IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016 189 Demand Response Management in the Smart Grid in a Large Population Regime Sabita Maharjan, Member, IEEE, Quanyan Zhu, Member, IEEE, Yan Zhang, Senior Member, IEEE, Stein Gjessing, Member, IEEE, and Tamer Başar, Life Fellow, IEEE Abstract—In this paper, we introduce a hierarchical system model that captures the decision making processes involved in a network of multiple providers and a large number of consumers in the smart grid, incorporating multiple processes from power generation to market activities and to power consumption. We establish a Stackelberg game between providers and end users, where the providers behave as leaders maximizing their profit and end users act as the followers maximizing their individual welfare. We obtain closed-form expressions for the Stackelberg equilibrium of the game and prove that a unique equilibrium solution exists. In the large population regime, we show that a higher number of providers help to improve profits for the providers. This is inline with the goal of facilitating multiple distributed power generation units, one of the main design considerations in the smart grid. We further prove that there exist a unique number of providers that maximize their profits, and develop an iterative and distributed algorithm to obtain it. Finally, we provide numerical examples to illustrate the solutions and to corroborate the results. Index Terms—Consumer welfare, demand response management (DRM), large population, profit optimization, Stackelberg game. I. I NTRODUCTION HE GROWING demand of electricity, the aging infrastructure, and the increasing greenhouse gas emission are some of the challenges with the traditional power grid. Recent blackouts [1] have further corroborated these issues, and have fueled the need to transform the traditional power T Manuscript received August 10, 2014; revised September 22, 2014, January 22, 2015, and April 26, 2015; accepted April 29, 2015. Date of publication June 1, 2015; date of current version December 19, 2015. This work is supported in part by the projects 240079/F20, funded by the Research Council of Norway, and in part by the European Commission FP7 Project CROWN under Grant PIRSES-GA-2013-627490. The work of Q. Zhu was supported in part by the Department of Energy Grant, and in part by the National Security Agency through the Information Trust Institute, University of Illinois. The work of T. Başar was supported by the U.S. Air Force Office of Scientific Research Multi-Disciplinary University Research Initiative under Grant FA9550-10-1-0573. Paper no. TSG-00800-2014. S. Maharjan is with Simula Research Laboratory, Fornebu 1364, Norway. Q. Zhu is with the Department of Electrical and Computer Engineering, Polytechnic School of Engineering, New York University, Brooklyn, NY 11201 USA (e-mail: quanyan.zhu@nyu.edu). Y. Zhang and S. Gjessing are with Simula Research Laboratory, Fornebu, Norway, and also with the Department of Informatics, University of Oslo, Oslo 1325, Norway (e-mail: sabita@simula.no; yanzhang@simula.no; steing@simula.no). T. Başar is with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA, and also with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: basar1@illinois.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSG.2015.2431324 grid into a more responsive, efficient, and reliable system. The smart grid [2], widely believed to be the future power grid, offers improved efficiency, reliability, and environmental friendliness in power generation, transmission, distribution, consumption, and management, by integration of advanced information and communication technologies. Demand response management (DRM) is the response system of end users to changes in electricity prices over time or to other forms of incentives. In the smart grid, DRM plays a key role in improving different aspects of both supply and demand sides. For instance, DRM can result in lower bills and higher utility efficiency for end users. DRM can also reduce the cost of power generation or improve the revenues to retailers or utility companies (UCs). Existing studies on DRM have mostly focused on either adjusting the demand side by load shedding schemes when the supply is given, or on improving the profit for the supply side when aggregate user demand is available. Efficient DRM relies on both demand and supply sides, and an integrated framework is needed to consider DRM in a holistic manner. In this paper, we holistically investigate a planninglevel problem for both supply and demand sides, to cover a large population of end users. We study regional level demand response, incorporating the welfare of multiple providers and a large number of consumers. We establish a DRM model to capture the strategic behaviors of the UCs and power generation units from the supply side, and the consumers from the demand side, in one single framework as a Stackelberg game model. The DRM model consists of the following: 1) power generation units at the top level; 2) UCs at the middle level; 3) end users at the lowest level. The UCs and end users interact through the unit price determined by the UCs at the middle level based on the consumer parameters and the power generation costs. In order to capture different reliability and efficiency requirements of the end users, we break them into two groups: 1) residential; and 2) industrial consumers. The model allows us to develop insights into the outcome of the strategic interactions with different types of players in the power system. We analytically characterize the Stackelberg equilibrium (SE) solution to the game and provide closed-form expressions for the equilibrium. Based on the Stackelberg game model, we explore the DRM problem in a large population regime, and introduce a new dimension of improving providers’ profits and consumers’ welfare by adding UCs. Since, multiple distributed power generation units form one of the key components of the smart grid, such an alternative c 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 1949-3053 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 190 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016 is an important design consideration in the smart grid. Our contributions in this paper can be summarized as follows. 