An Optimal Battery Energy Storage Charge/Discharge Method

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1 An Optimal Battery Energy Storage Charge/Discharge Method Stephen M. Cialdea, MIEEE, John A. Orr, LFIEEE, Alexander E. Emanuel, LFIEEE, Tan Zhang Electrical and Computer Engineering Department Worcester Polytechnic Institute Worcester, MA USA Abstract-- Implementing electric energy storage in distribution networks requires that the input and output of the storage units be controlled in a manner to achieve maximum economic benefit. This paper outlines an algorithm that ensures that the battery operate near or at its highest economic potential given predictions of Locational Marginal Pricing (LMP) and the time profile of demand. This is accomplished by using the application of the dynamic programming algorithm. This algorithm is increasingly accurate with complexity and is best coupled with computer simulation. Index Terms battery storage, BESS charge/discharge, storage economics, dynamic programming I. INTRODUCTION A Battery Energy Storage System (BESS) must be electrically and economically viable. Substantial improvements in battery technology are being reported [2] [3] [4] and optimal BESS charge/discharge algorithms are essential to ensure that the BESS operates at the optimum economic charge/discharge plan over a given time period. The algorithm developed here requires accurate predictions of Locational Marginal Pricing (LMP) [5] and of the load profile of the system (typically a distribution feeder) that the BESS affects. It is assumed that the LMP is set for the region and is not changed by the dynamics of the load. This algorithm makes use of dynamic programming, a discrete mathematical optimum path algorithm developed in the 1950 s at Princeton University [6] [7] that can be applied to BESS operation [8]. Dynamic programming is a broadly applicable optimization technique that is more efficient than exhaustive calculation of all possible solutions. This approach requires that the necessary decisions be discrete. For the BESS situation there are two aspects to be discretized: the times at which charge/discharge decisions are made, and the possible rates of charge or discharge. In determining the optimum overall strategy the algorithm organizes the search to re-use previous calculations and hence is significantly more efficient than a simple exhaustive search. BESS characteristics such as capacity and charge/discharge rate that differ between technologies [9] [10] are taken into account in the application of dynamic programming. The algorithm proposed in this paper is thoroughly explained and then illustrated in practical distribution system applications with MATLAB [11] used as the programming environment. II. DYNAMIC PROGRAMMING This section provides an overview of dynamic programming and how it can be applied to BESS operation. First the basic theory of dynamic programming will be presented, followed by the application to BESS. A. Dynamic Programming General Algorithm Dynamic programming can be used to find the optimum path through a directed graph, whether the optimum for the given problem is a maximum or a minimum. A directed graph consists of nodes and directional edges, as seen in Figure 1. Fig. 1: Graph with nodes denoted by the top letters and edges denoted by vectors with numerical values. The directional edges are the arrows connecting the nodes, which denote direction and value, the value being denoted by the number on the edge. The nodes are the points at which the edges are connected and are denoted using capital letters within the circles. The nodes A and I indicate the starting and Financial support for this work was provided by Premium Power Corporation, National Grid, and the US Department of Energy IEEE Electrical Power & Energy Conferenc (EPEC) /13/$ IEEE

2 ending points, respectively. Consequently, the path must start at node A and traverse the graph to adjacent nodes in the directions connected an edge arrow until reaching node I. The summation of the values of each edge traversed from node A to I indicates the value that would be achieved by seeking that particular path through the graph. By inspection, it can be seen that there are various path options that can be taken with total cumulative value varying with each path option. In the particular graph shown above the decisions are denoted by the D1, D2, D3 and D4, which lie between adjacent groups of vertical nodes. The method of dynamic programming works backwards, in this case, from the end of the graph (node I) to the start (node A) to determine the optimum path. Each node has an optimum path from itself to the end of the graph. Working from the end of the graph (node I), the first edge option is seen under D4. Since the edges traversing D4 connect directly to the end node, it can be deduced that these edges will be the optimum path for each node one step away from the end node. In other words, being one step away from the end node, there is only one possible decision that can be made to connect to the end node, making it the optimum. Next, the analysis will move to the node(s) one more step away from the end of the graph (two steps from