Reducing Price Volatility of Electricity Consumption for a Firm's Energy Risk Management

Aparna Gupta is an Associate Professor in the Lally School of Management and Technology at Rensselaer Polytechnic Institute. Her research interest is in risk management, financial engineering and financial decision support. She has published in leading quantitative finance and operations research journals and conducts U.S. National Science Foundation-funded research. She received her Ph.D. from Stanford University. Sreekanth Venkataraman is a Professor and Vice President – Business Incubation & Acceleration at the University of Petroleum and Energy Studies, India. His research interest is in the economics of renewable energy and energy storage technologies in the context of the utility industry. He received his Ph.D. from University of Madras and M.B.A. from Rensselaer Polytechnic Institute. The authors thank Xiaohua Wu for his research assistantship support in implementing the optimization problem in Matlab and helping to create the displays of the results.

April 2013,

Vol. 26, Issue 3

Reducing Price Volatility of Electricity Consumption for a Firm’s Energy Risk Management The authors develop a framework to assess strategies for optimally shifting peak load consumption using distributed storage systems. Risk management is achieved by optimal investment in storage systems and peak load shifting under stochastic electricity prices. The economic feasibility of storage systems on the end-user side is evaluated based on the optimized benefits. Aparna Gupta and Sreekanth Venkataraman

I. Introduction Energy demand is increasing in many countries as a result of economic and industrial developments; consequently, many governments are working toward providing reliable electricity to their populace. However, problems related with restrictions in electricity prices by means of a price ceiling and flat rates have produced a difference between marginal electricity

generation costs and energy consumption cost of electricity. This increases the growth of demand faster than the growth of generation capacity. In addition, the volatility of wholesale electricity prices affects the retailer’s ability to generate profit and increases their investment uncertainty. n order to deal with the volatility in electricity prices, there is a need for an effective load management strategy. One

I

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

89

of the measures to effectively manage the load is to induce changes in the electricity consumption patterns of end users to reduce the simultaneity of high demand with high electricity prices. Such measures are typically demand-side measures, and there has been a significant focus on examining the effectiveness and feasibility of implementation of demand-side measures. These measures adopted by utilities are more commonly known as demand response (DR) programs. A change in consumption pattern can be made by means of change in the price of electricity (pricebased programs) or incentive payments. Time-of-use (TOU) is a particular type of price-based program, whereby peak periods have higher prices than prices charged during off-peak periods; consequently, the users are expected to change their use of electricity. n recent years, utilities are beginning to employ a number of load-shifting strategies as part of a concerted attempt to reduce the congestion on the grid. Time-of-use pricing, a strategy still in its early stages, has begun to gain a lot of interest and attention amongst both the industry and the policymakers in the more recent past. Currently, the time-of-use pricing is predominantly implemented on a limited (and voluntary) basis, but there is enough reason to believe that it will increasingly see greater acceptance as a viable load management strategy on a

I

90

mandatory basis. At present, Pacific Gas & Electric utility in the United States is one of the few utilities that currently offer a mandatory time-of-use electricity tariff for their customers. The time-of-use tariffs are typically a reflection of a load profile of an enterprise. The tariffs are high during the period when the aggregate load profile reaches its peak and are low during the off-peak hours of the day.

Utilities are beginning to employ a number of load-shifting strategies as part of a concerted attempt to reduce the congestion on the grid. However, the effectiveness of time-of-use pricing as a load shifting strategy is likely to be influenced by a couple of factors: the availability and development of technology to store electricity; and the difference between peak and off-peak electricity prices. The importance of energy storage technologies for grid reliability has been recognized by the policymakers. In addition, the willingness on the part of the utilities to adopt them is an increasing evidence of their acceptance among the key stakeholders. Apart from their ability to improve grid stability, storage technologies also help

increase utilization of power plants, reduce the need for large capital expenditures on additional power plant generation capacity and allow existing plants to run at higher operating efficiency. The peakshifting storage can also function to enhance the reliability by reducing the peak load of the system (Figure 1). Energy storage predominantly consists of pumped hydro systems, most of which were built in the 1970s and 1980s in response to increases in oil prices and natural gas prices. The interest in pumped hydro stations waned after the 1990s. However, a confluence of many factors, such as the anticipated increase in the construction of new renewable energy generation sources, wind and solar, and their integration into the grid, and the market for new applications has renewed the interest in energy storage technologies. Currently, there are several energy storage technologies, such as pumped hydro, compressed air energy storage (CAES), batteries of various kinds, and flywheel. While there has been a continuous reduction in cost and improvement in technology of these energy storage technologies, battery systems in particular are generating a lot of interest for their potential in small-scale applications, and particularly their role in load shifting strategies. hile there has been a lot of research on the viability of energy storage technologies as

W

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

[(Figure_1)TD$IG]

Figure 1: The Possibility of Load Shifting Through Use of a Storage System

a load management strategy, the focus of this article is on electricity markets and the arbitrage opportunities available to an enterprise by using a merchant storage device, rather than on storage technologies themselves. This article describes the formulation of a mathematical model to find the best operating strategy for storage systems by an end-user enterprise. The selection of charging–discharging schedule takes into account the continuous market price, demand evolution and the technological parameters of storage devices. Since the market price of electricity exhibits different trends on a daily, weekly, and seasonal basis, optimal strategies are designed keeping in mind each of those time periods. Strategies under different price and demand patterns are compared, and key parameters of the model, such as capacity, maximum charging speed and discharging speed, and storage efficiency factor, are analyzed to determine their impact on the resulting optimal benefit from load shifting. April 2013,

Vol. 26, Issue 3

T

he remainder of this article is structured as follows. Section II develops an overview of the related research literature. In Section III, we will begin formulating the load shifting problem as a dynamic optimization problem from an enterprise perspective. Section IV discusses the results for the different settings of the problem. In the last section, we summarize the key conclusions of the article and provide suggestions for further work on this topic.