1) We introduce a Stackelberg game model, incorporating power generation units and UCs from the supply side and end users from the demand side. 2) We derive closed-form expressions to characterize the unique SE of the game and provide an iterative algorithm to reach the SE in a distributed manner. 3) We propose an extended framework of the Stackelberg game model in the large population regime. We prove that there exist a unique number of providers that maximizes the providers’ profits, and present a distributed algorithm to obtain the optimal number of providers. The rest of this paper is organized as follows. Related work is described in Section II. We introduce the system model in Section III. In Section IV, we formulate the problem as a Stackelberg game and prove the existence and uniqueness of the SE. The distributed algorithm to reach the SE is presented in Section V. In Section VI, we study the DRM problem for a large number of end users, and develop a distributed algorithm to find the optimal number of providers for maximizing providers’ revenues. We provide numerical results and discuss them in Section VII. Section VIII concludes this paper. II. R ELATED W ORK Several studies on demand side management and DRM have focused on either only one utility or a number of utilities treated as one entity [3]–[8]. Mohsenian-Rad et al. [3] have formulated an energy consumption scheduling problem as a noncooperative game among the consumers for increasing and strictly convex cost functions. Fan [4] has considered a distributed system where price is modeled by its dependence on the overall system load. Based on the price information, the users adapt their demands to maximize their own utility. In [5], a robust optimization problem has been formulated to maximize the utility of a user, taking into account price uncertainties at each hour. Wang and Groot [6] have exploited the awareness of the users and proposed a method to aggregate and manage end users’ preferences to maximize energy efficiency and user satisfaction. In [7], a dynamic pricing scheme has been proposed to provide incentives for customers to achieve an aggregate load profile suitable for UCs, and the demand response problem has been investigated for different levels of information sharing among the consumers. In [8], a multiresolution two-layer game is studied using mean-field game approach to incorporate inner interactions between users in the region and outer interactions between regions for dynamic distributed demand response in the smart grid. References [9] and [10] have also incorporated electric vehicles into the DRM framework. Some recent studies have explored the DRM problem with multiple providers (see [11]–[14]). In this paper, we study a planning-level problem in a large population regime by considering the demand side together with the market and supply side, bringing them into one single game framework. We note that, there is rich literature using Stackelberg games for congestion control, revenue maximization, cooperative transmission, and DRM [11], [14]–[17]. Our approach is similar to those in congestion control and power control, but our game model involves multiple providers and multiple consumers. Maharjan et al. [11] introduced a Stackelberg game with the supply of each UC fixed. When the number of consumers is large, the demand response problem unfolds new and unique challenges, e.g., very high costs and poor revenues for the UCs. In fact, the SE solution may not even exist for a given number of UCs. In this paper, we introduce a new dimension of improving DRM performance by bringing in additional providers to the smart grid. In addition, in our model, we have included demands from both residential and industrial consumers, and we have optimized power generation also. The model as well as the results are more generic than those in [11]. Our framework and game model can be related to [12] and [14]. Bu and Yu [14] have formulated a four stage Stackelberg game, but with only one provider, where as our model accommodates multiple providers. In [12], the interaction among multiple residential users is formulated as an evolutionary game. Each residential user has to choose one UC to purchase power. In reality, consumers do not choose the source of electricity or the power plant, and hence, do not require to purchase electricity from specific suppliers, especially for a time resolution of every hour. In fact, in a perfect competition market, the unit price charged by all the providers is the same, thus, not requiring consumers to buy power from a particular UC. In our model, the consumers play optimal response to the unit price without choosing a particular supplier, which is more realistic and closer to the current electricity market. To this end, compared to related existing literature, what differentiates this paper can be summarized in three main points. First, our focus is on a planning level problem, where the supply demand equilibrium concerns the optimal generation and consumption in a region, and not on the hourly or daily scheduling of individual appliances or individual consumers as in [3] and [13]. Second, different from the existing studies on power systems (see [17], [18]), we derive analytical and closed-form expressions for the unit prices, and consequently, the supply from the providers and the demand from the consumers, which enable us to provide meaningful insights into the behavior of and interactions between players at different levels in the system. Third and most important, we concentrate our analysis on a large population regime and introduce a new dimension of improving the system welfare by increasing the number of providers to accommodate the additional need of the consumers. III. S YSTEM M ODEL Fig. 1 depicts our hierarchical system model, which consists of three levels: 1) power generation units at the top level; 2) UCs at the middle level; and 3) residential and industrial consumers at lowest bottom level. The framework is motivated by the hierarchy of the real power grid system. The power generation units or power plants supply power, the UCs determine the unit price and optimal amount of power to supply, and MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME 191 However, our formulation can be easily modified to the case of imperfect competition. As the analysis for an imperfect competition-based framework can be built by modifying only certain parts of the formulation in Section IV, we provide a sketch of the analysis for the imperfect competition scenario, mainly emphasizing what should be modified, in the Appendix. IV. P ROBLEM F ORMULATION : S TACKELBERG G AME A PPROACH Fig. 1. Illustration of the interactions among the UCs and end users. the bottom level represents the demand response to the price signal from the residential consumers. The power generation units, UCs and the consumers have bidirectional communications support to exchange price and demand information. The data communication is carried out through the communication channel using wireless technologies. The solid lines in the figure represent power flow, whereas the dashed lines represent information flow. Now, to formulate our model in precise terms, we consider N end users, which we also call consumers, and K UCs, K := {1, 2, . . . , K}. Each UC receives power from one or multiple sources. Let Ck (Pk ) represents the cost of supplying Pk amount of power to the consumers, including the cost of generating the power. We assume Ck (Pk ) is increasing and convex ∀k, k ∈ K [3], [12]. The consumers’ side consists of NR residential users, NR := {1, 2, . . . , NR } and NI industrial users, NI := {1, 2, . . . , NI }, i.e., N = NR + NI . Let xR,n denotes the demand of residential user n. The power demand of the consumers depends on electricity price and consumer type. for each consumer n, the gain Specifically, function UR,n xR,n represents the satisfaction the consumer gets as a function of its power demand xR,n , which is nondecreasing and concave [10], [12]. The power demand from industrial consumers is normally on a different scale and they have relatively stricter power requirements. Thereby, we consider the total demand from all industrial consumers m ∈ NI , to be given: PI ≥ 0. The framework and the basic model for problem formulation and analysis, is that of a perfectly competitive market. In a perfectly competitive market, no market participant has the ability to influence the market price through its individual actions, i.e., the market price is a parameter over which the firms have no control. Consequently, each firm should increase its production up to the point where its marginal cost equals the market price. When each individual entity of a finite number of market participants (UCs) has noninfinitesimal influence in the market, it leads to imperfect competition. In an imperfect competition scenario, the UCs can charge consumers with different unit prices. In this paper, we concentrate on DRM analysis and modeling in a large population regime where the number of providers is usually large. In such a scenario, it is reasonable that a provider is likely not to have the ability to influence the market price through its individual action. Motivated by such a consideration, a perfect competition-based model is more realistic, and hence, relevant for our scenario. The end users are indirectly coupled through the unit price as a result of the competitions among the UCs at the market level. The generation units are also indirectly coupled through the price signal. These couplings between the decisions of the players make game theoretic approach an appealing one for cross-level understanding of a multiplayer multilevel complex system. Since the consumers respond to prices after UCs announce them, the hierarchical decision making process can be modeled as a Stackelberg game [19] where the UCs behave as leaders and the users react optimally to their strategies as followers. A. Demand Side Analysis Let y be the price per unit power. For given y, user n (n ∈ NR ) calculates its optimal demand response by solving the user optimization problem to maximize its welfare WR,n as follows: (1) max WR,n := UR,n xR,n − yxR,n xR,n s.t. xR,n ≥ xR,n,min (2) where xR,n,min is the minimum power requirement of consumer n. The above, that is, (1) and (2) characterizes a strictly convex optimization problem for given y. Hence, the stationary solution is unique and optimal. The first-order optimality condition for the optimizing residential user leads to (∂WR,n /∂xR,n,k ) = 0, ∀n ∈ NR , that is −1 UR,n = y, ⇒ xR,n = UR,n (3) (y). The condition required for constraint (2) to be satisfied can be established by substituting (3) into (2), which requires ] y ≤ [UR,n xR,n =xR,n,min, ∀n ∈ NR . This can be ensured if y ≤ ymax := UR,n . (4) min x =x n∈NR R,n R,n,min For the purpose of illustration and to provide functionspecific insights, we employ two widely adopted gain functions for residential consumers: 1) piecewise quadratic function [12]; and 2) logarithmic function [10]. We define the piecewise quadratic gain function of residential user n, (n ∈ NR ), as ⎧ 2 zR,n xR,n ⎪ vR,n ⎪ , if xR,n ≤ ⎨ vR,n xR,n − 2 zR,n UR,n xR,n = zR,n vR,n ⎪ ⎪ if xR,n > ⎩ 2vR,n zR,n (5) 192 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016 where vR,n and zR,n are user-specific parameters ∀n ∈ NR . In this case, (3) and (4), respectively, take the form vR,n − y (6) xR,n = zR,n and y ≤ ymax := min vR,n − zR,n xR,n,min . n∈NR (7) The logarithmic gain function can be defined for residential user n, (n ∈ NR ), as UR,n xR,n = αR,n ln βR,n + xR,n , ∀k ∈ K (8) where αR,n and βR,n are user-specific parameters. In this case, for given y, (3) and (4), respectively, take the form αR,n xR,n = (9) − βR,n y and αR,n . n∈NR βR,n + xR,n,min y ≤ ymax := min Then, the optimization problem for each UC is as follows: max RUC,k 0≤y≤ymax ,Pk ∈R+ s.t. PI + xR,n ≤ Pk k∈K (12) where Pk,max is the maximum power UC k can supply. When Pk,max is sufficiently large, PI , Pk << Pk,max . Then the second constraint can be relaxed. For a given y, (12) is a convex optimization problem. Given y, the first-order optimality condition for the UCs, (∂RUC,k /∂Pk ) = 0, gives the optimal amount of power to be supplied by UC k as −1 (13) Pk = Ck (y); ∀k ∈ K . Normally, if the power supplies of different UCs are given, the unit prices would be calculated based on the given power supplies. However, we are considering here a planning level problem where both unit price and optimal power to be supplied, are the variables. Thus, the optimal power of each UC is calculated by backward induction, based on the optimal unit price, a parameter obtained as a result of the profit optimization of the UCs. With the optimal demand response of residential users (3) and UCs’ power supply (13) in response to the price y, the objective of the UCs is to set the optimal price y. For supply demand equilibrium, it is required that Pk = PI + xR,n . (14) k∈K n∈NR Substituting (3) and (13) into (14) we obtain −1 −1 Ck UR,n (y) = PI + (y). k∈K n∈NR (15) y = (G1 )−1 (PI ) (16) −1 −1 where G1 (y) = k∈K (Ck ) (y) − n∈NR (UR,n ) (y). We employ a quadratic cost function for power generation [12], [13]. Let ak > 0 and bk , ck ≥ 0 be the coefficients of the cost function Ck (Pk ). Then, if the total power supplied by UC k is Pk , then the cost incurred to the UC is Ck (Pk ) = ak P2k + bk Pk + ck . (17) When the gain functions of the residential consumers are piecewise quadratic as given by (5), (16) takes the form bk vR,n + n∈NR 2ak zR,n . 