the end node). This area is denoted D3. Each edge value to adjacent node(s) in the direction of the end node will be added to the optimum path of the node it connects to, determining the optimum path from the node two steps away to the end node. These values can be compared and the highest or lowest path direction will be chosen to be the optimum depending on whether the desired outcome is to maximize or minimize the path value. Iterating this process for all decisions in the graph, regardless of size, and for each node will produce the final optimum direction from the start node of a graph given the same information. Finally, by following this optimum direction at the start node and each subsequent optimum direction, the optimum path through the graph can be determined. The end result will yield a graph with the optimum path direction and value at each node known, as seen in Figure 2. Fig. 2: Graph with each node s optimum path direction and value to the end node calculated. The second row inside each circle indicates these values. The numerical portion represents the value of the optimum path from that particular node to the end node and the letter represents the node to move to in order to follow the optimum path to the end node from that particular node. In this example, the maximum path value is desired. It can be seen that the last decision to be made is D4 and the path options for D4 stem from G and H to the end node, I. Since this is the first iteration, there is only one option for each node (G and H), which is to go to node I. Therefore, the optimum path from either node G or H is to move to I, which is summarized in Table I. TABLE I Table for D4 D4 Edge Value to I I G 2 2 I H 3 3 I The next decision, working backwards, is decision three (denoted by D3). The nodes that have edges traversing toward the end of the graph in D3 are D, E, and F. Each edge value is added to the previously calculated optimum path of the node it connects to (in Table I) to determine the corresponding path values if that direction were to be taken. Once all options are gathered, the highest yielding path direction is chosen. The outline of their decisions can be seen in Table II. TABLE II Table for D2 D3 Edge Value to G H G H D 6 N/A 8 N/A G E H F N/A 5 N/A 8 H It can be seen that the edge values between the adjacent node and the options are found (seen in the Edge Value to column). These values are added to the values of the best path of each option node, which was reported in the previous decision s table s in the row corresponding to the node option. This value is then reported in the current decision s Previous Best column for the corresponding node being left from. s that do not have an edge connecting them are marked N/A and not taken into account. The decision is then made depending on which node yields the highest or lowest sum of values and is reported in the last column. In this example, the goal is the maximum path value, so the maximum value is used. The second decision (D2) is seen in Table III and the same process that was used for D3 is used.

3 TABLE III Table for D2 D2 Edge Value to D E F D E F B 1 4 N/A 9 12 N/A E C N/A 3 1 N/A 11 9 E The last step shows the initial direction of the path from the start node (node A). This is seen in Table IV having only two edges and one node to leave from in the direction toward the end of the graph (node I). TABLE IV Table for D3 D1 Edge Value to B C B C A C It is seen that the optimum initial direction is to move from node A is to node C and the path value will be 15 units. Table III reports that the optimum path from node C is to move to node E. The third decision table demonstrations that the optimum direction from node E is to node H. Finally node H must traverse to node I. Therefore, the overall path is found to be C-E-H-I for a total value of 15 units. B. Algorithm Applied to BESS Charge/Discharge The method of dynamic programming can be applied to BESS charge/discharge to find the optimum output vs. time. A graph similar to the one seen in Figure 3 will represent the possible paths of the BESS input/output. traverse the graph with positive slope, then the BESS is acquiring charge. Therefore, the nodes are representative of state of charge. It can be seen that the nodes, in this case, are increasing by 0.5 MWh. It can also be seen that the BESS s maximum capacity is set at 2MWh. This gives the graph a pyramid characteristic with the top cut off at the BESS capacity. The edge values corresponding to the BESS operation can be calculated to a dollar amount by multiplying the rate of change by the regional instantaneous LMP. Since the operation of the BESS impacts load on the feeder, the costs associated with these changes must also be taken into account. Therefore, the instantaneous load value with the BESS as an addition (charging) or subtraction (discharging) can be multiplied by the instantaneous LMP and be applied to the edge values. Therefore, the optimum path from the left most node (time zero) to the right most node (ending time of duration under investigation) will be representative of the cost of the various options of the feeder given different BESS operation. Using the previously