Dynamic Control of Electricity Cost with Power ... - Computer Science

Dynamic Control of Electricity Cost with Power Demand Smoothing and Peak Shaving for Distributed Internet Data Centers Jianguo Yao, Xue Liu, Wenbo He and Ashikur Rahman School of Computer Science McGill University Montreal, Canada Email: {jianguo,xueliu,wenbohe,ashikur}@cs.mcgill.ca

Abstract—Internet based service providers, such as Amazon, Google, Yahoo etc, build their data centers (IDC) across multiple regions to provide reliable and low latency of services to clients. Ever-increasing service demand, complexity of services and growing client population cause enormous power consumptions by these IDCs incuring a major part of their running costs. Modern electric power grid provides a feasible way to dynamically and efficiently manage the electricity cost of distributed IDCs based on the Locational Marginal Pricing (LMP) policy. While recent works exploit LMP by electricity-price based geographic load distribution, the dynamic workload and high volatility of electricity prices induce highly volatile power demand and critical power peak problem. The benefit of cost minimization via geographic load distribution is counterbalanced with the high cost incurred by violating the peak power. In this paper, we study the dynamic control of electricity cost to provide low volatility in power demand and shaving of power peaks. To this end, a Model Predictive Control (MPC) electricity cost minimization problem is formulated based on a time-continuous differential model. The proposed solution minimizes electricity costs, provides low variation in power demand by penalizing the change in workload and alleviates the power peaks by tracking the available power budget. By providing extensive simulation results based on real-life electricity price traces we show the effectiveness of our approach.

I. I NTRODUCTION The computing needs of large-scale Internet service providers are often supported by dedicated Internet data centers (IDC) containing massive number of servers, largescale storage and heterogeneous networking elements together with an infrastructure to distribute power and provide cooling. As computation and storage continue to move into the cloud IDCs computing infrastructure sizes and energy consumptions are rapidly increasing in concert. Recent studies show that large-scale IDCs consume a power as high as dozens of MW incurring a cost of dollar 5.6M [1], [2]; the Environmental Protection Agency (EPA) of USA estimated the annual data center electricity consumptions to be over 100 billion kWh at a cost of dollar 7.4 billion by 2011 [3]. Thus minimizing electricity consumptions and costs of various IDCs are of paramount importance and provide great benefits to the Internet based service operators.

Although research on minimizing power consumptions has passed almost a decade, the research on minimizing power costs is still in its early stage. Some of the recent works on minimizing power costs [5], [6], [7] are based on an interesting observation on electricity prices which is as follows. The price of electricity in United States deregulated markets exhibit both temporal and spatial diversity across multiple regions. As IDCs of Internet service operators are also geographically distributed across multiple regions (in order to provide better QoS and reliability of services), a potential electricity cost savings could be achieved if workload is distributed based on the electricity price information of the regions where IDCs are located. Based on this observation various optimization frameworks for load balancing over geographically distributed data centers are developed. However these solutions can only partially minimize the power costs and bring some new problems as we address next. First of all, due to electricity-price based biased workload distribution, the IDCs located at relatively cheaper electricity price regions tend to meet high power peaks. As the electricity prices are highly volatile in multi-region electricity markets this high power peak problem becomes a frequent phenomena for some of the IDCs. With high power peaks, those IDCs need to subscribe for higher power delivery capacity which greatly increases their capital cost for building the infrastructure. Moreover, high spikes in power demand causes occasional power outages and potential hazards to physically constrained smart power grid. In addition, some electricity suppliers impose a peak power limit on the amount of power draw from the grid that arises due to some transmission limitations and penalize those IDCs heavily if this limit is exceeded [10]. Power peak problem also makes power demand of different IDCs difficult to predict in advance; consequently IDC operators become unable to qualify for price rebates by signing up advance-contracts with the power retailer or hedge against uncertainty in order to minimize operational risks. Secondly, migrating instantaneous workload from high price regions to low price regions does not necessarily reduce total electricity cost due to the inter-dependency of

