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Abstract: Conventional battery energy storage system (BESS) dispatching algorithms [1-6] assumes cost-based operation. In a deregulated power industry where utilities absorb the ultimate consequence of their decision making, a price-based approach should be developed to examine the benefit of BESS. In this paper, price-based BESS dispatching algorithm is introduced. Based upon the average marginal cost/market price function, the approach aims to increase the profit of utilities. In addition to a clearer objective, the developed algorithm includes various features of battery characteristics that have been ignored by some experts [1-6]. Instead of determining the amount of energy to be charged/discharged, the algorithm decides the number of battery cells to be charged/discharged. Furthermore, the algorithm considers the storage losses and full charge/discharge characteristics of batteries to better represent the BESS.

Keywords: load management, battery energy storage system (BESS), price-based operation, linear programming.

I. INTRODUCTION

Energy storage system is one of the load management strategy introduced in the 1970s. It allows both utilities and customers to store and consumer electric energy during scheduled periods. Within energy storage system, BESS has seen significant technological advancement during the past few years. This papers examines a price-based, utility-owned BESS using linear programming.

In a series of journal papers, Lee and Chen [1-3] accomplished the following tasks: coordination of BESS with pumped storage to reduce short-term production costs; investigation of BESS optimal capacity in a power system; and study of BESS effects on time of use (TOU) rates for industrial customers. Maly and Kwan [4] minimized the electricity bill for a given battery capacity, while reducing stress on the battery and prolonging battery life. These two groups approached the problem using multi-pass dynamic programming.

Alt et al. [5] used a heuristic unit commitment algorithm to determine dynamic operating cost benefits of BESS for utility application. They studied the effects of BESS on load frequency control, spinning reserve, and load leveling. However, it should be noted that a heuristic approach did not guarantee optimal scheduling. Daryanian et al. [6] discussed customer response to spot prices using a generic energy storage model. Even though the model presented was approached using linear programming, storage losses was ignored.

Several common features are found in the mentioned dispatch algorithms for BESS [1-6]: within each period, the amount of energy for charging and discharging is chosen as the variables; full charge/discharge is not assumed to be necessary; the objective of the algorithms were to reduce the system production cost.

Reviewing the conceptual designs of [7-8], the battery portion of the plant consists of battery cells with full charge/discharge characteristics. Also, the upcoming deregulation in the electric power industry and the corresponding competition in the marketplace [9] forces the utilities to absorb the ultimate consequence of their decision making. Beside the need for a more efficient algorithm to represent BESS, an explanation on how BESS may benefit the utilities is needed.

This paper introduces price-based BESS using linear programming to include various features that have been ignored by previous models [1-6]: to provide physical feel to BESS, the developed algorithm decides the number of cells to be charged/discharged at each period; full charge/discharge is assumed, partial charge/discharge is prohibited because it is not expected to occur; a price-based algorithm tries to increase utilities profit, rather than reducing system production cost.

II. NOMENCLATURE

: variable which determines the number of cells type i to be fully charged for k periods beginning at period j

: variable which determines the number of cells type i to be fully discharged for k periods beginning at period j

: variable which determines the increase in load at period j

: variable which determines the decrease in load at period j

: per MW average marginal cost/market price at period j

: required storage energy for cell type i at v charge stage for a k periods of charging duration

: required discharge energy for cell type i at w discharge stage for a r periods of discharging duration

: minimum charge duration for cell type i

: maximum charge duration for cell type i

: minimum discharge duration for cell type i

: maximum discharge duration for cell type i

n: maximum number of cell types

m: maximum period under study

: unit step function, where

= 1 if *0

= 0 if *<0

Additional parameters are explained when introduced.

III. MODEL

Utilities usually adopt a market-based pricing mechanism, i.e., customers are categorized and charged differently. when energy is regenerated, there is no certainty of which customer will use the energy. In fact, the regenerated energy will be shared by the system load during the discharge period. Since a reference to the rate structure is needed when a price-based approach is discussed, an averaged rate structure, , is introduced. It can be derived from Equation (1). represents the load of customer i at period j. represents the rate charged on customer Iiat period j.

(1)

Reviewing the conceptual design of [7,8], the battery portion of the plant consists of battery cells. In addition, as Figure 1 shows, there eight phases in the operation of battery, where full charge/discharge cycle is assumed. A new algorithm that may more closely mimic the physical structure with the following features are included:

· Number of cells to be charged/discharged is used as variable, rather than the amount of energy.

· Full charge/discharge is assumed. Partial charge/discharge is prohibited because it is not expected to occur.

Fig. 1. Operation of battery [7,8]

Instead of considering all phases as in Fig. 1, phases 1, 6, and 7 are ignored. Including these phases increases the variables to be studied but nothing else. More phases could beeasily incorporated without changing the underlying concept. Also, for simplicity, phases 4 and 5 are combined as one phase.

IV. FORMULATION

During the charge and discharge phases, the energy required during each period is fixed by the current and the length of charging. Denoting to be the required storage energy, i represents the cell type, k represents the total length of charging, and v represents the current charge stage. Similarly, is the required discharge energy, i represents the cell type, r represents the total length of discharging, and w represents the current discharge stage. Detailed information on how to determine and could be found in [1-5].

The profit for charging the energy at earlier periods, BDP, is shown in (2):

(2)

where the first term of the right side of (2) represents the revenue of stored energy and the second term represents the cost of increasing the load. For each control choice , i denotes the differences in group characteristics, j denotes the period when the full charge phase begins, and k denotes the number of periods needed for a full charge.