described method of dynamic programming, the optimum path through the graph represents the optimum charge/discharge of the BESS vs. time. The algorithm can become increasingly accurate by decreasing the time step (distance between nodes horizontally on a time scale) and increasing the options of rate of change between nodes (creating more edge values coming from each node). III. ALGORITHM RESULTS This section will present twenty-four hour results from a representative feeder with an approximate maximum peak demand of 10MW and a base load of roughly 7.2MW in an area with the LMP seen in Figure 4. Fig. 3: Graph applied to BESS charge/discharge with the value of each edge representing the charge/discharge rate of the BESS and the nodes corresponding to state of charge represented by vertical height. The gray path depicts the charge/discharge path plan determined to be optimal. Each black dot represents a node and the arrows represent the edges. The edge s slopes indicate the rate of charge (positive slope), discharge (negative slope), or charge hold (slope of zero). It is assumed that if the BESS holds a constant charge there is a negligible amount of loss due to the short duration of time before the next operation. If the edges Fig. 4: LMP and load curve for representative feeder to be used in dynamic programming examples

4 The demand and LMP curves represent the statistical average of all days in 2011 in the eastern Massachusetts LMP. The demand has been scaled down from all of eastern Massachusetts to a level that would be more appropriate to a feeder, while still keeping the curve shape. The daily value of the energy supplied to the feeder without the BESS was found to be $9, Applying the algorithm at a resolution of 30 minute time steps, five BESS input/output options (two charge and discharge rates, and hold ), a 75% round trip efficiency, a maximum discharge rate of 500kW, and a capacity of 2.5MWh, the energy supplied is found to be worth $9,228.20, representing a savings of $ Figure 5c shows the optimal BESS operating strategy given the LMP and demand seen in figures 5a, and 5b, respectively. The BESS output is a negative value when charging, and a positive value when discharging. More energy is consumed by the BESS than is output by a ratio that equals the round trip efficiency. to the feeder with the BESS is found to be $6,378.30, or a savings of $250 over the 24 hour period. An important aspect of BESS operation is how efficiency affects the benefit of the system. Figure 6 shows the BESS with round trip efficiencies of 100% (ideal system with no losses), 70%, and 50% using the same LMP and demand curves depicted in Figure 5a and 5b, respectively. It can be seen that the algorithm takes into account the inefficiencies of the BESS and alters the charge/discharge path accordingly to maximize the benefit. Figure 6 shows the state of the BESS at each corresponding round trip efficiency. It can be seen that the 50% efficiency battery only charges during the lowest LMP and does not use its full capacity because it would not be worthwhile to do so. Meanwhile, the BESS with no losses uses its full capacity and even charges in between the two peaks to benefit from another price differential in energy. Fig. 6: Optimum BESS output given multiple round trip efficiencies attained from the same LMP and demand curves seen in Figure 5a and 5b, respectively Fig. 5: (c) Output of optimum BESS charge/discharge applied to example feeder given (a) and LMP, (b) and demand curve. The LMP is adjusted to have a diminished LMP during wind generation operation (d) that results in a new BESS operation path (e) Applying the algorithm to a case where with wind generation the LMP goes to zero overnight (seen in Figure 5d) produces the results shown in Figure 5e. The energy to the feeder with the wind generation and no BESS is found to be $6, The value of energy supplied The magnitude of the effect of round trip efficiency can be investigated by parameterizing the efficiencies of the BESS system for given demand and LMP curves (perhaps typical curves for a feeder) and plotting the benefit vs. efficiency. Figure 7 shows the results of this study for an average daily benefit using the demand and LMP curves presented in Figure 4. Obviously, as efficiency increases, it would be expected there would be a positive trend. However, here it can be seen there the trend is nonlinear. By using a feeder s actual or predicted daily demand and LMP curves over a full year this study can be annualized, yielding important information in determining the necessary BESS efficiency to achieve an acceptable payback period for the capital investment. This approach offers an optimum charge/discharge plan based on a prediction. It is important to realize that the path may not be exactly the optimum path for the real time LMP and demand. Figure 8 shows day ahead LMP and demand predictions for the eastern Massachusetts region on July 9 th 2010 from ISO New England vs. the corresponding actual curves [5]. The demand curve has been scaled to an acceptable feeder value while maintaining its shape. The difference between the predicted and ideal charge/discharge profiles can be seen.