1 1 + k∈K n∈NR 2ak zR,n PI + y= If the total power supplied by UC k is Pk , the profit of provider k is defined as RUC,k = yPk − C(Pk ). (11) n∈NR (10) B. Supply Side Analysis Pk ≤ Pk,max ; ∀k ∈ K From (15), we can obtain k∈K (18) Proposition 1: When the gain functions of the residential consumers are piecewise quadratic as given by (5), (18) is the unique feasible solution to the profit maximization problem (12) only if PI ≤ PI,max := min vR,n − zR,n xR,n,min n∈NR ⎛ ⎞ 1 1 ⎠ + ×⎝ 2ak αR,n k∈K n∈NR ⎛ ⎞ bk vR,n ⎠. −⎝ (19) + 2ak αR,n k∈K n∈NR Proof: Since ak > 0, bk , ck ≥ 0, ∀k ∈ K , vR,n , zR,n > 0, ∀n ∈ NR , and PI ≥ 0, (18), implies that y > 0. For given ak , bk , ck ∀k ∈ K , and vR,n , zR,n ∀n ∈ NR , substituting (18) into (7), we obtain bk vR,n + n∈NR 2ak zR,n ≤ min vR,n − zR,n xR,n,min . 1 1 n∈NR + n∈NR k∈K 2ak zR,n (20) PI + k∈K Further simplification of (20) yields (19). Remark 1: Note that, UCs may impose their own limits on the unit price, and usually there is a maximum limit the market imposes, i.e., yk,min ≤ y ≤ ym,max ∀k ∈ K . Without loss of generality, we consider ymax ≤ ym,max and y ≥ yk,min ∀k ∈ K . Proposition 2: When the gain functions of the residential consumers are logarithmic as given by (8), a unique feasible solution of (12) is −T1 + T1 2 + 8AAR , if (21) y= 2A AαR,n AR BA − PI ≤ min + BR − α 2 n∈NR 2 βR,n + xR,n,min min βR,n +xR,n R,n,min n∈NR (22) MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME where T1 = 2BR − BA − 2PI , A = k∈K 1/ak , AR = n∈NR αR,n , BR = n∈NR βR,n , and BA = k∈K bk /ak . Proof: Substituting Ck , UR,n from (17) and (8) into (15) and further simplification yields αR,n y − bk (23) = PI + − βR,n . 2ak y k∈K n∈NR The solution of (23)is y = (−T1 ± T1 2 + 8AAR /2A). Since A > 0 and T1 2 + 8AAR > T1 , the root y = (−T1 + T1 2 + 8AAR /2A) is the only real, positive one, and hence, feasible solution for y. Now, substituting (21) into (10) leads to αR,n −T1 + T1 2 + 8AAR . (24) ≤ min 2A n∈NR βR,n + xR,n,min Simplification of (24) yields αR,n 2A min + T1 ≥ T1 2 + 8AAR . (25) n∈NR βR,n + xR,n,min Squaring both sides of (25) and upon further simplification, (25) takes the form (22). Remark 2: If for any of the UCs, (Ck )−1 (y) > Pk,max , then instead of using (13), UC k supplies Pk = Pk,max . The power supply from UC k can, therefore, be expressed as Pk = min((Ck )−1 (y), Pk,max ). C. Stackelberg Equilibrium: Existence and Uniqueness Theorem 1: A unique SE exists in the hierarchical Stackelberg game. Proof: Given the prices y := {yk = y, ∀k ∈ K } and power supplies P := {Pk , ∀k ∈ K }, the optimal responses x̂ := {xR,n , ∀n ∈ NR } to y can be determined by solving the strictly convex problem (1) in Section IV-A for residential consumers. Given these unique responses, an equilibrium exists for the price setting game between the UCs if the following conditions hold. C1: y is a nonempty, convex, and compact subset of some Euclidean space R K . C2: RUC,k (y) is continuous in y and concave in yk , ∀k ∈ K . Here, yk ∈ [0, ymax ] and ∀k ∈ K . Thus, the strategy set is a nonempty, convex, and compact subset of the Euclidean space R K . From (11), we see that RUC,k is continuous in yk . Next, the second-order derivative of RUC,k with respect to yk is ∂ 2 RUC,k = −2ak < 0, ∀k ∈ K . ∂y2k (26) Hence, RUC,k (y) is concave in yk , i.e., in y. Note that, the supply from each UC is not dependent on the supplies of other UCs when y and x are fixed, and it is strictly concave in Pk , which leads to a unique solution (13). Since the optimal response of users are given by (3), using C1 and C2, we can conclude that an SE exists for the game. As proven in Section IV-B, there exists only one feasible solution for the price given by (16), provided that (19) or (22) is satisfied for the piecewise quadratic and the logarithmic gain functions, respectively. Therefore, the solutions (3), (13), and (16), constitute the unique SE of the hierarchical DRM scheme. 193 Algorithm 1 Distributed Algorithm for Optimal DRM 1: For t = 1, control unit: arbitrarily choose y1 and announce it to the UCs and the consumers. 2: Repeat for t=2,3, . . . 3: User n = 1, 2, . . . , NR , Do 4: Find xR,n,t from (3) for given yt , and report the demand to the control unit. 5: end 6: UCs k = 1, 2, . . . , K, Do 7: Update Pk,t = min((Ck )−1 (yt ), Pk,max ) for given yt , and report the power supply to the control unit. 8: end 9: Calculate yt+1 using (27). 10: If yt+1 = yt , 11: Send a no-change signal to the UCs and the consumers. 12: break. 13: else 14: Send the new value of price to the UCs and the consumers. 15: Go to 3. 16: end V. D ISTRIBUTED A LGORITHM In order to obtain the optimal demand response solution: (3), (13), and (16), a centralized solver needs to know the exact cost functions of the UCs and the gain functions of the consumers. Usually, UCs as well as consumers prefer not to share their private information [20], [21]. Moreover, distributed algorithms offer scalability. In this section, we present an algorithm in order to obtain the optimal demand response solution in a distributed and iterative way, starting from an arbitrary initial value [22]. For the distributed algorithm, subscript t is appended to all variables defined in the previous sections, as the time index, i.e., t ≥ 1 indicates the iteration number. We consider that an intermediate entity between the supply and demand sides such as a control unit, finds the optimal demand response solution without knowing the cost and gain functions. The control unit starts with an initial unit price y1 ≥ 0 and announces it to the UCs and the consumers. Based on the unit price, the residential consumers compute their demands xR,n,t using (3), and is sent to the control unit. The power demand from the industrial consumers PI is known to the control unit. Each UC also computes its power supply Pk,t as Pk,t = min((Ck )−1 (yt ), Pk,max ), and announces its power supply to the control unit. Then, the control unit updates the unit price for the next iteration based on the difference between the total power supply and the total power demand as yt+1 = yt + n∈NR xR,n,t + PI − σ k∈K Pk (27) where σ , a sufficiently large positive number, adjusts the convergence speed of the algorithm. The process repeats until the unit price (and consequently the supplies and demands) remains the same as in the previous iteration. The details are presented in Algorithm 1. 