II. Literature Review Demand response has been an active area of research. Recently, many studies have been carried out to determine how end-users can adjust their load level according to a determined DR program. Juan et al. [10] present an optimal load management strategy for residential customers that utilizes communication infrastructure of the future, a Smart Grid. The optimal load management strategy entails the use of activity scheduling by the

user (expressed as the power purchase of energy) to negotiate an optimized agreement between the utility and the end use using heuristic optimization techniques. Using heuristic methods, authors obtain reasonable solutions with low computational burden. Conejo [3] presented a demand response model that minimizes the cost of energy consumption considering the load-variation limits, hourly load, and price prediction uncertainty. Sianiki [17] proposed a methodology for demand response that incorporates customer preferences for peak load electricity consumption by Analytic Hierarchy Process, and concludes that it achieves the twin goal of an improvement in both the consumer’s budget and global electricity consumption on the grid. Various research studies have been conducted over the years in an attempt to improve the accuracy of heuristic optimization techniques to find a global solution. Molderink [14] developed an algorithm for the control of energy streams based on one-day head predictions, but concluded that in such cases forecasting errors affect the outcomes of this approach. Joo [9] introduced a multi-layered multidirectional information exchange framework for integrating the adaptation of demand to the changing market conditions in order to include better information of the demand side in system optimization. In the model, which is formulated as a Lagrange dual decomposed

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

91

optimization problem, the three different layers communicate with each other via the Lagrangian multiplier or the price at each layer to create individual demand functions of the end users at the primary layer in order to enable better information of the demand side. Tanaka [19] presented a methodology that reduced electricity consumption and the cost of electricity by minimizing the power flow fluctuation in the Smart Grid with a fleet of houses optimally using a battery bank and a heat pump. iven the increasing relevance of demand response programs over the years, the role of energy storage technologies in demand response has assumed increased importance, where some of the evidence lies in the significant body of literature that has focused on an overview of various applications of energy storage technologies (including peak shaving). Cook et al. [4] provided an overview of various benefits (including peak shaving) accrued by electric utilities, and concluded that battery energy systems can be an economically attractive addition to power generation capacity. Oudalov et al. [16] highlighted the combinations of technologies and applications with the highest benefit for the owner of the energy storage systems (ESS). The analysis shows that frequency regulation and integration of renewables are likely to be the most lucrative applications of ESS. The technologies that are most

G

92

suitable for the applications are estimated to offer payback times of five to 10 years, and are unlikely to be restricted by their geographical footprint. Miller et al. [13] discuss the possibilities for some of the applications of energy storage, and with regards to peak shaving. They note that given the current price levels of battery storage technologies and the overall total system costs, it is very challenging to develop an

The battery system break-even costs were found to be higher for smaller battery penetrations, thus favoring the utility phase-in process and the manufacturing process development. economic justification for utilities to store energy during off-peak times using batteries and redispatch this energy later at peak times, unless they are combined with other applications. ignificant research on the possible applications of the energy storage technologies led to a body of literature that focused on economic feasibility of battery energy storage systems. Sobieski [18] provides estimates of battery system break-even costs for a representative U.S. utility system considering a full spectrum of tangible battery economic credits. The battery system break-even costs were found to be higher for

S

smaller battery penetrations, thus favoring the utility phase-in process and the manufacturing process development. Nieuwenhout et al. [15] evaluated a number of business cases for financially beneficial operation of electricity storage in case of small users in the built environment in Netherlands. Financially, most attractive is the case in which generated electricity is stored for own use at a later time. Based on the prevailing tariffs level, the mechanism to prevent feed-in of self-generated electricity was found to be financially the most attractive option for the recovery of investment cost for storage systems. Gross benefits at the current tariff levels amount to about $100–150 per year, depending on different technologies chosen, assuming a payback period of 10 years. While the economics of energy storage system depends on several factors, such as, cost of battery, electricity price spread, the charge–discharge schedule is one of the most important factors that impacts the economics of energy storage systems. Optimization techniques have been used quite extensively to assess the economic impacts of energy storage systems over the years. Maly [12] used a multi-pass dynamic programming approach for the optimal charging– discharging schedule for a battery energy storage system (BESS). The study noted that sizing and charge scheduling of BESS depends on utility rates, and called for a balance between

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

smaller size batteries and larger size batteries, since while smaller batteries give faster payback times; the larger ones were necessary for reliable load forecasts. Graves et al. [8] examined business cases for generating benefits from smallscale energy storage systems and concluded that grid-connected energy storage is most attractive if the cost of the battery falls down to less than $300 per kWh. inear optimization models have been extensively used in estimating the economic benefits of the storage systems over the years. Ahlert [1] uses a linear optimization model based on ex ante known values of price and load data to estimate the economic benefits of a distributed energy storage system at the enduser level by determining the cost of optimal charge–discharge schedule. The price spreads and the distribution of the market price and load curves are the primary factors that influence the economics of the storage system. Recognizing that forecasts based on the known ex ante values of price and load data represented a flaw in the simulation model, Ahlert [2] presented a simulation model to determine the economic impact of price forecast accuracy variations on the operations of a distributed storage system that is located at the end-user location. It was found that price forecast errors have a significant impact on the achievable savings of a distributed storage application that aims at arbitrage accommodation.