5 Fig. 7: BESS benefit as a function of efficiency on an example feeder with known demand and LMP curves For a given feeder, demand will vary from day to day. However, the shapes of the daily demand curves are typically similar within a season. It is the shape of the demand curve, rather than its absolute value, that determines the charge/discharge activity. This is illustrated in Figure 9 where the demand curves are normalized to a peak value and the LMP from Figure 5a is used to determine the BESS charge/discharge path. The top curve depicts the feeder at 100% loading, the middle curve represents 80% loading, and the bottom curve represents 60% loading. Note that it seems as if there is only one BESS output shown since all three are coincident. This yields the conclusion that the BESS charge/discharge plan can be set for various load profile shapes regardless of their peak values. IV. CONCLUSION The discrete mathematical topic of dynamic programming was introduced and applied to BESS charge/discharge strategy. This is shown to be a viable option to determine the optimum charge/discharge plan of a BESS given accurate predictions of LMP and demand impacted by the BESS operation. This algorithm becomes increasingly accurate with decrease in time step and increase in available charge/discharge rate options. Several examples depicting the capabilities of the algorithm were presented on a twentyfour hour time scale. ACKNOWLEDGMENT The authors gratefully acknowledge the contributions of Clayton Burns and Justin Woodard of National Grid and Stephen Rourke and the staff of ISO New England. Fig. 8: The top two plots depict day ahead predictions and corresponding actual LMP and demand curves for July 9 th 2010 from ISO New England. The bottom plot shows the path based on the prediction and the ideal path if the prediction had been perfect (the same as the actual LMP and demand). Fig. 9: Demonstration that changes in system load while maintaining load shape vs. time has no impact on charge/discharge profile. LMP depicted in Figure 5a is used. REFERENCES [1] Electric Energy Storage Technology Options: A White Paper Primer on Applications, Costs, and Benefits. EPRI, Palo Alto, CA, [2] Midwest Independent Transmission System Operator (MISO) Energy Storage Study. EPRI, Palo Alto, CA, [3] I. Gyuk, P. Kulkarni, J.H. Sayer, J.D. Boyes, G.P. Corey, and G.H. Peek The United States of storage [electric energy storage], Power and Energy Magazine, IEEE, vol. 3, pp , Apr [4] D. Chartouni, C. Ohler, and G. Linhofer, Value Analysis of Battery Energy Storage Applications in Power Systems, in 2006 IEEE PES Power Systems Conference and Exposition, pp [5] ISO New England. Accessed < [6] R.E. Bellman, Dynamic Programming. Princeton University Press, Princeton, N.J., 1957 [7] R.E. Bellman, S.E Dreyfus, Applied dynamic programming. Princeton University Press, Princeton, N.J., 1962 [8] D.K. Maly, K.S. Kwan, Optimal battery energy storage system (BESS) charge scheduling with dynamic programming in 1995 IEE Proc.-Sci. Meas. Technol., vol. 142, no. 6 [9] K.C. Divya, Battery energy storage technology for power systems An overview, Electrical Power Systems Research, vol. 79, pp , Apr [10] G.L. Soloveichik, Battery Technologies for Large-Scale Stationary Energy Storage, The Annual Review of Chemical and Biomolecular Engineering, Mar. 23, 2011 [11] The MathWorks, Learning SIMULINK 5, ISBN: