule (x n, n N ). Next we sow te following key result on te performance at Nas equilibrium of Game 1. Teorem : Te unique Nas equilibrium of Game 1 is te optimal solution of problem (1). Te proof of Teorem is given in Appendix B. From Teorem, as long as te cost functions C ( ) are increasing and strictly convex for eac H and also te price model satisfies te axiomatic requirements (14) and (17), te. subscribers ave all te incentives to cooperate wit eac oter in order to solve te energy consumption management problem in (1) leading to te best possible energy consumption sceduling wit load-sifting and low PAR properties. IV. DISTRIBUTED ALGORITHM From te results in Section III, te subscribers would be willing to cooperate and allow teir devices scedule teir ouseold energy consumption to pay less. In particular, we sowed tat te unique Nas equilibrium of te energy consumption game among te subscribers is indeed te same as te global optimal solution of energy consumption sceduling problem (1). In tis section, we provide a simple algoritm to be implemented in eac device to reac te Nas equilibrium of Game 1 and acieve te optimal performance. Consider an arbitrary subscriber n N. Given x n and assuming tat all oter subscribers fix teir energy consumption scedule according to x n, subscriber n can maximize its own payoff by solving te following local problem: max x n P n (x n ; x n ) s.t. =α n,a x n,a = E n,a, a A n, γn,a min x n,a γn,a max, H n,a, a A n, x n,a =, H\H n,a, a A n. (1) Notice tat ere x n is te only vector variable. Since κ a An E n,a a Am E is fixed and does not depend on te coice m,a of x n, te maximization in (1) can be replaced by a minimization over H C ( a A m xm,a). Terefore, we can replace te maximization in problem (1) equivalently wit te following minimization ( ) H min C x m,a. () x n a A m We notice tat Problems () and (1) ave te same objective functions. Problem () as only local variables to subscriber n. Problem () is convex and can be solved by IPM [1]. Te above observations motivate us to propose Algoritm 1 to solve problem (1). Algoritm 1 works based on te coordinate ascent metod [14], were we fix sceduling variables across all subscribers except for te subscriber n, and minimize te total cost H C ( ) a A m x m,a only wit respect to x n as in (). Tis procedure is repeated, leading to an iterative algoritm across te subscribers. Next, we explain ow Algoritm 1 works. In Line 1, eac subscriber starts wit random initial conditions. Ten, te loop in Lines to 11 continues until te algoritm converges. Witin tis loop, eac device solves te local problem () using IPM in Lines 4 and 5 and ten announces its updated scedule to oter devices in Line 6. It also updates its local memory wenever it receives a control message from oter subscribers in Line 9. Let T n denote te set of time instances at wic subscriber n N solves local problem (). We assume tat:

5 Algoritm 1 : Executed by eac subscriber n N. 1: Initialization. : repeat 3: if time t T n ten 4: Solve local problem () using IPM [1]. 5: Update x n according to te solution. 6: Broadcast a control message to announce x n to te oter devices across te LAN. 7: end if 8: if a control message is received ten 9: Update x n accordingly. 1: end if 11: until no device announces cange of scedule. (a) For any subscriber n m, we ave T n T m = {}. Tat is, te iterative local maximizations are carried out successively as in te Gauss-Seidel mapping [14, p. 1]. (b) Tere is a constant T max suc tat for eac subscriber n N, tere exist time instances t 1, t T n suc tat t 1 t T max. In oter words, all subscribers update teir transmission probabilities at least once every T max seconds. Tese assumptions guarantee te asyncronous convergence of Algoritm 1 to some fixed point [14, Proposition.5, p. 8]. Te convergence property is directly resulted from te coordinate ascent structure of te algoritm and te Gauss- Seidel updates. Now te questions are: (1) Starting from different randomly selected initial scedules, does Algoritm 1 always converge to te same point? () Wat is te performance of a fixed point tat Algoritm 1 may converge to? Since eac subscriber updates its energy consumption sceduling variables in Algoritm 1 to maximize its own payoff P n (x n ; x n ), te fixed point of Algoritm 1 is te Nas equilibrium of Game 1. From Teorem 1, te Nas equilibrium of Game 1 is unique. Tis directly answers our first question: Algoritm 1 always converges to te unique Nas equilibrium of Game 1. Moreover, from Teorem, te unique Nas equilibrium of Game 1 is te optimal solution of problem (1). Tis also answers te second question: Algoritm 1 reaces te optimal performance wit respect to solving te energy consumption sceduling problem in (1). V. SIMULATION RESULTS In tis section, we present te simulation results and assess te performance of our proposed algoritm. In our model, te example power system at Fig. 1 is assumed to ome 1 load subscribers, N = 1. For te purpose of study, eac subscriber is selected randomly to ave between 1 to appliances wit ard energy consumption sceduling constraints. Suc appliances include refrigerator-freezer (daily usage: 1.3 kw), electric stove (daily usage: 1.89 kw for self-cleaning and.1 kw for regular), ligting (daily usage for 1 standard bulbs: 1. kw), eating (daily usage: 7.1 kw), etc. [15]. Moreover, eac subscriber is selected randomly to also ave between 1 to appliances wit soft energy consumption sceduling constraints. Recall tat te devices may scedule only te appliances wit soft Energy Consumption (kw) Cost (Dollars) (a) Day (b) Nigt Fig. 3. Sceduled energy consumption and corresponding cost wen devices are not used. In tis case, PAR is.1 and total daily cost is $ energy consumption sceduling constraints. Suc appliances include diswaser (daily usage: 1.44 kw), clotes waser (daily usage: 1.49 kw for energy-star 1.94 kw for regular), clotes dryer (daily usage:.5 kw), and PHEV (daily usage: 9.9 kw), etc. [3], [15]. Te sceduling durations (i.e., valid stop and ending sceduling times) are selected randomly witin a 4 our period. Te simulation time is also 4 ours, starting from 7: AM in te morning at one day until 7: AM in te next morning. As discussed in Section II-B, we assume tat te cost functions are increasing and strictly convex as depicted in Fig.. We select te cost functions to be quadratic: C (L ) = φ Day (L ) during te day and C (L ) = φ Nigt (L ) at nigt, were < φ Nigt φ Day are constant. Witout loss of generality, we select φ Nigt = 1 φ Day and φ Day =.1875 cents/kw. In tat case, te cost function during te day becomes as in Fig. (b). Assuming a budget-balanced system, we set parameter κ = 1. Last but not least, we assume tat day-time cost applies to te first 16 ours of te simulation period (i.e., from 7: AM to 11: PM) and te nigt-time cost applies to te last 8 ours of te simulation period (i.e., from 11: PM to 7: AM on te next day). Te simulation results on total sceduled energy consumptions and total cost in te system witout and wit te deployment of devices are sown in Figs. 3 and 4, respectively. As sown ere, wen te devices are not used, te PAR is.1 and te total energy cost is $ At te same time, wen te devices are used, te PAR reduces to 1.3 (i.e., 38.1% less) and te total energy cost reduces to $53.81 (i.e., 37.8% less). In fact, we ave more even load in te latter case. Note tat eac subscriber consumes te same amount of energy in te two cases, but it simply scedules its consumption more efficiently in te case tat te devices are used. In tis case, all subscribers will even pay less to te utility company as sown in Fig. 5. Terefore, te subscribers would be willing to participate in te proposed automatic demand side management system.

6 Energy Consumption (kw) Cost (Dollars) (a) Day (b) Nigt Fig. 4. Sceduled energy consumption and corresponding cost wen devices are deployed. In tis case, PAR is 1.3 and total daily cost is $ Daily Utility Carge (Dollars) Fig No Deployment Deployment Load Subscriber Daily carges for eac subscriber witout and wit deployment. VI. CONCLUSIONS In tis paper, we proposed an optimal, autonomous, and incentive-based energy consumption sceduling algoritm to balance te load among residential subscribers tat sare a common energy source. Te proposed algoritm is designed to be implemented in energy consumption sceduling () devices inside smart meters in a smart grid infrastructure. We also proposed a simple pricing and billing model wic provides te incentives for te subscribers encouraging tem to actually use te devices and run te proposed distributed algoritm in order to be carged less. Simulation results confirm tat our proposed algoritm significantly reduces te PAR as well as te total energy cost in te system. A. Proof of Teorem 1 APPENDIX We first notice tat since C ( ) is strictly convex for eac H, te payoff function P n (x n ; x n ) is strictly concave wit respect to x n. Terefore, Game 1 is a strictly concave N- person game. In tis case, te existence of a Nas equilibrium directly results from [16, Teorem 1]. Moreover, te Nas equilibrium is unique due to [16, Teorem 3]. B. Proof of Teorem We first sow tat te global optimal solution of problem (1) forms a Nas equilibrium for Game 1. For notational simplicity, let x 1,..., x N denote te optimal solutions for problem (1). We also define ) H C. (3) C ( a A m x m,a By definition of optimality, for eac subscriber n N and n we for any arbitrary x, ave H C C m,a +. (4) x \{n} a A m a A n x n,a After multiplying bot sides in (4) by negative constant κ a An E n,a, it becomes a Am E m,a P n (x n; x n) P n (x n ; x n), x n. (5) Comparing (5) and (), we can conclude tat te optimal solution x 1,..., x N forms a Nas equilibrium for Game 1. However, from Teorem 1, Game 1 as a unique Nas equilibrium. Tus, te optimal solution of problem (1) is equivalent to te Nas equilibrium of Game 1. REFERENCES [1] U.S. Department of Energy, 8 Buildings Energy Data Book. Energy Efficiency and Renewable Energy, Mar. 9. [] Energy Conservation Committee Report and Recommendations, Reducing Electricity Consumption in Houses. Ontario Home Builders Association, May 6. [3] A. Ipakci and F. Albuye, Grid of te future, IEEE Power and Energy Magazine, pp. 5 6, Mar. 9. [4] M. Farioglu and F. L. Alvardo, Designing incentive compatible contracts for effective demand managements, IEEE Trans. Power Systems, vol. 15, no. 4, pp , Nov.. [5] B. Ramanatan and V. Vittal, A framework for evaluation of advanced direct load control wit minimum disruption, IEEE Trans. Power Systems, vol. 3, no. 4, pp , Nov. 8. [6] M. A. A. Pedrasa, T. D. Spooner, and I. F. MaxGill, Sceduling of demand side resources using binary particke swarm optimization, IEEE Trans. Power Systems, vol. 4, no. 3, pp , Aug. 9. [7] R. Krisnan, Meters of Tomorrow, IEEE Power and Energy Magazine, pp. 9 94, Mar. 8. [8] U.S. Department of Energy, Te Smart Grid: An Introduction, 9. [9] A. Vojdani, Smart integration, IEEE Power and Energy Magazine, pp. 7 79, Nov. 8. [1] L. H. Tsoukalas and R. Gao, From smart grids to an energy internet: Assumptions, arcitecrures, and requirements, Apr. 8. [11] D. Fudenberg and J. Tirole, Game Teory. Te MIT Press, [1] S. Boyd and L. Vandenberge, Convex Optimization. Cambridge University Press, 4. [13] BC Hydro, Electricity Rates, 9. [14] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Metods. Prentice Hall, [15] Office of Energy Efficiency, Natural Resources Canada, Energy Consumption of Major Houseold Appliances Sipped in Canada, Dec. 5. [16] J. B. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, vol. 33, pp , 1965.