IDCs’ power demand and electricity price levels in the multiregion electricity market. The price of electricity based on real time pricing (RTP) varies during the day depending on usage pattern. Consequently, being a massive consumer of electricity, the IDCs are in a position to influence the electricity price levels during the day. When the power demand of an IDC is adjusted in one time instance, it affects the price levels in the wholesale market for the next time instance. Therefore the data center operators no longer act as simple passive consumers rather they are active consumers in a sense that they can also influence the price level by manipulating their demands. This behaviour was first observed in [10]. In addition, we like to argue that geographical load balancing creates a vicious cycle among electricity demand, cost and price which can be described as follows. The volatility of electricity price level causes IDC operators to adjust their power demands across multiple regions to minimize the cost which is achieved via geographic load balancing. The newly adjusted power demand causes the price level to become more volatile which in turn requires further adjustment of power demand via redistribution of workload based on new price levels. This mutual interactions of demand, cost and price paradoxically continues to create a vicious cycle. As a result the price levels tend to oscillate and some times turn to be unacceptably high. In this paper we consider both power peak problem and high volatility (i.e. fluctuations) of power demand problem under the same framework. The volatility of power demand is formally defined as the rate of change in power demand and the power peak is the power demand at peak load during a day. We propose a novel approach of dynamic electricity cost control for minimizing the electricity cost with low volatility in power demand and power peak shaving for distributed IDC under real-time electricity price market. To this end, a Model Predictive Control (MPC) is exerted where we formulate the electricity cost model using timecontinuous differential equations. Based on this model the total electricity cost function is derived as a combination of electricity cost and the volatility in power demand. Then, the dynamic control is achieved by the closed-loop MPC based control by minimizing this cost function. The power peak is shaved by setting a control reference of MPC at the target power budget for each IDCs. The feasibility and stability of this scheme is also illustrated and proved. Moreover, through extensive simulations based on real-life electricity price traces we evaluate the efficacy of the proposed scheme. The rest of this paper is organized as follows. Section II describes related research works. Section III presents backgrounds, models and problem formulation. Section IV provides a feedback control solution scheme with detailed description of control model formulation, two-time scale control architecture, MPC control design, optimal control reference solution and stability analysis. Section V evaluates the proposed architecture and shows that the effectiveness of

the proposed solution in reducing volatility of power demand and shaving power peaks. Finally, Section VI concludes the paper. II. R ELATED WORKS Data center power management is an active area of research over the past decade. Based on the general trend of all related research works, we can broadly classify them into two groups:–(i) power consumption management, and (ii) power cost management. The research works within the first group operate at the intra-data center level and aims at reducing power consumptions by the server firms within a data center. Most commonly proposed H/W and S/W level power optimization techniques include dynamic voltage and frequency scaling, processor sleep scheduling, core parking, memorybank parking, hard drive segment parking, virtualization, server consolidation, load balancing etc. However, minimizing power consumptions at the data center level is beyond the scope of this work. For a detailed overview of design challenges for power consumption management at IDCs please see [12] and the references therein. The second group of works targets minimizing total power cost and mostly works at the inter-data center level. There are myriad of works focusing on electricity cost minimization. Qureshi et al. [13] present the cost minimizing techniques in a whole-sale market environment for large distributed systems. Rao et al. [5] studied the same problem but in multi-region electricity markets to better capture the spatial diversity of electricity prices across multiple regions. Zhang et al. [10] introduce the idea of capping electricity cost of Internet-scale data centers by classifying users into premium and ordinary category and providing QoS to premium users and best-effort services to occasional users. Yao et al. [9] proposes a solution to reduce power cost of delay tolerant workloads. The target applications that can generate delay tolerant workloads are based on MapReduce programming and include searching, social networking, data analytics etc. By exploiting temporal and spatial variations of both workload and electricity prices they provide a power cost-delay trade off which is further exploited to minimize power expenses at the cost of service delay. In [4] an online algorithm for migrating jobs between data centers based on electricity prices is proposed. The job migration requires bandwidth intensive migration of the application’s state and data, consequently, they incorporate bandwidth cost in the electricity cost optimization problem. In [8] Sankaranarayanan et al. exploit the heterogeneity of data centers for achieving additional energy efficiency and power savings via intelligent scheduling of requests to heterogeneous servers at each data center. In [7] the authors propose an optimization framework for throughput-intensive applications like web search engine. They propose a workload shifting algorithm considering both electricity prices, (i.e. to reduce the energy

IDC 1

cost) and workload of data centers at the time of shifting (i.e. to reduce response time). Liu et al. [6] address whether the geographic load balancing can additionally encourage the use of green energy and reduce the use of brown energy. To solve this issue they develop a simple power cost model which is a linear combination of variable electricity prices and lost revenue due to reduced response time of the system arising from both network propagational delay and service delay. However, none of these works have studied the dynamic control of electricity costs that can provide controlled variation in power demand and power peak shaving under the real-time pricing policy for modern power grid which is the main focus of this paper.

C

L1

λ1 = ∑ λi1

λ11

i =1

M1 = 6 λ1 j

λ1N

m1 = 4

IDC j

λi1

C

λ j = ∑ λij

λij

Li

i =1

Mj =5

λiN

λC1 LC

mj = 3

IDC N λCj

λCN

C

λN = ∑ λiN i =1

MN = 4 mN = 2

Figure 1.

Workload allocation architecture for IDCs.

III. M ODELS AND PRELIMINARIES In this section, we summarize the models to capture the behaviour of electricity consumption and cost for Internet Data Centers (IDCs). A. Workload allocation architecture for IDCs The workload allocation architecture for IDCs is shown in Fig. 1. This architecture comprises two types of elements: the front-end Web portals and IDCs. When a front-end Web portal receives requests from clients, it distributes the requests to different IDCs for processing. Without loss of generality we may assume that there are C front-end Web portals, and N IDCs located across different regions in the world. In each IDC j, there are Mj servers. We further assume a homogeneous system, where each server has the similar configuration in terms of CPU, memory and power requirements etc. Each IDC turns ON mj servers to deal with the client requests. Then we have the following constraints: mj ≤ Mj , ∀j = 1, 2, · · · , N.

(1)

The client requests offer a workload Li for each front-end Web portal i = 1, 2, · · · , C. Only a portion of the workload λij , i = 1, 2, · · · , C, j = 1, 2, · · · , N is forwarded from the front-end Web portal i to IDC j. The total workload constraints for each front-end Web portal is given by: Li =

N ∑

λij , ∀i ∈ 1, 2, · · · , C.

(2)

j=1

λij ≥ 0.

(3)

For each IDC j, the total received workload from all the front-end Web portals is denoted by λj which is represented as follows: λj =

C ∑ i=1

λij , ∀j = 1, 2, · · · , N.

(4)

B. Energy consumption for each IDC The power consumption P for an individual active sever mainly depends on two variables: CPU utilization Ucpu and frequency f . Other variables indirectly affect the power consumption through CPU utilization and/or frequency. To derive a model, mapping CPU utilization and frequency to power consumption, the curve fitting method is exerted [14] through a set of experiments, in which the power is measured by running a server under different CPU utilization and frequencies. After the experimentation it was shown that the the power consumption has a linear relationship with CPU utilization and frequency when other variables are kept constant. The derived power consumption model for each server in [14] is given by1 : P (f, Ucpu ) = a3 f Ucpu + a2 f + a1 Ucpu + a0 ,

(5)

where a0 , a1 , a2 and a3 are the fitting parameters using a specific curve fitting method. CPU utilization can be approximated by the workload λ using the equation Ucpu = λ/f [14]. Hence, the mapping relationship between the power consumption P (λ) and the received workload λ for each server is given by: P (λ) = b1 λ + b0 ,

(6)

where the new fitting parameters b0 and b1 are deduced by b0 = a2 f + a0 and b1 = a3 + a1 /f , respectively. Assuming that the frequency of CPU for each server is fixed and all the frequencies of CPUs are the same in each IDC, we derive a linear model of total power consumption Pj (λj ) for IDC j (mj active severs) with respect to offered total workload λj : Pj (λj ) = b1 λj + mj b0 .

(7)

1 The total power is consumed by servers, cooling systems and environment components (e.g. network devices and UPS). This paper only considers the power consumption by servers since traditional design separates the three subsystems.

0

i=1

C. Total Electricity Cost for IDC in modern electric power grids The modern electric power grid in North America allows dynamic pricing of electricity based on different regions, time of the day and the power demand. The dynamic electricity price is a promising mechanism to improve the power grid efficiency, reduce the peak load, and mitigate wholesale price volatility. Most large-scale IDCs are built in regions where dynamic electricity prices are available under competitive electricity market. The dynamic price could be a function of different regions, peak load, and different hours in a day etc. In this paper, we use a bottom-up bid-based stochastic price model [17] to represent the dynamic price: Prj = function (region, time, load) .

(9)

Note that here we assume that the electricity price only depends on regions, power load and different hours in a day. The real-time prices are shown in Fig. 2 for three different regions in North America namely Michigan, Minnesota and Wisconsin on October 03, 2011. The electricity prices are adjusted every hour according to current power load. 100 Michigan Minnesota Wisconsin

80

Hence, the total electricity cost for a system comprising N number of IDCs is summarized as follows: ¯ = C(t)

N ∑

Cj (t).

(11)

j=1

D. Workload modelling Modelling Internet workload is fundamental in predicting the future workload arrival to take appropriate actions in advance. The workload is usually modelled as a stochastic process which captures the future uncertain evolution. Many class of random processes, such as Markov Modulated Poisson Process (MMPP) [15] and Markovian Arrival Processes (MAP) [16], are used for fitting the actual web service workload. In this paper, we use a time-varying pth-order autoregressive model to model the arrival of future workload, which is a widely used technique to capture the Markov process behaviour: µi (k) =

p ∑

αpi µi (k − s) + ε(k),

(12)

s=1

where α1i (k), · · · , αpi (k) are the fitting parameters, and the innovations ε(k) are independent and identically distributed white noise. The prediction precision mainly depends on the accuracy of the workload model. To this end, we use Recursive Least Square (RLS) method for online estimation of the workload arrival. The RLS method is described in [18]. 2000 original predicted 1500 Request rate

The electricity energy consumption is the time integral of power consumption. Hence we formulate the electricity energy consumption Ej (t) through the time integral of total power consumption Pj (λj ), which is time varying due to the varying nature of workload λj . The formulation of Ej (t) is represented by: ∫ t Pj (λj )dt Ej (t) = 0 ) ∫ t( ∑ C b1 λij + mj b0 dt. (8) =

1000

500

Price ($/MWh)

60

0

40

0

5

10

15

20

25

Hours

20 0

Figure 3.

Original workload vs. predicted workload.

−20 −40

0

5

10

15

20

25

Hour

Figure 2.

Real-time electricity prices.

The electricity cost for each IDC j is the time integral of the product of real-time electricity price and electricity energy consumption: ∫ t Cj (t) = Prj Ej (t)dt. (10) 0

We assume that the estimated workload model is represented as: p ∑ µ ˆi (k) = α ˆ pi µi (k − s), (13) s=1

α ˆ 1i (k), · · ·

,α ˆ pi (k)

where are parameters estimated with RLS method. Given the previous p samplings of workload Li (k− s), s ∈ 1, · · · p, we can predict the workload for the next period. To evaluate the workload prediction method, a numeric simulation is conducted based on the workload data to an

EPA server on Aug. 30th 19952 . The original and predicted workloads are drawn in Fig. 3, which clearly shows that the prediction model can accurately capture the workload characteristics. E. Service latency model In the Internet service, the latency is a major parameter to evaluate the Quality of Service (QoS). For each IDC, different queueing models result in different QoS. We use the M/M/n queueing model to process the incoming workload from front-end Web portal. According to this model, the average latency D is represented as D = PQ /(nµ − λ) where n is the number of servers , λ is the workload arrival rate, µ is the service rate and PQ is the probability of clients waiting in the queue. Without loss of generality, we may assume that there are always client requests waiting in the queue and the servers at IDCs are always busy to handle the requests. Hence, we have PQ equals 1. The actual average latency for IDC j is represented as follows: Dja =

1 mj µj − λj

.

(14)

where µj is the service rate for each server in IDC j. To meet the QoS requirements for all Internet clients, the following constraints are put on the average processing latency for IDC j as: Dja ≤ Dj .

(15)

where Dj is a latency bound for IDC j. In this section, we formulate the control problem of electricity power consumption and cost. The detailed controller design is described and analyzed. A. Control modelling formulation The time-continuous model of electricity consumption for IDC j is represented as Ej (t) = b1 λij + mj b0 . dt i=1

(16)

The time-continuous model of electricity cost for IDC j is represented as C¯j (t) = Prj Ej (t). (17) dt The time-continuous model of total electricity cost for all IDCs is represented as: N ∑ ¯ C(t) = Prj Ej (t). dt j=1 2 The

Internet Traffic Archive http://ita.ee.lbl.gov/.

˙ X(t) = AX + BU (t) + F V, Y (t) = W X(t),

(18)

(19) (20)

where the system model matrices A ∈ ℜ(N +1)×(N +1) , B ∈ ℜ(N +1)×N C , F ∈ ℜ(N +1)×N and W ∈ ℜ1×(N +1) are given by:   0 Pr1 Pr2 · · · PrN  0 0 0 ··· 0     0 0 0 ··· 0  A= ,  .. .. .. ..  ..  . . . . .     B= 

0

0

···

0

O1×C ′ b1 I1×C .. .

O1×C O1×C .. .

··· ··· .. .

O1×C

O1×C

···

    F =   W =

IV. F EEDBACK C ONTROL S OLUTION

C ∑

The state vector is chosen as X(t) = ¯ [C(t), E1 (t), E2 (t), · · · , EN (t)]T , the control input is given by U (t) = [λij ]T ∈ ℜ(N C)×1 and ¯ V = [m1 , m2 , · · · , mN ]T , and the output is Y (t) = C(t). The state space model is represented as follows:

(

0 b0 0 .. .

0 0 b0 .. .

0

0

1

0

··· ··· ··· .. .

0

O1×C O1×C .. .

   , 

′ b1 I1×C

0 0 0 .. .

    ,  

· · · b0 ) ··· 0 ,

′ where I1×C = [1, · · · , 1]1×C and O1×C = [0, · · · , 0]1×C . For the purpose of simplifying design and analysis while facilitating the implementation, we convert the continuoustime system model to the discrete-time form using digitization method. Most of the engineering applications using a ZOH with a sampling period Ts , can be represented as follows:

¯ (k − 1) + ΓV (k − 1), X(k) = ΦX(k − 1) + GU Y (k) = W X(k),

(21) (22)

where X(k) , X(kTs ) is the system state at the kth sampling period, and the discrete-time system model matrices are given by: Φ = eATs , ∫ h ¯= G eAs Bds, 0 ∫ Ts Γ= eAs F ds.

(23) (24) (25)

0

The control system suffers from the input constraints which comes from workload and service latency. The workload constraints defined in (2) is represented in matrix form

as: HU = h, where the matrix H ∈ ℜC×N C are given by:  H1 H1  H2 H2  H= . ..  .. .

(26)

and the vector h ∈ ℜC×1 ··· ··· .. .

H1 H2 .. .

   , 

HC · · · HC H i (j) = 1, i = j; H i (j) = 0, i ̸= j, )T L2 · · · LC .

(27)

HC { i H1×C

h=

(

= L1

i=1

(28) (29)

To facilitate representing the delay constraints with respect to inputs, we rewrite (15) as follows: C ∑

λij ≤ µj (mj −

i=1

1 ). µj Dj

(30)

The matrix form of (30) is given by: ΨU ≤ φ,

(31)

where the matrix Ψ ∈ ℜN ×N C and the vector φ ∈ ℜN ×1 are represented as:    Ψ= 

′ I1×C O1×C .. .

O1×C ′ I1×C .. .

··· ··· .. .

O1×C · · · 1 )]. φ = [µj (mj − µj Dj O1×C

O1×C O1×C .. .

   , 

(32)

′ I1×C

(33)

In addition, the assignment workload must be nonnegative. Hence, the following constraint is added: U ≥ ON C×1

subject to the constraints in (1). Sleep (ON/OFF) Controllability Condition: This condition is derived from constraints perspective. The workload 1 ¯j = capacity for IDC j is λ Mj µj −Dj with the assurance of service latency bound Dj , where Mj indicates that all available servers are turned ON. The IDCs are capable to deal with the arriving workload with the assurance of service latency bound Dj if the arriving workload is less than the C N ∑ ∑ ¯j . λ µi ≤ sum of workload capacity for all IDCs, i.e.

(34)

B. Server Sleep (ON/OFF) control design In the dynamic control problem of electricity cost for distributed Internet data centers, two types of tunable variables are used to adjust the electricity energy consumption and cost online, i.e. mj denoting the number of activated servers in IDC j, and λij denoting the allocated workload in each IDC j. The most effective way to save power is turning OFF servers that are not used. Note that a change in mj is decided based on the current workload assigned to the IDC j. The inputs mj of the server sleep control is decided by the received workload λj for IDC j. We use the following equation to get the slow loop inputs mj : ∑  C λij  i=1 1   mj =  (35)  µj + µj Dj  , ∀j = 1, · · · , N.    

j=1

C. Dynamic workload control design Model predictive control (MPC) is predication and optimization based control theory which targets to achieve high performance for complex systems. MPC predicts the future changes of output using the current output measurement, the current dynamic models and the current control inputs. By solving an optimization problem to minimize the errors between the references and the predicted outputs, MPC derives the future control input series and implement the first one as current control input. After repeating the calculation, MPC will achieve the convergence stability and good tracking performance. Before designing the controller for the workload control loop, we verify the workload control loop controllability condition, which is described as follows. Workload Loop Controllability Condition: This condition comes from the control theory perspective. The system (19) is completely controllable based on the condition that, given an initial state X0 (t0 ) and a destination state Xf (tf ), there exists a tf ≥ t0 and a control solution U (t) to make the system run from the initial state to the destination state. The sufficient and necessary condition )for the fast loop ( controllability is rank B AB · · · AM B = M + 1, which is ensured since Prj > 0 and b1 > 0. Using sleep (ON/OFF) control (35), the system (36) is represented as: X(k) = ΦX(k − 1) + GU (k − 1) + Ω,

(36)

where the new input matrix G is F + Γ¯ µΨ, and µ ¯ = diag[ µ11 , µ12 , · · · , µ1N ]. The disturbance Ω is [ µ11D1 , µ21D2 , · · · , µN1DN ]T The MPC controller computes the control input U (k) to minimize the following cost function, which is given by:

J(k) =

+

β1 ∑

∥Y (k + s|k) − r(k + s|k)∥2Q(s)

s=1 β∑ 2 −1

∥U (k + s|k) − r(k + s − 1|k)∥2R(s) ,

(37)

s=1

where β1 and β2 denote the prediction horizon and the control horizon, respectively. X(k + s|k) is the state prediction for next s sampling period based on the current

state X(k). Similarly, r(k + s|k) and U (k + s|k) gets their corresponding meaning. Q(s) and R(s) are the weighting matrices penalized on the tracking errors and control inputs. Power Demand Smoothing Through Penalizing Inputs: The power demand jumping is produced by dynamic changing of allocated arriving workload and online turned ON/OFF servers for each IDC, and it can be represented by: ∆P = b1 ∆λj + ∆mj b0 . where ∆λj =

C ∑

(38)

∆λij is the change of allocated arriving

i=1

workload for IDC j. ∆mj is the change of turned ON severs for IDC j. Note that the power demand can be smoothed by slowly adjusting the allocated arriving workload ∆λj smoothly, i.e. penalizing inputs U (k). The cost function 37 can be interpreted as a compromise of electricity cost and power demand jumping. The relative magnitudes of Q and R provide a way to trade-off minimizing electricity cost for smaller changes in volatile power demand. To transform the MPC problem (37) to a standard leastsquares problem, we rewrite the system model (36) as follows: ′

′

′

′

W ′ = diag ¯= Ω 

    Y ′ (k) =     





 X(k)   ..    .       X(k)  Y (k + β2 |k)  ¯   , X(k) =  X(k)  , Y (k + β2 + 1|k)       .  ..   ..  . Y (k + β1 |k)

( Ξ(k) =

G ···

β∑ 2 −1

G

s=0



G 2G .. .

     β  ∑2 Θ=  s=0 G   ..  .   β∑ 1 −1 G s=0

X(k) β2 ∑

G

···

s=0

O G .. . β∑ 2 −1

··· ··· .. .

O O .. .

G ···

2G

s=0

..

G

s=0

s=0

.. . β∑ 1 −2

β∑ 1 −1

)T

.

G ···

.. . β1∑ −β2 s=0



G

      ,      

···

··· Ω

W )T

)

,

,    . 

Now, we define a new vector Π ∈ ℜ(N +1)β1 ×1 which is as follows: ¯ Π(k) = Υ(k) − W ′ X(k) − W ′ ΞU (k − 1) − W ′ Ω(k), (40) where Υ(k) is the reference trajectory with the prediction horizon β1 defined by: ( )T Υ(k) = r(k + 1|k) r(k + 2|k) · · · r(k + β1 |k) (41) Then we transform the MPC problem (37) to a standard least-squares problem as follows: (β ) β∑ 1 2 −1 ∑ min ∥W ′ Θ∆U (k) − Π(k)∥2Q(s) + ∥∆U (k)∥2R(s) , ∆U (k)

s=1

s=1

(42) subject to constraints:

¯ ¯ ∈ ℜ(N +1)β ×1 , where the matrices Y (k) ∈ ℜβ1 ×1 , X(k), Ω 1 Ξ(k) ∈ ℜ(N +1)β1 ×N C , Θ ∈ ℜ(N +1)β1 ×N Cβ2 , ∆U (k) ∈ ℜN Cβ2 ×1 and W ′ ∈ ℜβ1 ×(N +1)β1 are given by: Y (k + 1|k) .. .

