A Distributed Cooperative Power Allocation Method for Campus Buildings He Hao, Yannan Sun, Thomas E. Carroll, and Abhishek Somani Abstract— We propose a coordination algorithm for cooperative power allocation among a collection of commercial buildings within a campus. We describe a typical commercial building Heating, Ventilation, and Air Conditioning (HVAC) system, and utilize Model Predictive Control (MPC) to characterize its power flexibility. The power allocation problem is formulated as a cooperative game using the Nash Bargaining Solution (NBS) concept, in which buildings cooperatively maximize the product of their utilities subject to their local flexibility constraints and a total power limit set by the campus coordinator. To solve the optimal allocation problem, a distributed protocol is designed using dual decomposition of the Nash bargaining problem. Numerical simulations are performed to demonstrate the efficacy of our proposed allocation method. Index Terms— Building-to-grid, power allocation, commercial buildings, Nash bargaining theory, dual decomposition

I. I NTRODUCTION There are about 5.6 million commercial buildings in the United States, collectively consuming approximately 35% of the nation’s electricity [1], [2]. The enormous thermal storage capability of commercial buildings makes them an ideal demand-side dispatchable resource for providing flexibility in power system operation and control. Therefore, buildingto-grid integration has recently attracted significant attention in the smart grid research [3]–[10]. Demand response capabilities of commercial buildings have been extensively studied. For instance, [8], [11] demonstrated precooling of commercial buildings for peak load shaving. Various control strategies were identified for commercial buildings to reduce power consumption in response to grid requirement [7]. An assessment framework was proposed in [5] to evaluate demand response capability of commercial buildings. More recently, several researchers have examined control of commercial buildings to provide ancillary services to the grid [4], [9], [10], [12], [13]. However, most of the work in literature focus on characterizing the flexibility of a single commercial building [4], [5], [11]– [13]. In this paper, we study how to coordinate responses from a collection of commercial buildings on a campus to provide grid services, while respecting their preferences and leveraging their flexibility. A central campus-wide coordinator that manages the power consumption of a large collection of complex building systems is untenable and unscalable. The information set required to solve a centralized campus-wide power allocation problem may become prohibitively expensive. To this end, a distributed intelligent multi-agent system (MAS) in which The authors are with Pacific Northwest National Laboratory, P.O. Box 999, 99352, Richland, Washington, USA. Email: {He.Hao, Yannan.Sun, Thomas.Carroll, Abhishek.Somani}@pnnl.gov.

agents cooperatively1 interact with one another to coordinate buildings’ activities is strongly preferred. The MAS represents a natural problem decomposition and can find innovative solutions to problems that would be difficult–if not impossible–for a centralized monolithic system to devise. In this paper we present the application of the gametheoretic Nash bargaining solution to a multi-agent system, for the cooperative power allocation among buildings within a campus. The campus coordinator decides the optimal power limit for the campus based on an external dispatch signal from the grid, and the buildings then decide how to cooperatively adjust their power consumption to meet the campus coordinator’s power limit, while respecting their local constraints. The buildings reach a solution by “negotiating” amongst themselves, in order to maximize the product of their utility functions. In essence, the Nash bargaining approach resembles a distributed resource allocation problem. Nash bargaining solution (NBS) [14] is an attractive approach for solving such cooperative resource allocation problems as it balances fairness with efficiency [15]. Unlike non-cooperative game-theoretic approaches in which agents make decisions independently, NBS is a unique, Paretoefficient solution that maximizes social welfare, which is defined in this context as the product of the buildings’ utilities. Using dual decomposition [16], [17] and distributed consensus algorithm, a distributed protocol is developed in this paper for buildings to cooperatively compute their power consumption levels to reach the optimal power allocation. In this paper, we assume the baseline operation of buildings is under optimal control using MPC to minimize their energy consumption. With certain allowable degree of violation on the comfort constraints such as temperature, we utilize a MPC approach to characterize their upward and downward power flexibility. The proposed distributed protocol is then used to optimally allocate power to buildings while respecting their flexibility and preferences. We conduct numerical simulations to demonstrate the efficacy of our allocation strategy. We show that a successful negotiation for a 43-building campus is achieved after a small number of bargaining iterations. Additionally, a 24-hour long simulation shows that they could successfully achieve peak load shaving with a minimal impact on occupants’ comfort. The rest of this paper unfolds as follows. Section II presents the problem statement. We describe a typical commercial building HVAC system, and characterize its power flexibility in Section III. In Section IV, a distributed protocol is developed to solve the Nash bargaining problem. Numer1 It is reasonable to assume that buildings owned and operated by a single entity act in a cooperative manner to collectively maximize the common objective set by, in this instance, the campus coordinator. Buildings owned by separate entities may act in a more non-cooperative manner.

Chiller Chilled Water

max pi ’s

subject to:

n Y

(ui (pi ) − ui (pdi ))

i=1 n X

pi ≤ Q,

i=1 p− i ≤

pi ≤ p+ i ,

(1b) ∀ i ∈ {1, · · · , n},

(1c)

where the objective function is referred to as the Nash product, pdi ’s are the threat points if a deal among the players + is not made, and p− i , pi are respectively the minimum and maximum power requirements of a building. In this paper, we set the threat point as pdi = p− i , which implies if the negotiation is not successful the default power allocation for each building is their minimum power requirement. III. C HARACTERIZING HVAC P OWER F LEXIBILITY In this section, we describe a typical commercial building HVAC system, and present a MPC method to characterize the minimum and maximum power requirements of a building. The architecture of a commercial building HVAC system that services multiple zones is shown in Fig. 1. A typical

Duct

VFD

Air Handling Unit (AHU)

Zone 2 Zone 3

Zone 6 Zone 5

Cooling Coil

Fan Mixed Air

(1a)

Zone 8

Valve

II. P ROBLEM F ORMULATION Consider a campus comprising of n buildings. Let {i : i = 1, 2, · · · , n} denote the ith building. At each time period t (e.g., every 15 minutes), a campus coordinator is responsible for allocating power to all buildings subject to a total power limit Q(t), which could either be determined by a dispatch signal from the grid or a limit imposed by the campus coordinator due to budgetary considerations. The campus coordinator first computes aP default power allocation pdi (t) for each building such that i pdi (t) ≤ Q(t), and pdi (t) ≥ − p− i (t), where pi (t) is the minimum power requirement of the ith building to maintain its operational functionality. The information of default allocations pdi (t)’s and total power limit Q(t) is then broadcast to all the buildings Pon campus. In the presence of power surplus, i.e., Q(t)− i pdi (t) > 0, each building then negotiates on a new allocation profile (p1 (t), p2 (t), ..., pn (t)) with other peer buildings, so that the new allocation pi (t) increases its own utility function or preference, ui . A feasible Pnallocation is achieved when pi (t) ≥ pdi (t) for all i, and i=1 pi (t) ≤ Q(t). For ease of explanation, we consider in this paper a single period power allocation, and drop the dependence of notations on time t. A multi-period power allocation problem is a subject of our future work. From a game theoretic perspective, each player in a negotiation must always keep in mind that a strategy of trying to unilaterally improve its own return at the expense of the other players will typically be self-defeating [18]. Therefore, we consider a cooperative game, in which each building is motivated to negotiate with others in a collaborative way to jointly maximize their utility functions. In this paper, we consider a Nash bargaining solution to solve the power allocation problem. Formally, the Nash bargaining problem is formulated as

VAV Box

Zone 1

ical simulations are presented in Section V. The paper ends with conclusions and future work in Section VI.

Zone 4

Zone 9

Zone 10

Zone 7

Zone 11

Return Air Damper

Fig. 1.

Configuration of a typical commercial building HVAC system.

configuration includes air handing unit (AHU), chiller, and variable air volume (VAV) boxes. A detailed description of these main components and their functionalities can be found in [4], [19]. The thermal dynamic model of a m-zone building is constructed by interconnection of RC-network models of individual zones [4], [20]. The major electric power consumers of a HVAC system are its supply fan, and chiller, whose models are respectively given in [4] and [19]. In this section, we use a Model Predictive Control (MPC) approach to calculate online estimates of the power flexibility of a commercial building HVAC system over a predicted time horizon. MPC is a powerful tool to handle constrained multiple input multiple output systems such as a commercial building HVAC system [5], [19]. In this paper, we assume all commercial buildings are in the cooling mode. Let ∆ = 15 minutes be the discretization step, and t index the current time step. The thermal model of a multi-zone building can be written compactly as xt+1 = f (xt , ut , wt ),

(2)

where the state vector xt = [T1 (t), . . . , Tn (t), T(i,j) (t)0 s] ∈ Rm+w collects the temperatures of m zones and w equivalent separating walls, the input vector u = [m1 (t), . . . , mn (t), ∆Tc (t), δ(t)] ∈ Rm+2 collects the mass flow rate into each zone, the temperature deviation across the cooling coil, and the damper position, and the disturbance vector wt = [Q1 (t), . . . , Qn (t), Toa (t)] ∈ Rm+1 contains the external disturbances of each zone, and the outside air temperature. Interested readers are referred to [4], [19] for more details. Let N be the number of prediction steps, and we denote K = {0, 1, · · · , N −1}. We aim to characterize the maximum and minimum power requirements of the HVAC system over the predicted horizon. First of all, we consider the characterization of minimum power consumption over the predicted horizon. More specifically, min

ut→t+N

N −1 X

wt+k pt+k

(3a)

k=0

subject to: xt+k+1 = f (xt , ut , wt ), ∀ k ∈ K, xt+k ∈ Xt+k , ut+k ∈ Ut+k , ∀ k ∈ K, xt+N ∈ Xt+N ,

(3b) (3c) (3d)

where wt+k ’s are non-negative weights negotiating the importance of pt+k ’s at different time steps, and Xt+k , Ut+k are respectively the feasible sets of the systems states xt+k and control inputs ut+k at time step t + k, which are determined by user specified temperature band, and operational

constraints on the air flow rate mj , temperature drop across the cooling coil ∆Tc , and ventilation requirement on the damper position δ. Similarly, we can characterize the maximum power consumption over the predicted horizon. The characterized upward and downward power flexibility at time step t − are denoted by p+ t and pt respectively. The set F = − + {pt+k |pt+k ≤ pt+k ≤ pt+k , ∀ k = 0, · · · , N − 1} represents the flexibility of the building over the predicted horizon. + Given any dispatch power trajectory p¯t+k ∈ [p− t+k , pt+k ] for k = 0, · · · , N − 1, we use the following algorithm to find a control action to track it, N −1 X

min

ut→t+N

(kpt+k − p¯t+k k2 + γkxt − x ¯ t k2 )

Second, due to privacy reasons, each building might not want to reveal their minimum/maximum power requirements, and their utility functions to the other buildings or the campus coordinator. As an alternative, we design a distributed protocol for buildings to “negotiate” among themselves to reach the optimal power allocation [15]. Upon observing the separability of the cost function and constraints, our insight is to take advantage of the dual decomposition technique [16], [17]. The partial Lagrangian of the optimization problem (5) is given by L(qi0 s, λ) = −

n X

i=1

i=1

(4a)

where λ is the dual variable. The Lagrange dual function is

k=0

subject to: xt+k+1 = f (xt , ut , wt ), ∀ k ∈ K, xt+k ∈ Xt+k , ut+k ∈ Ut+k , ∀ k ∈ K, xt+N ∈ Xt+N ,

(4b) (4c) (4d)

where γ is a positive weight. The first term in the cost function penalizes the tracking error, the second term aims to keep the temperatures of each zone as close as possible to target values x ¯t . For instance, in case of peak load shaving, we want the temperatures to be as close as possible to their lower bounds with the dispatched power p¯t+k . For the considered scenario of single period power allocation, we are interested in the upward and downward power flexibility for the current time step only, which are denoted − respectively as p+ t and pt . For ease of notation, we drop the time-dependent subscript for the remainder of this paper.

L(qi0 s, λ)

g(λ) = min 0 qi s

= − λQ +

Consider a collection of n buildings with each building + − + having a power requirement pi ∈ [p− i , pi ], where pi , pi are respectively the upward and downward power flexibility of the ith building. With a change of variable, q i = (pi − + − p− i )/(pi − pi ), we scale the power requirement of each building so that q i ∈ [0, 1] for all i ∈ {1, · · · , n}. For each building, its comfort or preference is modeled by a utility function ui (qi ) : [0, 1] → R+ with power allocation qi . In this paper, we assume each utility function is concave, differentiable, and ui (qi ) = 0 if qi = 0. With the above change of variables and the property of logarithmic functions, the original Nash bargaining problem (1) can be reformulated as max f (q1 , · · · , qn ) = 0 qi s

n X

log(ui (qi ))

(5a) (5b)

i=1

0 ≤ qi ≤ 1,

p+ i

p− i .

∀ i ∈ {1, · · · , n},


− gi∗ (−λβi ) ,

i=1

max g(λ) = −λQ + λ

n X

− gi∗ (−λβi )




i=1

subject to: λ ≥ 0.

A subgradient of −g(λ) is given by s=Q−

n X

(βi qi∗ + p− i ),

(6)

i=1

where qi∗ is a solution of the sub-problem min gi (qi ) + λβi qi qi

subject to: 0 ≤ qi ≤ 1.

(7a) (7b)

We use a projected sub-gradient method to solve the dual problem λ := max{λ − as, 0},

(8)

where a > 0 is a step size. Given an initial value of λ, each building solves the local optimization problem (7), and locally updates the value of λ using (8). At each iteration, lower and upper bounds on the optimal objective function are given by f− =

n X

log(ui (qi∗ )),

f + = −g(λ).

i=1

i=1 n X (βi qi + p− i ) ≤ Q,

n X

∗ where gi (qi ) = − log ui (qi ) + λp− i , and gi (−λβi ) is ∗ its conjugate function, which is given by gi (−λβi ) = supqi (−λβi qi − gi (qi )). The dual problem then becomes

IV. C OOPERATIVE P OWER A LLOCATION S TRATEGY

subject to:

n X
(βi qi + p− i )−Q ,

log(ui (qi )) + λ

(5c)

where βi = − In this paper, we aim to find a fully distributed protocol for buildings to “negotiate” among themselves to reach a Nash bargaining solution. There are several reasons for using a distributed power allocation strategy. First, in absence of a campus coordinator, the buildings have to jointly maximize the Nash product using a distributed algorithm by sharing part of their information with some of the other buildings.

The above iterative process repeats until convergence is achieved, i.e., |f + − f − | ≤  for a small positive number . Remark 1: The above iterative process exactly describes a bargaining process for buildings to cooperatively share a constrained resource. It resembles the scenario that a group of building owners sit on the same table negotiating on a deal. The initial proposal corresponds to the default power allocation (pd1 , · · · , pdn ) and an initial price λ on the power. Based on this price, each building evaluates its own interest by solving (7), and bargain for a new allocation p∗i = βi qi∗ + p− i . They then jointly propose a new allocation profile (p∗1 , · · · , p∗n ). If there is still power surplus, i.e.,

80 Baseline Power Upward Power Downward Power

Power (kW)

60

40

20

0 00:00

03:00

Fig. 3.

Pn Q − i=1 p∗i > 0, then a new price is constructed using (8), and buildings repeat the previous bargaining process. This process is repeated until all buildings are satisfied with their current proposals or there is no more power surplus. In practice, we aim to implement this bargaining protocol in an automated way with an embedded controller at each building’s energy management system.  In the above iteration process, in order to calculate the common sub-gradient given in (6), each building needs the information of the other buildings’ current desired power consumption, i.e., (βi qi∗ + p− i )’s. In this paper, we are interested in a fully distributed protocol for power allocation. Therefore we choose to estimate the total desired power consumption using a distributed consensus algorithm. The communication network of n buildings is modeled by a graph G = (V, E) with vertex set V = {1, . . . , n} and edge set E ⊂ V × V. We use (i, j) to represent a directed edge from i to j if node i can receive information from j. For each edge (i, j) ∈ E in the graph, we associate a weight Wi,j > 0 to it. The set of neighbors of i is defined as Ni := {j ∈ V : (i, j) ∈ E}. A linear consensus protocol is an iterative update law: zi (k + 1) = Wi,i zi (k) +

X

Wi,j zj (k),

i ∈ V,

(9)

j∈Ni

with initial conditions zi (0) = N p∗i , where k = {0, 1, 2, · · · } is the discrete time index. Clearly, the convergence rate of the consensus algorithm is critical to how fast a successful negotiation is achieved. Several papers have studied how to design the graph weights or design new consensus protocols to improve the consensus rate [21]–[24]. V. N UMERICAL S IMULATIONS In this section, we consider a fictitious campus comprising of 43 buildings. Its campus map and information graph are depicted in Fig. 2, which are adapted from [25]. For a centralized power allocation strategy, the campus coordinator has to characterize the power flexibility for each building, and optimally allocate the power to buildings to maximize the social welfare. To characterize each building’s flexibility, at each time step, there are (m + w) × (N + 1) state variables, (m + 2) × N decision variables, and m × N × 2 + (m + 2) × N × 2 constraints. The computational burden and communication requirements are overwhelming when the number of buildings is large. Therefore, a distributed allocation strategy is strongly preferred.

09:00

12:00 Time

15:00

18:00

21:00

24:00

Power flexibility of the reference building.

We first construct a reference building, whose model parameters are taken from [4], which describes a commercial building HVAC system that services a half of a real commercial building with an area of 50, 000 square footage. We then use MPC approach to characterize its power flexibility. Fig. 3 shows its baseline power as well as the upward and downward power limits. In particular, the baseline power corresponds to a daily power profile when MPC is minimizing the building’s power consumption, and the upward and downward power limits are the maximum and minimum power the building can consume at each time step if its temperature is allowed to deviate from the prespecified band for one degree in Celsius. Note that if no violation is allowed, the downward power limit will be the same as the baseline power. For the application of peak load shaving, we let the upper and lower power requirements of each building to be the baseline power and the downward power limit respectively. In the simulations, we assume the upper and lower power requirements of the 43 buildings in each period can be obtained by scaling those of the reference building by the ratios of their areas to the serviced area of the reference building, which is 25, 000 square footage. The architectural information of the 43 buildings can be found in [25]. Additionally, we assume the utility function for each building is a power function ui (qi ) = qiαi , where αi ∼ U(0.7, 1), and U denotes continuous uniform distribution. Additionally, we assume the total dispatched power is Q = 7 MW. Moreover, we take the initial value of λ to be 0.2. We use the distributed protocol developed in Section IV to find the optimal allocation. The plot in Fig. 4 describes the negotiation process in a period. At the beginning, each building claims a small fraction of the total power, whose summation is much smaller than the power limit Q. Since there is power surplus, the buildings begin to negotiate on sharing the power surplus to increase their own utility. As the number of negotiations increases, the power surplus progressively converges to zero. In the end, each building is satisfied with its current allocation, and a deal among them is achieved. Moreover, we compare our allocation strategy 7.5 Total Allocated Power (MW)

Fig. 2. Graphic representation of a campus, and a fictitious information graph for its buildings. Only highlighted buildings are considered.

06:00

7 Power Limit Q

6.5 6

Default Total Allocated Power

5.5 5 0

2

Fig. 4.

4

6

8

10 Iterations

12

14

16

18

Progress of negotiation on power allocation.

8 Uncoordinated Case

Power (MW)

7.5 7 6.5

Power Limit Q

Coordinated Case

6 5.5 5 4.5 00:00

03:00

06:00

09:00

12:00 Time

15:00

18:00

21:00

24:00

Fig. 5. Aggregate power consumptions of all buildings on campus with and without coordination.

with P a heuristic allocation, which allocates power surplus + − Q − i p− i proportionally to pi − pi . It is shown that such a heuristic allocation results in much lower social welfare. Additionally, this allocation strategy completely ignores each building’s preference on power consumption. Moreover, in order to show the effectiveness of our allocation strategy, we conduct a 24-hour long simulation comparing the total power consumption of the 43 buildings with and without coordinated power allocation. In the uncoordinated case, each building consumes the baseline power. We can see from Fig. 5 that with our proposed power allocation method, the total power consumption of buildings can be well caped within the total power limit Q, while the power consumption without coordination exceeds the power limit for about 6 hours. Moreover, it is interesting to see that although cooperative power allocation helps to shave the peak load, it requires the total power to stay close to the power limit slightly longer than the uncoordinated case. Additional simulations show that the shavable peak power depends on the amount of degrees allowed to deviate from the default temperature range. A sensitivity analysis will be included in future work. VI. C ONCLUSIONS AND F UTURE W ORK We proposed a cooperative power allocation strategy for a population of commercial buildings within a campus. Nash bargaining theory and dual decomposition technique were exploited to develop a distributed protocol for commercial buildings to negotiate on sharing a limited power resource. Numerical simulations showed that our allocation strategy was very effective to achieve a power allocation for buildings while respecting their own preference and flexibility. We are currently investigating a multi-period power allocation strategy for buildings, which allows buildings to trade their power capacity rights in the considered period. We are also considering coordinating buildings to provide fast ramping capacity to facilitate renewable integration. In the future, we are interested in implementing a small-scale demonstration in a real campus environment. R EFERENCES [1] “Buildings energy data book.” [Online]. Available: http: //buildingsdatabook.eren.doe.gov/default.aspx [2] “Commercial buildings energy consumption survey (CBECS): Overview of commercial buildings,” Energy information administration, Department of Energy, U.S. Govt., Tech. Rep., December 2012. [Online]. Available: http://www.eia.gov/consumption/ commercial/reports/2012/preliminary/index.cfm

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