194 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016 VI. L ARGE P OPULATION R EGIME In Section IV, we derived closed-form expressions for the unit price y: (16), optimal demands from residential consumers xR,n : (3), optimal power supply Pk from the UCs: (13), and provided that certain conditions are valid: (19) and (22). In the low to moderate demand markets, such conditions may be readily satisfied. However, in a large population regime the total demand from residential and industrial consumers can be very high. This happens if either there is a large population of residential consumers, or their minimum power requirements are high; the number of or demand from industrial consumers increases significantly; or both are true. In this section, we concentrate our discussions on such scenarios and extend the hierarchical DRM model to accommodate the large population needs. We introduce proportionality between power supply units and consumers’ demands and show that the demand response performance can be significantly improved in the large population regime by adding extra providers. We start with the case when the number of UCs is given, and then move on to investigate the DRM problem with the provision of additional providers. Since the results can be similarly derived and analyzed for quadratic and logarithmic gain functions, we limit our discussion to the piecewise quadratic gain function of the residential consumers. When (19) is not valid, y = ymax . In this scenario, the supply from each UC, Pk , cannot be obtained directly from (13). We introduce a virtual unit price y such that the optimal power supply Pk can be computed for each UC k, k ∈ K as follows. Replacing y by y in (13), and then substituting (13) into (15), we obtain −1 −1 Ck y = PI + UR,n (28) (ymax ). k∈K n∈NR Upon simplification of (28), we get ⎛ ⎞ −1 −1 y = G2 ⎝PI + UR,n (ymax )⎠ (29) n∈NR −1 where G2 (y ) = y . For the quadratic gain k∈K Ck function, (29) takes the form vR,n − ymax bk PI + n∈NR + k∈K zR,n 2ak y = (30) 1 k∈K 2ak where ymax is obtained using (7). Now, Pk and xR,n can be obtained as −1 y − bk y = ; ∀k ∈ K (31) Pk = Ck 2ak and −1 xR,n = UR,n (32) (ymax ). Note, however, that the unit price charged to the consumers will be ymax , not y . Thus, the UCs have to meet increased demand requirements without increasing the unit price, consequently, reducing their profits. Since our focus is on the planning level problem, the power generation, supply, and demand are at a macroscale covering the population of the electricity consumers in a region or a state. Therefore, hourly shifting and scheduling of load is not a feasible option. Such a condition physically indicates the need of cheaper or more power resources to accommodate the power requirements of the consumers. Distributed local or regional power sources are one of the major components in the smart grid. In line with this objective, we introduce a new dimension of optimizing DRM by adding providers if necessary. Let K0 denotes the set of UCs available. We introduce a set of UCs J := {1, 2, . . . , J} that can supply additional power to meet the demand requirements of a large number of consumers. Then, K := K0 ∪ J . Each additional power supply unit is associated with the cost coefficients aj > 0, bj , cj ≥ 0 ∀j ∈ J . Suppose, with given K0 , Pk is calculated using (31) but the unit price charged to the consumers is ymax < y . Then (18) indicates that with every j added to the list of the providers, the optimal y decreases for bj < 1. This means, with every additional j, the gap between y and ymax reduces. Consequently, the amount of power supply from each UC will be closer to Pk for given price y. Thus, with more UCs, the profit of each UC improves, which serves as an incentive for the supply side to add UCs. However, after a certain number of additional UCs, the optimal unit price starts decreasing because of excessive competition among them and the demands from the residential consumers start increasing in response to the cheaper unit price. As a result, the profits start degrading and the welfare of the consumers start increasing. Thus, in the large population regime, it is necessary to scale the supply side proportionally with the consumer population (and their demands), but for a given population, there exists a fundamental tradeoff between the UC profit and consumer welfare with respect to the number of UCs. Theorem 2: Suppose J is the number of UCs that can be added to the supply side with the associated cost coefficients a > 0, b, c ≥ 0. For the piecewise quadratic gain function-based residential user-welfare model, let E1 := + 2 (v − y /z ) + 2P max R,n n∈NR R,n k∈ K0 bk /ak , E2 := I := 1/a , E + b /a +2 2P k 3 I k∈K0 k∈K0 k k n∈NR (vR,n /zR,n ), and E4 := k∈K0 1/ak + 2 n∈NR (1/zR,n ). Then, provided bk (J + aE2 ) − aE2 ) <b< J bk (J + aE2 ) − aE1 ) < if J E1 E2 E1 E2 or E1 bk (J + aE2 ) − aE1 ) <b< E2 J bk (J + aE2 ) − aE1 ) E1 < if E2 J and bk (J + aE4 ) − aE3 ) <b< J bk (J + aE4 ) − aE3 ) < if J E3 E4 E3 E4 (33) MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME Algorithm 2 Distributed Algorithm to Optimize Provider Profits in Large Population Regime or E3 bk (J + aE4 ) − aE3 ) <b< E4 J bk (J + aE4 ) − aE3 ) E3 < if E4 J 195 (34) there exists a unique number of UCs for which the profit of each UC and the total profit are maximized. Proof: As described earlier, the consumers will be charged at the unit price of ymax as long as y > ymax and will be charged at the unit price of y when y ≤ ymax . The profit of UC k, k ∈ K when it supplies a power given by (31) is RUC,k, = ymax y − bk y 2a k y − bk 2 y − bk + bk (35) − ak + ck 2ak 2ak 1: For t = 1, K = K0 , j = 0 and for given NR , compute yj , ymax , yj using (18), (7), (30), respectively. Compute Pk,j and RUC,k,j from (13) or (31) as required, and (11), respectively. Sort J with the cheapest source first, i.e., Jsorted = {1, 2, . . . j, . . . J}. 2: Repeat for t = 1, 2, 3, . . . 3: If (19) is true 4: break 5: else if j > J 6: break 7: else 8: for K = K0 ∪{1, 2, . . . , j} compute yj+1 , ymax,j+1 using (18), (7). 9: j ← j + 1 10: end 11: Go to 2. where y is obtained from (29). When y ≤ ymax , the profit of UC k, k ∈ K takes the form RUC,k, = y y − bk y 2ak y − bk 2 y − bk + ck + bk (36) − ak 2ak 2ak where y is given by (18). Differentiating (35) and (36) with respect to J yields ∂|RUC,k |y a(E1 − bE2 )(bJ + aE1 − (J + aE2 )ymax ) = ∂J 2ak (J + aE2 )3 ∂|RUC,k, |y a(E3 − bE4 )(bJ + aE3 − (J + aE4 )bk ) =− . ∂J 2ak (J + aE4 )3 Fig. 2. Convergence of the distributed algorithm. (a) y1 = 0.2, σ = 300. (b) y1 = 1, σ = 300. (37) VII. N UMERICAL R ESULTS (38) When y > ymax , the profit of UC k, k ∈ K , |RUC,k, |y is an increasing function of J if (∂|RUC,k |y /∂J) > 0. From (37), we observe that (∂|RUC,k |y /∂J) > 0 if (33) is valid. When y ≤ ymax , the profit of UC k, k ∈ K , |RUC,k, |y is a decreasing function of J if (∂|RUC,k, |y /∂J) < 0. Equation (38) indicates that (∂|RUC,k, |y /∂J) < 0 if (34) is true. Therefore, if (33) and (34) are valid, |RUC,k, |y is an increasing and |RUC,k, |y is a decreasing function of J, which implies that a unique J exists that maximizes the profit. We design a distributed algorithm to determine the optimal number of UCs for maximizing their profits. For the distributed algorithm, the additional UC index j ≥ 0 is appended as a subscript to all variables defined in the previous sections. For given parameters of the residential and industrial consumers, if (19) is true, the unit price and consequently, the profits of the UCs are optimal, i.e., no additional providers should be added. If (19) is not valid, yj , ymax , and yj are computed from (18), (7), and (30), respectively. The power supply of each UC Pk,j is obtained using (31) and the profits RUC,k,j are calculated by substituting ymax and Pk,j into (11). The cheapest additional UC j is added, i.e., j = j + 1 and RUC,k,j are computed for all k ∈ K0 ∪ {1, 2, . . . j} again. The process will be repeated as long as every additional UC improves the profits. The details of the algorithm are presented in Algorithm 2. We begin with three UCs. In practice, power generation costs of the UCs can vary for different kinds of energy sources. However, if the UCs supply power from similar kinds of power plants, the costs can be similar or within a certain range. For the purpose of illustration, we choose the cost coefficients of these UCs as a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, and a2 = 0.05 cents, b2 = 0.1 cents, c2 = 0, and a3 = 0.02 cents, b3 = 0.05 cents, and c3 = 0. We have selected the quadratic and linear coefficients (ak , bk ) significantly different for each provider, to represent general scenario and to illustrate the difference in their profits. The constant coefficient (ck ) are chosen as zero, as we are focusing on a planning level problem, where the energy sources that do not have highstarting costs, are preferred. Note that, the same costs for each UC or similar costs are special cases of our formulation and analysis, and hence, are covered in our model. We consider piecewise quadratic gain function for the residential consumers with zR,n = 2, vR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR unless otherwise mentioned. A. Stackelberg Game Fig. 2(a) and (b) shows the convergence of Algorithm 1. Starting from a different initial values, the unit price, and consequently, the residential user-demand converge to the same optimal values in Fig. 2(a) and (b), within 16 iterations. 196 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016 (a) (b) Fig. 3. Variation with respect to the population of residential consumers (a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, a2 = 0.05 cents, b2 = 0.1 cents, c2 = 0, a3 = 0.02 cents, b3 = 0.05 cents, c3 = 0, PI = 50 MWh, vR,n = 2, zR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR ). (a) Unit price and welfare of residential and industrial users. (b) Total supplies and profits of the UCs. (a) (b) Fig. 4. Variation with respect to the demand from industrial consumers (a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, a2 = 0.05 cents, b2 = 0.1 cents, c2 = 0, a3 , a4 , . . . , a10 = 0.02 cents, b3 , b4 , . . . , b10 = 0.05 cents, c3 , c4 , . . . , c10 = 0, NR = 100, vR,n = 2, zR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR ). (a) Unit price and welfare of residential and industrial users. (b) Total supplies and profits of the UCs. Since, there exists a one-to-one mapping of the power supply from the UCs, their profits, and consumer welfare with the unit price, it is clear that all these measures also converge. Fig. 3(a) shows the variation of unit price, and welfare of the residential and industrial consumers with respect to the number of residential users for PI = 50 MWh. We observe that the unit price increases as the number of residential consumers increases, until NR = 100. Consequently, the residential users’ demand reduces in response to the increasing price, and the residential user welfare also decreases. When NR = 100, the price reaches the maximum value (ymax ). As a result the unit price remains constant after this point. The total cost to the industrial consumers increases with the increasing unit price and then remains constant after y becomes ymax , as expected. In Fig. 3(b), the increase in supply from all UCs after NR = 100 is normal with increasing NR when y is constant. In the region NR ≤ 100, the increase in the unit price y causes the supplies to increase, as the increase in residential-user population is dominant over the decrease in their demands. The profits on the other hand attain their maximum values when NR = 100. Until NR = 100, the increase in the unit price y improves the profit with higher supply of power. After NR = 100, even though the UCs need to increase their supply which consequently increases their costs, the unit price remains the same, thus, reducing their profits. As UC 3 is the cheapest one, it supplies the maximum power, and its profit is the highest. In Fig. 4(a) and (b), we illustrate the performance of our model with respect to the power demand from the industrial consumers. For this plot, we used NR = 100. Other parameters were the same as used for Fig. 3(a) and (b). Fig. 4(a) shows a similar pattern to Fig. 3(a). For given NR , PI = 50 is the point until where (19) is valid. As a result, the unit price and the demand and welfare of the residential consumers decreases until PI = 50. After this point, the price is constant, the demand of the residential consumers is their minimum power requirement and their welfare is also constant. The cost paid by the industrial users increases with increase in the total MAHARJAN et al.: DRM IN THE SMART GRID IN A LARGE POPULATION REGIME 197 B. Large Population Regime (a) (b) (c) Fig. 5. Profits with respect to number of UCs (a1 = 0.1 cents, b1 = 0.2 cents, c1 = 0, a2 = 0.05 cents, b2 = 0.1 cents, c2 = 0, a3 , a4 , . . . , a10 = 0.02 cents, b3 , b4 , . . . , b10 = 0.05 cents, c3 , c4 , . . . , c10 = 0, vR,n = 2, zR,n = 1, xR,min = 0.2 MWh, ∀n ∈ NR ). (a) NR = 500, PI = 50 MWh. (b) NR = 500, PI = 100 MWh. (c) NR = 1000, PI = 50 MWh. industrial demand, as expected. The pattern of supplies and profits for the UCs in Fig. 4(b) is similar to that in Fig. 3(b), The profits are maximized at PI = 50. In this section, we extend the scope of our solutions to cover a large population of residential consumers and a higher demand from industrial consumers. In Section VII-A, we observed that the residential users can only fulfill their minimum power requirement and that the profits of the UCs decrease after certain values of NR and PI . Therefore, we start with three UCs as specified in Section VII-A but we added 17 UCs in the list of providers, which can be deployed if they can improve the provider profits or the consumer welfare. Without loss of generality, for the purpose of illustration, we choose the cost coefficients for all of the additional UCs same as that of UC 3, i.e., a4 , a5 , . . . , a20 = a3 , b4 , b5 , . . . , b20 = b3 , and c4 , c5 , . . . , c20 = c3 . The results can be easily extended for any set of cost coefficients associated with the additional UCs. Fig. 5(a)–(c) depicts the profits of the UCs and the total welfare of the residential consumers for different combinations of NR and PI . The figures show that the profits of all UCs increase as new UCs are added, but the profits start degrading if there are too many UCs due to excessive competition among them, clearly indicating the optimal number of UCs for given NR and PI . Let Kopt be the number of UCs that maximizes the profits. Let us call the region where K < Kopt as small-K region and the region where K > Kopt as large-K region. For PI = 50 MWh, as NR increases from 500 to 1000, Kopt changes from 5 to 7. When NR = 500, change in PI from 50 to 100 changes Kopt from 5 to 6. On the other hand, the consumer welfare unfolds a different perspective. The residential-consumers’ welfare is less for the small-K region and it starts improving in the large-K region. For example, in Fig. 5(a)–(c), the welfare starts drastically increasing after K = 5, K = 6, and K = 7, respectively. Note that, the profits of all the UCs start decreasing after these points. Thus, the figure illustrates a fundamental tradeoff between maximizing UC profit or optimizing consumer welfare, with respect to the number of UCs, in the large population regime. When the unit price is constant (ymax ) and the number of participating UCs increases in the small-K region, the supply from each UC decreases, thus, reducing the extra cost incurred due to the power supply higher than the optimal value for given y. As a result, the profits improve. The increase in the welfare in the large-K region is governed by the cheaper unit price. Intuitively, each UC supplies less power at a reduced unit price, thus, lowering the profit of all UCs. Despite this tradeoff between the provider profits and the consumer welfare, closer observation reveals some interesting facts. In Fig. 5(a), when K changes from 5 to 6, the total profit decreases from 58.41 to 53.67, a reduction of about 8%. The welfare on the other hand grows from 11.09 to 19.78, a gain of about 78.5%. These measures for Fig. 5(b) and (c) are, respectively: when K changes from 6 to 7, 8.4% profit reduction versus 80% improvement in welfare and when K changes from 7 to 8, 3.6% profit reduction versus 35% improvement in welfare. It can be argued that the gain in the welfare with a UC additional to Kopt yields about ten times more gain in the residential consumer welfare compared to the loss in the total profit for given set of chosen parameters. 198 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 1, JANUARY 2016 Another interesting observation from Fig. 5(a)–(c) is that, every additional UC in the small-K region, brings drastic improvement in the profits of the UCs without worsening the residential consumers’ welfare. This implies that additional UCs essentially improve both supply and demand performances in the small-K region, a natural incentive for the supply side to add UCs proportional to consumer population if necessary. In the large-K region, the number of additional UCs should be determined according to the specific requirements of the system, which prioritize the UC profits or the consumers’ welfare. We have proposed a hierarchical system model incorporating multiple processes from power generation to market activities and to power consumption into one framework for a large population of residential and industrial consumers. We have modeled the decision processes of both supply and demand sides as a multileader multifollower Stackelberg game, where the UCs act as the leaders to maximize their profits, and the residential users are the followers that optimize their individual welfare. We have developed closed form solutions for the SE of the game and have proved that a unique SE exists. Moreover, we have presented a distributed algorithm to reach the optimal solution. We have introduced a new dimension of improving the UC profits in a large population regime by adding UCs, and developed a distributed algorithm to reach the optimal number of UCs for profit maximization. We have also explored the tradeoff between the provider profits and consumer welfare with respect to the number of providers. Finally, we have provided extensive numerical results to illustrate the performance of the proposed solutions. A PPENDIX S KETCH OF THE A NALYSIS FOR I MPERFECT C OMPETITION -BASED S CENARIO In an imperfect competition-based model, let yk be the unit price charged by UC k, and let xR,n,k be the power consumer n obtains from UC k. Then, the user optimization problem, i.e., (1), takes the form yk xR,n,k . (39) WR,n := UR,n xR,n − max xR,n :={xR,n,k ,∀k∈K } k∈K For illustration, considering the logarithmic gain function (8) for the consumers, for given {yk , k ∈ K }, the optimal demand of consumer n from UC k can be expressed as αR,n xR,n,k = − βR,n , ∀k ∈ K . (40) yk Then, the optimization problem for UC k, k ∈ K , (12), can be expressed as yk Pk − C(Pk ) s.t. PI + xR,n,k ≤ Pk 0≤yk ≤yk,max ,Pk ∈R+ n∈NR k∈K Pk ≤ Pk,max ; ∀k ∈ K . n∈NR Substituting (40), (42), and Pk,I = PI /K in (43), and after simplification, we obtain VIII. C ONCLUSION max For a quadratic cost function for power generation, the optimal power to be supplied by each UC (13), takes the form yk − bk ; ∀k ∈ K . (42) Pk = 2ak Let Pk,I be the power that the industrial consumers get from UC k, such that k∈K Pk,I = PI . For simplicity, we consider P1,I = P2,I = · · · PK,I = PI /K. For supply demand equilibrium, it is required that xR,n,k , ∀k ∈ K . (43) Pk = Pk,I + k∈K (41) Ky2k + K(2ak BR − bk )yk − 2ak (PI + KAR ) = 0. 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(SmartGridComm), Gaithersburg, MD, USA, Oct. 2010, pp. 415–420. Sabita Maharjan (S’09–M’13) received the M.Eng. in wireless communication from the Antenna and Propagation Laboratory, Tokyo Institute of Technology, Tokyo, Japan, in 2008, and the Ph.D. degree in network and distributed systems from the University of Oslo, Oslo, Norway, and the Simula Research Laboratory, Fornebu, Norway, in 2013. She is currently a Postdoctoral Fellow with Simula Research Laboratory. Her current research interests include wireless networks, network optimization, security, game theory, smart grid communications, and cyber-physical systems. Quanyan Zhu (S’04–M’12) received the B.Eng. (Hons.) degree in electrical engineering with distinction from McGill University, Montreal, QC, Canada, in 2006; the M.A.Sc. degree in electrical engineering from the University of Toronto, ON, Canada, in 2008; and the Ph.D. degree from the University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 2013. He is an Assistant Professor with the Department of Electrical and Computer Engineering, Polytechnic School of Engineering, New York University, Brooklyn, NY, USA. From 2013 to 2014, he was a Postdoctoral Research Associate with the Department of Electrical Engineering, Princeton University, Princeton, NJ, USA. His current research interests include optimal control, game theory, reinforcement learning, network security and privacy, resilient control systems, and cyber-physical systems. Dr. Zhu was a recipient of the Natural Sciences and Engineering Research Council of Canada Graduate Scholarship, the Mavis Future Faculty Fellowships, and the NSERC Postdoctoral Fellowship. He spearheaded the INFOCOM workshop on Communications and Control on Smart Energy Systems, and the Midwest Workshop on Control and Game Theory. 199 Yan Zhang (SM’10) received the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 2005. From 2006, he was with Simula Research Laboratory, Fornebu, Norway. He is currently the Head with the Department of Networks, Simula Research Laboratory, and an Adjunct Associate Professor with the Department of Informatics, University of Oslo, Oslo, Norway. His current research interests include wireless networks, cyber physical systems, and smart grid communications. Dr. Zhang is a Regional Editor, an Associate Editor, is on the editorial board, and a Guest Editor of a number of international journals. Stein Gjessing (M’90) received the Dr.Phil. degree in computer science from the University of Oslo, Oslo, Norway, in 1985. He is currently a Professor of Computer Science with the Department of Informatics, University of Oslo, Oslo, Norway, and an Adjunct Researcher with Simula Research Laboratory, Fornebu, Norway. His current research interests include network and transport protocols, network resilience, cognitive radio networks, and the smart grid. His original research was in the field of object-oriented concurrent programming. He has researched computer interconnects such as Scalable Coherent Interface (IEEE Standard 1596), and local area network/metropolitan area networks such as Resilient Packet Ring (IEEE Standard 802.17). Tamer Başar (S’71–M’73–SM’79–F’83–LF’13) received the B.S. degree in electrical engineering from Robert College, Istanbul, Turkey, in 1969, and the M.S., M.Phil, and Ph.D. degrees in engineering and applied sciences from Yale University, New Haven, CT, USA, in 1970, 1971, and 1972, respectively. He is with the University of Illinois at UrbanaChampaign, Urbana, IL, USA, where he is the Swanlund Endowed Chair. He is a Professor with the Department of Electrical and Computer Engineering, Center for Advanced Study, Urbana. He is a Research Professor with Coordinated Science Laboratory, Urbana, and the Information Trust Institute, Urbana. He is the Director of the Center for Advanced Study, University of Illinois at Urbana-Champaign. His current research interests include stochastic teams, games, and networks; security; and cyber-physical systems. He has over 700 publications in systems, control, communications, and dynamic games, including books on noncooperative dynamic game theory, robust control, network security, wireless and communication networks, and stochastic networked control. Prof. Başar was a recipient of several awards and recognitions over the years, including the highest awards of the IEEE Control Systems Society (CSS), the International Federation of Automatic Control (IFAC), the American Automatic Control Council (AACC), the International Society of Dynamic Games (ISDG), the IEEE Control Systems Award, and a number of international honorary doctorates and professorships. He is a Member of the U.S. National Academy of Engineering, European Academy of Sciences, and a Fellow of the IFAC, and the Society for Industrial and Applied Mathematics. He was the President of the IEEE CSS, the ISDG, and the AACC, and an Editor-in-Chief of Automatica from 2004 to 2014. He is currently an Editor of several book series.