Ω Ω

W

U (k + β2 − 1|k) − U (k + β2 − 2|k)

′



W

U (k|k) − U (k − 1) U (k + 1|k) − U (k|k) .. .

  ∆U (k) =  

′¯

¯ Y (k) = W X(k) + W ΞU (k − 1) + W Θ∆U (k) + W Ω, (39)

(

(

¯ I∆U ¯ ¯ (k − 1) + φ, Ψ (k) ≤ −ΨU ¯

(43)

−∆U (k) ≤ U (k − 1), ¯ ¯ I∆U ¯ ¯ (k − 1) + h, H (k) = −HU

(44) (45)

¯ ∈ ℜN β ×N C , I¯ ∈ ℜN C×N Cβ , φ¯ ∈ where the matrices Ψ 2 2 ¯ ∈ ℜCβ ×1 are derived based ¯ ℜN β2 ×1 , H ∈ ℜCβ2 ×N C and h 2 on (30) as follows:     Ψ O ··· O I1 O · · · O  O Ψ ··· O   I1 I1 · · · O     ¯ = Ψ  .. .. . . ..  , I¯ =  .. .. . . ..  ,  .   . . . .  . . .    φ¯ =     ¯= h  

O ··· Ψ φ(k) φ(k + 1|k) .. .

I1 I1 · · · I1   H O ··· O     ¯  O H ··· O   , H =  .. .. . . ..  ,   . . .  . φ(k + β2 − 1|k) O O ··· H  h(k)  h(k + 1|k)  . ..  .

O



h(k + β2 − 1|k) After transforming the MPC formulation to a standard constrained least-square optimization problem described by (42),(43), (44) and (45), we can use the standard leastsquares solvers to solve the electricity cost control problem. It is worth noting that the solution is based on the electricity

cost model and the workload constraints. The electricity price and the arriving workload are the parameters in the model.

W ORKLOAD FOR i Li

1 30000

FIVE

2 15000

Table I F RONT- END 3 15000

PORTAL SEVERS

4 20000

5 20000

D. Optimization solution of control reference The objective of MPC controller for electricity cost is to track the electricity cost reference value as accurately as possible. For cost savings, the minimum electricity cost and consumption references are preferred with the given electricity price and workload. Hence, an optimization problem is needed to derive the minimum electricity cost and consumption references. In this optimization problem, the price penalty of power peak is set to 0 to minimize the electricity cost. This problem has been studied in [5], which minimizes the electricity cost through solving a linear programming formulation given by: min

mj ,λij

N ∑

Prj Pj (λij ),

(46)

j=1

subject to the constrains (1), (2), and (15). This problem is solved using leverage Brenner’s fast polynomial-time algorithm. For more details please see [5]. Note that the prediction of control reference is needed in MPC controller. To this end, the optimization is conducted based on the predicted workload as discussed in Sec.III-D. Power Peak Shaving Through Setting Control References: To enforce actual power demand to fall under desired budget, we set the control power reference as Pr = Pro if Pro ≤ Prb and as Pr = Prb if Pro > Prb , where Pro is the control power reference derived by the optimization method (46), and Prb is the maximum power budget constrained by power grid market. E. Stability analysis The problem formulation in this paper is a standard MPC with constraints. The stability for such a constrained MPC closed-loop system does not rely on the poles of the closed-loop transfer matrix of linear systems. Mayne et al. proved the closed-loop stability for constrained MPC using the contraction mapping theorem [21]. For more detailed stability proofs see [21]. V. S IMULATION EXPERIMENTS In this section, we present the evaluation of the proposed control scheme using simulation experiments. A. Experimental setup To evaluate the proposed dynamic control of electricity cost minimization, we set up a simulation environment consisting of five front-end Web portal servers and three IDCs (i.e. C = 5, N = 3) located at three different locations: Michigan, Minnesota and Wisconsin. The Internet clients send requests to the front-end Web portal servers which then forward the requests to different IDCs for processing.