L

April 2013,

Vol. 26, Issue 3

While the larger focus has been on determining the economics of energy storage systems, sufficient research has also been conducted to improve the algorithms for optimizing the charge–discharge schedule. Lee [11] presented a new algorithm for the solution of nonlinear optimal scheduling problem, called a multi-pass iteration particle swarm optimization (MIPSO). Taking into account the effects of wind

The price spreads and the distribution of the market price and load curves are the primary factors that influence the economics of the storage system. uncertainty and load, the MIPSO algorithm is used to solve the optimal operating schedule of a battery energy storage system (BESS) for an industrial time-ofuse (TOU) rate user with wind turbine generators (WTGs). The efficiency and feasibility of the algorithm in solving the scheduling problems of the power system is tested by a numerical example, where it is revealed that average solution of MIPSO was better than other algorithms, while the computational times was half that of other algorithms. Xu et al. [20] combined a load aggregator with an energy storage device to determine the

energy savings by optimally scheduling the imported power in the day-ahead market. They concluded that the energy savings were optimal if the forecasts of imported power during operation in the real-time balancing market were perfect. Garcia-Gonzalez [7] conducted a similar analysis, where he combined a wind electricity generator with a pumped hydro energy storage system to compare the two formulations (LP and MILP) of a two-stage stochastic model constructed to solve the problem of coordinating the joint operation of a wind farm and a supporting hydro pumpedstorage unit. Dukpa [6] proposed an optimal participation strategy for a wind electricity generator by pairing it with an energy storage device to maximize its operational benefits. Specifically, maximization of profits and returns and minimization of risk and penalties due to uncertainty in wind forecast, and showed that maintaining an energy reserve sufficient to ensure that the risk of optimal participation schedule is the key to optimizing the performance of the energy storage system. Daryanian [5] used a fast non-simplex algorithm to show that the benefits of energy storage systems to utilities can be increased by close to 50 percent under real-time pricing (RTP) rates compared to the time-of-use (TOU) rates based control, and argued that the additional costs of RTP based retrofitting was justified.

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

93

While the existing literature has focused on the feasibility of distributed energy storage systems as a load management strategy and optimization of the storage performance through charging–discharging schedule, the stochasticity of electricity prices has been neglected in the various analysis. In this article, we develop a framework to assess strategies for optimally shifting peak load consumption using distributed storage systems under stochasticity of electricity prices. We examine how an enterprise end user can optimize their investment in energy storage systems in order to minimize the risks associated with electricity price stochasticity.

electricity price and load profile pattern. It is assumed that there is a single charge and discharge cycle per day. Multiple stochastic price and demand profiles are generated by the month (summer vs. winter) of the year and the day (i.e., weekend, weekday, holiday) of the week. The charging and discharging schedules, i.e., the starting and ending times of

Charging–discharging feasibility constraints

III. Problem Formulation In this section, we first describe the overall optimization problem, combining all the models that constitute the formulation of the problem. We then discuss the two major models of the problem, the electricity price evolution model and the enterprise load model. This decomposition of the problem is accompanied with a detailed description of market and consumer demand parameters. A. Optimization problem The objective of the optimization problem is to maximize the daily benefit for the enterprise from the storage system it invests in, for a given 94

words, the charging capacity constraint is defined as one where the charging duration cannot exceed the maximum capacity of the storage device. We must also impose a constraint that the discharging capacity cannot be greater than the charging capacity in any charging–discharging cycle, and this condition in the model is imposed by two types of constraints on the discharge decision for the storage system. These are the constraint by probability and constraint by the mean of charging–discharging cycle. These are explained below.

charging and discharging periods, are the four key variables that are likely to impact the benefits of the arbitrage opportunities created by energy storage system. e construct the objective function as the savings in electricity cost due to the discharging of electricity (at offpeak tariffs) net of the cost of charging during the off-peak hours (see Appendix I). This in effect, measures the net benefit of the energy storage system to the enterprise, which depends on its load profile. This is contingent on the assumption that the charging/ discharging operation is less than the maximum allowable rate of charging and discharging. In other

W

1. Feasibility constraint Probability constraint: This constraint implies that at any given time during the charge– discharge cycle, the discharging capacity cannot exceed the charging capacity of the storage device. Furthermore, the probability that the charging capacity accumulated in a charge–discharge cycle is more than the discharging done in that cycle is enforced to be not lower than 99 percent. By imposing such a condition, we are implying that the charge–discharge solution created for the storage system is physically feasible. Mean constraint: By imposing a mean feasibility constraint, we imply that on average the discharging capacity cannot be greater than the charging capacity of the storage system. The difference between the two constraints is that, while the first