C ONFIGURATION j 1 2 3

OF

µj 2 1.25 1.75

Table II IDC S IN THREE DIFFERENT LOCATIONS PjL 285 285 285

PjH 130 130 130

Mj 30000 40000 20000

Dj 0.001 0.001 0.001

Table III E LECTRICITY PRICE IN THREE DIFFERENT LOCATIONS Time 6H 7H

Michigan 43.2600 49.9000

Minnesota 30.2600 29.4700

Wisconsin 19.0600 77.9700

The workload is represented by the average number of requests per second and is summarized in Table I. The delay constraint at each IDC is assumed to be 1ms; the processing speed for each server at three locations are 2.0 requests per second, 1.25 requests per second and 1.75 requests per second, respectively. Each server in the IDCs consumes 150 Watts when idle and 285 Watts when running at peak processing speed [19]. The configuration of IDCs in three different locations is shown in Table. II. B. Performance evaluations of power demand smoothing The power consumption is emulated from 6:00 to 6:10 on October 03, 2011. The electricity price in three different locations is shown in Table III. To compare the performance of dynamic control with only power demand smoothing and the optimal allocation policy for electricity cost minimization presented in [5], we conduct numeric simulation and show the power consumption results in Fig. 4(a)-4(c) for three IDCs in three different locations: Michigan, Minnesota and Wisconsin, respectively. The electricity consumptions for optimal workload allocation policy are 2.1375MWH, 11.4MWH and 5.7MWH at 6H for three IDCs, respectively. The power demand of optimal policy jumps to 5.7MWH, 11.4MWH and 1.628775MWH at 7H, respectively. We can clearly see that the dynamic control effectively smooths the power demand when the workload and electricity price vary at each IDC. The number of turned ON servers for three different IDCs are plotted in Fig. 5(a)-Fig. 5(c). The number of turned ON severs are 7, 500, 40, 000 and 20, 000 at 6H for three IDCs, respectively. However the number of turned ON severs for optimal policy jumps to 20, 000, 40, 000 (no jump) and 5, 715, respectively. The sudden increase in number of turned ON servers for each IDC will drastically increase the power demand. To avoid power demand jumping, the dynamic control approach turns ON or turns OFF servers gradually as shown in the figures. Although from the figures

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it is apparently visible that by avoiding peak powers a system may end up with more total electricity costs, the modern smart grid prefers stable power consumptions and may ultimately charge much more electricity costs for large power demand jumping. C. Performance evaluations of power peak shaving As discussed in V-B, the optimal electricity power demand jumps to 5.7MWH, 11.4MWH and 1.628775MWH at 7H at Michigan, Minnesota and Wisconsin, respectively. Let us assume that the available power budgets at 7H are 5.13MWH, 10.26MWH and 4.275MWH at the three locations, respectively. Under such power budgets the power

demand of Michigan IDC is obviously attainable but the Minnesota IDC and Wisconsin IDC violate their corresponding power budgets. To evaluate the performance of proposed approach, the simulations with power budgets constraints are conducted and the power consumption for three different IDCs are drawn in Fig. 6(a)-Fig. 6(c). The results show that the proposed control scheme is able to successfully track the power consumptions of Michigan and Minnesota to keep below their corresponding budgets but power consumptions derived by the optimal policy exceed the budgets. The power consumption of Wisconsin converges to the value between its power budget and the power consumption values derived from the optimal policy. The number of turned ON servers

for three different IDCs are drawn in Fig. 7(a)-Fig. 7(c). VI. C ONCLUSION This paper studies the dynamic control of electricity cost considering the power demand smoothing and the power peak shaving for distributed Internet data centers. The electricity cost is minimized using the real-time electricity pricing policy in modern power grid. The formulated electricity cost problem using differential equations is solved by the standard constrained model predictive control approach, in which the power demand is smoothed through penalizing the control inputs, i.e. the change of workload, and the power peak is shaved by setting power control reference below the desired budget. The extensive simulations based on real-life electricity prices verify that the proposed control scheme is able to achieve three goals together: minimizing the electricity cost, smoothing the power demand and shaving the power peak. R EFERENCES [1] X. Fan, W. Weber, and L. Barroso. Power provisioning for a warehouse-sized computer. In Proceedings of the 34th annual international symposium on Computer architecture, New York, NY, USA, 2007.

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