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

[(Figure_2)TD$IG] constraint eliminates any possibility of a solution being physically infeasible, and hence is a more strict condition; the mean constraint on the other hand refers only to the average charging and discharging capacity, and in doing so is easier to work with. Bounds for decision variables: The start and end time for charging and discharging are the decision variables of the problem, and are defined to be within a 24-hour period. Time order constraint: We impose a constraint that the start time of charging happens before the end time of charging in a 24-hour period. Similarly, the start time for discharging happens before the end time for discharging in a 24-hour period. Moreover, end time for charging proceeds the start time for discharging. While the probability constraint clearly is more stringent than the mean feasibility constraint, we test the feasibility of the energy storage system usage under both constraints. B. Electricity price model: auto-regression model with trend and seasonality The objective of the problem we address in this article is to reduce the price risk due to fluctuating demand for electricity. As the spread between the peak and the off-peak price of electricity is an important factor driving the arbitrage opportunities for the end-user, the price profile (of electricity) becomes a key input in April 2013,

Vol. 26, Issue 3

Figure 2: Simulated Price Profile of Long Island 2009

the model. Inability to incorporate the spread between peak and offpeak prices, on the other hand, can lead to significant errors in estimation. However, simulation of stochastic price curves using price model calibrated from historical data sets is sufficient to determine the optimal strategy for the storage system based on the benefit-risk ratio. In this model, we plot the electricity price during peak and off-peak periods to demonstrate the price spread by type of month and day (holiday/weekday). These plots are then used to describe the behavior of the price in terms of two types of components, which are the deterministic trend and a stochastic variable that follows an auto-regressive trend. The model is calibrated from location based marginal price of Long Island (2009) at day-ahead market (see Appendix II). Based on the price profile, we extract the deterministic trend step by step. First, we assume the annual

average is the same as the annual price level, and similarly, the monthly average is assumed to be identical to the monthly price level. The average of weekday and holiday price level is used to characterize the weekday/holiday price trends. The remaining value of the prices is used to calibrate the stochastic parameters in the model (Figure 2). s an illustration, the simulated annual price profile calibrated from locationbased marginal price (LBMP) of Long Island (2009) at day-ahead market (DAM) is shown in Figure 3. The 10 simulations are generated for ‘‘January’’ and ‘‘Weekend’’ pattern. The graph shows high degree of variation in the price level of electricity. The prices are typically very high in the afternoon as a result of the increased demand due to heating needs. In the early and late evening, the prices are lower and reflect the decline in demand for electricity during those time periods.

A

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

95

[(Figure_3)TD$IG] Table 2: Monthly Levels of Load/Price. Month

Table 1: Annual and Weekday/Weekend Levels of Load/Price. Level

Annual

Weekday

Weekend

Load

537.11

44.43

110.03

Price

86.10

2.43

6.04

In order to better understand the structure of price model, we now examine the deterministic levels of price and the load profile based on the 24 load/price profile patterns that are generated (12 months and weekday/weekend). Table 1 shows that the aggregate load is much higher during weekends than during weekdays, which can be attributed to a greater residential demand for electricity (cooking and heating purposes). Conversely, the electricity price is lower during the weekends. The intuitive reasoning behind this is relatively simple. The industrial and commercial demand for electricity is much higher during the weekdays, and this translates into a higher power load on the grid and consequently higher electricity prices. The annual and weekday/weekend levels are listed in Table 1. The monthly levels of price and aggregate load offer a slightly 96

different intuitive perspective for their seasonal trends. While the electricity prices reach their peak during summer, primarily as a result of the increased demand for cooling, the load profile seems to reach its peak during the winter. The high demand for electricity from commercial and industrial users coupled with the increased heating needs by the residential end users drive the load profile during the winter upwards. If we decompose the price and the aggregate load trends on an hourly level, we can see that the hourly price level shows a similar pattern in both summer and winter, except that the daily peak price occurs during early evening in the winters (5–7 pm) and late afternoon in the summers (3– 4 pm). Increased demand for heating during the early evening hours is primarily responsible for the electricity price reaching its peak later than in the summer. The load profile, on the other

Price

1

34.954

4.0175

2 3

26.879 11.251

7.0763 2.3959

4 5

6.6393 46.987

6

Figure 3: Monthly Levels of Load/Price

Load

103.23

1.127 10.081 1.0791

7 8

88.151 0.65824

9 10

19.791 46.763

11

46.819

7.9694

12

68.479

8.0884

32.899 34.261 27.158 23.055

hand, does not seem to exhibit significant variability in pattern during both the summer and winter months. The load profile seems to peak early night and noon uniformly across all the months, though the peak load seems to be higher in the winter. Table 2 and Figure 4 exhibit the monthly levels of load and price profile while Figure 5 provides the hourly levels of the electricity load. Load model: time-dependent binomial model with jumps The consumption (or demand) of electricity by the consumer (or the end user) is the major factor that drives the pricing of electricity. As discussed earlier in the article, electricity prices typically reflect the load profile, which are different for different users. Electricity prices are high when the aggregate load is at its peak, and conversely, prices are low when the aggregate load decreases. While the historical data of these different load

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

[(Figure_4)TD$IG]

Figure 4: Hourly Level of Electricity Prices

profiles are important for the purposes of our analysis, building a model based on the historical load profile of the different enterprises represents a significant challenge. Consequently, we construct a time-dependent binomial model with jumps that will simulate the actual load profile of an enterprise end user. The time-dependent binomial model simulates the basic shape or evolution of the load along with volatility of the

[(Figure_5)TD$IG]

Figure 5: Hourly Levels of Electricity Load

April 2013,

Vol. 26, Issue 3

load profile. Time-dependent parameters that define the directional trend are flexible to generate all kinds of load patterns. he other component of the model, a jump model, captures sudden surge or drops along the time horizon, which is an important characteristic of the load profile (see Appendix III for details). The key feature of this model is that, both the price and demand profiles are simulated

T

discretely in 15-minute intervals. We use a spline interpolation process to calibrate the 15-minute interval from the hourly dayahead market (DAM) price data. As a result, we use the cubic spline function to transform the discrete profiles into more flexible continuous ones. This enables the charging and discharging schedule to be continuous from the perspective of the optimization problem. It needs to be noted here that while directional trend parameters characterize the load profile of the enterprise (shown in Appendix III), it is the scale of the step size that will eventually define the load profile of the enterprise. s an illustration, we use the characteristics observed from the real load data of a detached house in Quebec, Canada (1994–1995) to simulate the peaks in the late afternoon and troughs in the late night/early morning periods. Our model yields the following values: step size = 0.03 MW/15 minutes, jump size = 0.02MW, mean of jump size = 0, standard deviation of jump size = 0.001 MW. The simulated load profile and load profile from observed data are displayed in Figure 6. The real load data used here pertains to the measured load profiles of the Canadian detached house in Hydro Quebec (1994–1995). The parameters of our model are calibrated from the real data, and we can see that the simulated profile captures the key characteristics of the real load profile.

A

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

97

[(Figure_6)TD$IG]

Figure 6: Simulated and Real Load Profiles

IV. Results and Discussion In this section, we describe the process chart used to solve the optimization problem and demonstrate its validity for determining the optimal charge– discharge schedule through an example. Having demonstrated the validity of the solution algorithm, we test the physical feasibility of the energy storage system by imposing the two constraints developed in the model, i.e. the probability feasibility constraint and the mean feasibility constraint. Subsequently, we examine the impact that characteristics of the storage system, such as capacity and charging rate, and the load parameters, such as step size and jump size, have on the solution properties. Finally, we compare the results of the performance of the storage system during the summer and winter months, and by type of day (holiday/ weekday). 98

A. Solution algorithm Since all the daily patterns are dependent on the time of the day, the order between charging and discharging periods is not critical to our model. However, for the purpose of simplicity, we assume that charging occurs between 8 pm and 9 am, while discharging occurs between 9 am and 8 pm. A time interval of observation of 15 minutes is chosen based on practical consideration and data availability. The solution algorithm of this stochastic and

continuous optimization problem is illustrated by the flowchart given in Figure 7. he optimization problem is solved using MATLAB. The fmincon() function in optimization toolbox is used as solver, and sequential quadratic programming (SQP) is the core algorithm utilized for arriving at the solution. We use two valuesearching algorithms to arrive at a global optimum. During the experimental trials, the results are found to be very sensitive to the initial value of time

T

[(Figure_7)TD$IG]

Figure 7: Processing Chart of How the Storage Problem is Solved

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

period, ti’s. The optimal objective value gets easily trapped in local minima, with a random initial value of ti’s. The first search enhancement is by imposing grid search, where the time horizon [0,24]hr is divided into small time intervals of 30 minutes each. Subsequently, all combination of ti’s at the end of each of these subintervals is substituted as the initial points to yield global optimum values. The second search algorithm to further hone into a global optimum is by using particle swarm optimization (PSO) algorithm, where a number of stochastic price/load profiles are generated for each combination of decision variables and the combination with best mean is chosen as the optimal charging/ discharging operating time. B. Discussion of results We now solve the optimization problem by calibrating the load model for the weekend data for the month of January of Long Island (2009). The storage capacity is assumed to be 5 MWh and the charging capacity, V, is 1 MW/hour. The load profile of the enterprise has a step-size of 0.01 MW/15 minutes, and the average load is 1 MW/hour. The average jump size in the load model is calibrated at 0.001 MW. roviding input of these parameters into the algorithm yields that the optimal charging time to be from midnight to early morning, while the optimal discharging time is from late afternoon to late night.

P

April 2013,

Vol. 26, Issue 3

Table 3: Test of Actual Physical Feasibility under Different Cases of Probability and Mean Constraints. Probability (Charge > Discharge)

t1

t2

t3

t4

100% 90%

1.9 1.6

7 6.5

15.5 16

20.6 20.9

96.3 113.6

75%

1

7.1

14.9

21

120

50% E(Charge) = E(Discharge)

1.1 1.2

5.9 6.2

16.2 16.1

21.1 21.8

102.3 148

The intuitive reasoning behind these results is easy to understand, as they inherently are a reflection of the price profile. The price profile of an enterprise is likely to suggest that the cost of charging is likely to be the lowest during off-peak hours (midnight to early morning). This logically implies that any enterprise will choose to charge when the cost of charging can be minimized. Conversely, they will opt to discharge when the electricity prices are at their peak (due to high levels of aggregate load), which is likely to be during the period between afternoon and late evening. Discharging during the peak hours, i.e. when electricity prices are high, will enable the enterprise to avoid electricity consumption during peak hours, thus yielding savings that might be significant. Based on the inputs in our example, the estimated daily savings in electricity costs for the enterprise is estimated to be $147.20. This translates into an annual benefit of $46,368 when the charging occurs from 1:12 am through 6:12 am and discharging occurs from 4:00 pm to 9:36 pm. After solving the algorithm under specific parametric

Benefits ($)

configuration, we now test the actual physical feasibility of the energy storage system under different cases of probability and mean constraints (Table 3). When the probability that charging capacity exceeds discharging capacity is set at 50 percent, the time of charging and discharging is very similar to the case when the average charging capacity equals average discharging capacity. However, there is a significant difference in the benefits between the two cases. This shows that the physical feasibility of the energy storage system is likely to be very different from what the average configurations would lead one to conclude. The benefits associated with the energy storage system, however, increase up to the case when the probability of charging capacity exceeding the discharging capacity is set at 75 percent. However, when this probability becomes greater than 75 percent, the benefits starts to decline. For instance, when the probability is set at 90 percent, the daily benefit generated is only $113.6, which further declines to $96.3 when this probability is set at 100 percent. The decline in the benefit is due to

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

99

the fact that the time of charging and discharging under the cases when probability is set at 90 percent and 100 percent, respectively, is extremely strict. Since the charge and discharge times are not state-dependent, i.e. are independent of the actual scenario of load profile observed in a 24-hour period, the optimal charge and discharge time should work for all scenarios – in these cases 90 percent or 100 percent, respectively. This stringent requirement shrinks the window of charging, and hence discharging, thus reducing the advantage from the price spread between peak and off-peak electricity prices. nce the physical feasibility of the storage system is determined, it is important to

O

understand the impact the storage characteristics, such as capacity and charging rate, and load parameters, such as step size and jump size, are likely to have on the performance of the storage system. Increase in the capacity of storage results in an increase in the benefits due to storage, but only until a threshold is reached (Figure 8). Continued increase in the capacity of storage beyond the threshold level is likely to result in reduction in the benefits due to decreasing returns to scale. This implies that the increase in benefits due to a higher storage capacity is not likely to be justified by the increase in costs of the storage system. The relationship between the charging rate and the performance of the storage system, on the other hand is direct

(Figure 9). Low charging rates and discharging rates imply longer charging periods and consequently, higher costs of charging, which in turn undermines the efficiency of the load shifting strategy. n order to understand the impact of step size and jump size of the load profile on the solution characteristics, we examine the load profile of an enterprise. The load profile of an enterprise at a time is expressed as a sum of the load profile for the previous time period, the likely directional trend of the load profile in the current time period, and the sudden spike, if any, in the demand for electricity. The jump size for the sudden spike in load captures the volatility in the load profile. (See Appendix III for

I

[(Figure_8)TD$IG]

Figure 8: Impact of Storage Capacity C 100

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

[(Figure_9)TD$IG]

Figure 9: Impact of Maximum Charging Speed V

details.) In other words, the jump size determines the eventual shape of the load profile in the event of an upward or a downward trend in the load of the enterprise. As the performance of the storage system depends upon the load profile to a significant extent, the step size and the jump size become critical factors in impacting the solution characteristics. From Figure 10, we can see that the step size of the binomial model has little impact on the benefits of the storage system, even though this results in a greater volatility in the load profile. The minimal impact on the benefits of the storage technology is due to relative insensitivity of the storage system April 2013,

to the load profile as compared to the price profile. On the other hand, higher jump size results in a greater volatility in load profile due to losses in the discharging period. This in turn reduces the benefits of the storage system (Figure 11). In this section, we compare the benefits of the energy storage system in summer and winter. For the purposes of our analysis, we assume that price and load profile in January is representative of the winter months, and similarly, the price and load profiles in July is representative of the summer months. We also make a further distinction between the holidays and weekdays for both the

summer and the winter periods. Table 4 lists the optimization results for both January and July during weekdays and holidays. e first examine the difference in daily benefits due to storage in summer and winter on weekdays. The daily benefits are higher in summer on the weekdays and holidays, primarily because the time to start discharging is earlier in the summer than in the winter. This need to discharge earlier in summer is because of a longer duration of peak hours in summer than in winter. In the winter, the peak hours typically shift to later in the evening and lasts for shorter duration than in the summer. The daily benefits in the

W

Vol. 26, Issue 3 1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

101

[(Figure_10)TD$IG]

Figure 10: Impact of Step Size of Binomial Model V

summer during weekdays are much higher than the benefits during the holidays. This is attributed to higher demand resulting from industrial and commercial consumption of

[(Figure_1)TD$IG]

electricity on weekdays as compared to the holidays. The difference in daily benefits on weekdays and holidays during the winter, although relatively insignificant, can be attributed to

a higher share of the demand for heating by the residential end users, thus negating the possible impact of the demand from industrial and commercial end users.

Figure 11: Impact of SD of Jump Size 102

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

Table 4: Results of Performance of Energy Storage System during Summer and Winter by Type of Day.

Daily Benefits

Holiday

Holiday

Weekday

Weekday

(Summer – July)

(Winter – Jan.)

(Winter – July)

(Winter – Jan.)

779.9

108.86

351.15

95.57

c1 c2

01.00 06.00

02.00 06.00

23.15 04.15

01.00 06.00

d1

11.30

16.15

12.15

15.15

d2

20.00

20.00

20.00

20.00

[(Figure_12)TD$IG]

Figure 12: Arbitrage Benefit in Different Months

W

e plot the benefits of the energy storage benefits on a monthly basis. As we can see from Figure 12, the benefits due to arbitrage are highest during the summer, when the price spread between the peak and the off-peak prices are typically at their highest. The benefits due to arbitrage in the winter are also significantly high. The price spread in this case is also high due to high peak prices, which is due to increased heating needs. Conversely, the spring and the fall seasons offer comparatively fewer opportunities to benefit from arbitrage. It needs to be emphasized the benefits of the arbitrage are largely determined by price during times of charge

April 2013,

and discharge, thus affecting the charging schedule necessary for optimizing the performance of the energy storage system. he time of charge and discharge, we now plot the time of charge and discharge on a

T

monthly basis to develop a better intuitive understanding of how these in turn impact the benefits due to arbitrage. During weekdays (Figure 13), charging occurs around midnight and ends in the early morning hours uniformly across all months. However, we do notice a difference in the discharging pattern during the summer and winter months. The discharging begins three hours earlier in summer due to the shift in the peak time by four hours (3 pm in summer compared to 4 pm in winter). The plot of the charging and discharging time on the weekends reveals a higher degree of variability in the start times for charging and discharging across all months (Figure 14). This higher degree of variability is attributed to the differences in the optimal spread between peak prices and off-peak prices that is critical to optimizing the performance of the energy storage system. The optimal spread between peak and off-peak prices is different at various times due to the fluctuations in the pattern of

[(Figure_13)TD$IG]

Figure 13: Weekday Timing of Storage Charging/Discharging

Vol. 26, Issue 3 1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

103

[(Figure_14)TD$IG] Decision variables: The times at which the energy storage system are charged and discharged in a day are continuous variables, and are denoted as t1, t2, t3 and t4. In the above objective function, Eq. (1), they play the following role. t1 – start time of charging period, t2 – end of charging period, t3 – start time of discharging period, and t4 – end of discharging period.

Figure 14: Weekend Timing of Storage Charging/Discharging

consumption for electricity. In other words, the relative share of residential demand vis-a-vis commercial and industrial demand of electricity on weekend is likely to impact the electricity.

V. Conclusions and Future Work Based on our analysis, while we are able to conclude that the benefit from optimized load shifting strategy can be potentially significant, the profitability of the storage system needs to be evaluated by incorporating the capital cost and lifetime of storage system. Secondly, a stricter probability constraint on the discharge capacity seems to reduce the benefits relative to the equal means constraint. Thirdly, duration of charging time and the rate of charging speed both have a significant impact on the load shifting strategy, and are critical in maximizing the benefits due to arbitrage. Finally, parameters of characterized load model, such as the binomial tree step size and 104

sudden shock jump size, are important to the load shifting strategy and should be further calibrated in order to maximize the benefit by changing the enterprise’s load structure. However, there are some gaps in this study, which provide scope for further research. The scope for future work lies in assessing the economic feasibility of storage system by taking into account the investment cost and lifetime of storage, and by integrating the trend and seasonality of the load and price model into the problem. Further work should also strive to solve the optimization problem by different type of storage technology.

Appendix I. Optimization Model The optimization model is expressed as:

max E

t1;t2;t3;t4



Z

t2

t1

"Z

t4

t3

Pt minðLt ; VÞ dt #

Pt V dt  F

(1)

Appendix II. Electricity Price Model The electricity price model is expressed as: Pt ¼ a þ bk Dkt þ

12 X

ci Mit

i¼1 24 X þ d j H jt þ Xt ;

(2)

j¼1

Xt ¼ Xt1 þ Nð0; s 2 Þ;

(3)

where Pt = electricity price profile (including a deterministic and stochastic part), Xt = stochastic component of price profile, a = annual price level, bk = holiday or weekday price level, ci = monthly price level, dj = hourly price level, Dkt = 1 if t is in a holiday, Dt = 0 otherwise, Mit = 1 if t is in month i, Mit = 0 otherwise, Hjt = 1 if t is in hour j, Hjt = 0 otherwise, w = parameters that describe the autocorrelation of X,

1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

The Electricity Journal

N(0,s2) = i.i.d. normal random variables with mean zero and variance s2.

Appendix III. Load Model: Time Dependent Binomial Model with Jumps The time dependent binomial model is expressed as: Lt ¼ Lt1 þ ð put þ ð1  pÞdt Þ þ hl Lt1 dm;s 0

(4)

where Lt = load profile, p = random Bernoulli variables with 0.5 mean, ut, dt = time dependent step size of moving up or down, hl = Bernoulli variable with mean l for jumps, and d = random magnitude of jump with Normal distribution N(m,s). The parameters, ut and dt, are represented as: ut ¼ ð1 þ cosð2pt þ eÞÞv; dt ¼ ð1  cosð2pt þ eÞÞv: &

References [1] K.-H. Ahlert and C. van Dinther, 2009, Sensitivity Analysis of the Economic Benefits from Electricity Storage at the End Consumer Level, in: Proc. IEEE Bucharest PowerTech, 2009, pp. at 1–8. [2] K.-H. Ahlert and C. Block, 2010, Assessing the Impact of Price Forecast Errors on the Economics of Distributed Storage Systems, in: Proc. 43rd Hawaii Int System Sciences (HICSS) Conf., 2010, pp. at 1–10.

April 2013,

[3] A.G. Conejo, J.M. Morales and L. Baringo, 2010, Real Time Demand Response Model, IEEE Trans. Smart Grid, 1 (3) at 236–242. [4] G.M. Cook, W.C. Spindler and G. Grefe, 1991, Overview of Battery Power Regulation and Storage, IEEE Trans. Energy Convers., 6 (1) at 204–211. [5] B. Daryanian, Nov. 1991,R.E. Bohn and R.D. Tabors, Nov. 1991, Control of Electric Thermal Storage under Real Time Pricing, in: Int’l. Conf. on Advances in Power System Control, Operation and Mgmt., Vol. 1, Nov. 1991, pp. at 397–403. [6] A. Dukpa, I. Duggal, B. Venkatesh and L. Chang, 2010, Optimal Participation and Risk Mitigation of Wind Generators in an Electricity Market, IET Renew. Power Gen., 4 (2) at 165–175. [7] J. Garcia-Gonzalez, 2008, Hedging Strategies for Wind Renewable Generation in Electricity Markets, in: Proc. IEEE Power and Energy Soc. General Meeting: Conversion & Delivery of Elec. Energy in 21st Century, 2008, pp. at 1–6. [8] F. Graves, T. Jenkin and D. Murphy, 1999, Opportunities for Electricity Storage in Deregulating Markets, Electr. J., 12 (8) at 46–56. [9] J.-Y. Joo and M.D. Ilic, 2010, Adaptive Load Management (ALM) in Electric Power Systems, in: Proc. Int’l. Networking, Sensing and Control (ICNSC) Conf., 2010, pp. at 637–642. [10] L.M. Juan, M. Claudio, D.L. Rodolfo and L.B. Jose, 2012, Optimum Residential Load Management Strategy for Real Time Pricing (RTP) Demand Response Programs, Energy Policy, 45 (8) at 671–679. [11] T.-Y. Lee, Sept. 2007, Operating Schedule of Battery Energy Storage System in a Time-of-Use Rate Industrial User with Wind Turbine Generators: A Multipass Iteration Particle Swarm Optimization Approach, IEEE Trans. Energy Convers., 22 (3) at 774–782. [12] D.K. Maly and K.S. Kwan, Nov. 1995, Optimal Battery Energy Storage System (BESS) Charge Scheduling with Dynamic Programming,

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

IEEE Proc. Sci. Measure. Technol., 142 (6) at 453–458. N. Miller, 2010,D. Manz, 2010,J. Roedel, 2010,P. Marken and E. Kronbeck, 2010, Utility Scale Battery Energy Storage Systems, in: Proc. IEEE Power & Energy Soc. General Meeting, 2010, pp. at 1–7. A. Molderink, 2010,V. Bakker, 2010,M.G.C. Bosman, 2010,J.L. Hurink and G.J.M. Smit, 2010, A Three-Step Methodology to Improve Domestic Energy Efficiency, in: Proc. Innovative Smart Grid Technologies (ISGT), 2010, pp. at 1–8. F. Nieuwenhout, 2005,J. Dogger and R. Kamphuis, 2005, Electricity Storage for Distributed Generation in the Built Environment, in: Int’l. Conf. on Future Power Sys., 2005, pp. at, p. 5. A. Oudalov, 2008,T. Buehler and D. Chartouni, 2008, Utility Scale Applications of Energy Storage, in: Proc. IEEE Energy 2030 Conf. Energy 2008, 2008, pp. at 1–7. O.A. Sianiki, Sept. 2010,O. Hussain and O.R. Tabesh, Sept. 2010, A Knapsack Problem Approach for Achieving Efficient Energy Consumption in Smart Grid for EndUsers Life Style, 2010, in: IEEE Conference on Innovative Technologies for Better & Reliable Electricity Supply, Sept. 2010, pp. at 159–164. D.W. Sobieski and M.P. Bhavaraju, 1985, An Economic Assessment of Battery Storage in Electric Utility Systems, IEEE Trans. Power Apparatus Syst., 12 at 3453–3459. K. Tanaka, A. Yoza, K. Ogimi, A. Yona, T. Senjyu, T. Funabashi and C.H. Kim, 2011, Optimal Operation of DC Smart House System by Controllable Loads Based on Smart Grid Topology, Renew. Energy, 39 (1) at 132–139. Y. Xu, 2010,L. Xie and C. Singh, 2010, Optimal Scheduling and Operation of Load Aggregator with Electric Energy Storage in Power Markets, in: Proc. North American Power Symp. (NAPS), 2010, pp. at 1–7.

Vol. 26, Issue 3 1040-6190/$–see front matter # 2013 Elsevier Inc. All rights reserved., http://dx.doi.org/10.1016/j.tej.2013.03.005

105