The profit for discharging the energy at earlier periods, BIP, is shown in (3):

(3)

where the first term of the right side of (3) represents the revenue of charged energy and the second term represents the cost of decreasing the load. For each control choice , i denotes the differences in group characteristics, j denotes the period when the full discharge phase begins, and k denotes the number of periods needed for a full discharge.

Then, the objective of BESS is to store the energy at a high-profit period and discharge it at low-profit period for the utility:

(4)

subject to:

(5)

which requires non-negativity of the number of charging choices, ,

(6)

which requires non-negativity of the number of discharging choices, ,

(7)

which indicated that at any period, the total number of charged cells, , and discharged cells, , is limited by the available number of cells, ; the unit step functions act as ON/OFF switches to determine if the battery cells under control choice, , are at fully or partially discharge stage,

(8)

which indicated that at any period, the total number of dischargeable cells should not be more than the total number of available fully charged cells. The unit step functions act as ON/OFF switches to decide if the cells under control choice are fully charged at period j.

Also, (9) is included to relate the number of charging cells, , and discharging cells, , with the increase in energy, , or decrease in energy, , at period j:

(9)

where all energy charged/discharged by all control choices, and , at period j should be equivalent to or ; the unit step functions decide if the control choices, and , have extended to period j.

Graphical Representation:

To visualize the constraining equations (7), (8), and (9), a graphical representation is presented in Figure 2. The n pieces laying on top of each other represent cells with different charge/discharge characteristics. The charging phase, with variables , is shown on the left. The discharging phase, with variables , is shown on the right. Governing equations connect the charge and discharge phases for differently grouped cells.

(7) and (8) are individualized constraints while (9) is a group constraint. According to (6), the maximum number of charge and discharge cells of any group is restricted by the existing cells, . Equation (8) further restricts the available discharge cells to available fully charged cells. Finally, (9) is included to relate the total increase/decrease in energy at each period to the number of charge and discharge cells, and .

Fig. 2. BESS graphical representation

Iterative Procedure:

Even though the average marginal cost/market price function is nonlinear, it can be perceived as a constant slope for a range of change in load levels, as shown in Fig. 3. Thus, the nonlinear average marginal cost/market price function is reduced into a piecewise linear function. To obtain the final optimal solution, an iterative procedure is employed.

Defining to be the increased load at period j, then, the relationship between the value of i-th iteration increased load to previous iteration increased load is described in (10):

(10)

Define one set of variables, , to represent the integer part of divided by ( is the maximum decreaseable or increaseable load for period j at each iteration,)

(11)

and another set of variables, , to represent the remainder of divided by ,

(12)

Fig. 3. Piewise linearization of system cost function

With known , the average linearized average marginal cost/market price for -th increase in load, , is shown in (13):

(13)

where is the system production cost at * load level; is the initial system load level at period j.

To conduct iterative procedure, (9) needs to be modified as in (14):

(14)

which is formed by adding at the end of (9)

With the defined parameters, a six steps iterative procedure shown below could then be carried out to achieve the final optimal objective function.

STEP 1

Initializing all variables, where

iter = 1

STEP 2

Determine the values of and as defined in (11) and (12).

STEP 3

Finding/updating values for and :

STEP 4

Solve for linear program as described in (2) through (8), and (14), in addition to the constraints set in STEP 3.

STEP 5

Updating increased and decreased load as defined in Equation (10).

STEP 6

Checking for optimality:

if any or value is greater than 0

iter = iter +1

GO TO STEP 1

else

Optimal solutions are found

V. RESULTS

Successful clarification of the concept presented in Section III through IV is accomplished by an example consisting of two BESS cell types and six periods (in this case, 1 period = 1 hour) test system. was set to 50 kW for all periods. However, it should be noted that can be set differently for each period under consideration. Additional details of the model and relevant data used are listed in the Appendix.

Using the developed algorithm to test the system, Table 1 shows the resulting dispatching sequences and Table 2 shows all relevant results.

Table 1. Dispatching sequences for price-based BESS.

Control	Battery type 1		Battery type 2
choice	Charging	Discharging		Charging	Discharging
i,1,2	30	0		15	0
i,3,1	0	10		0	5
i,4,1	0	20		0	10

The discrepancy between increased utility profit and system cost savings, $46.91, is due to the energy being discharged at higher rate periods, period 3 and 4. However, it should be noted that when energy is stored, it is stored within the equipments of the utilities and when stored energy is discharged, customers are the consumers. Therefore, the customers are not being hurt for paying a higher bill. In fact, customers are paying the same amount of money for the electric usage. Using a price-based algorithm, utilities base on the profit margin of observed duration to decide the dispatching sequences of the battery cells.

Also, the results show that maximum energy efficiency, 90%, may not be desirable when system conditions are considered. The actual efficiency level achieved by the batteries is 86.49% and 86.59% respectively.

Table 2. Results from price-based BESS.

Description	Results
Energy efficiency of battery
type 1	86.49%
type 2	86.59%
Increased utilities’ profit	$123.03
Increased system cost savings	$76.12

Table 3 shows the system load before and after dispatching is conducted.

Table 3. Change in system load.

Period,	System load, kW
hour	before scheduling	after scheduling
1	7300	7472.5
2	7600	7792.5
3	12000	11900.5
4	12900	12701
5	7900	7900
6	8600	8600

The sum of system load after scheduling is 46.50 kW higher than the sum of system load before scheduling. The difference is the storage losses during energy conversion.

Table 4 shows the average marginal cost/market price before and after the dispatching is conducted. Table 5 shows the profit margin of utilities for additional increased load before and after the dispatch is conducted.

From Table 4, the cost differences between periods 1 and 2 and periods 3 and 4 are large, as high as 27.625 cents/kW (34.02 cents/kW - 6.395 cents/kW). However, from Table 5, the profit margins between periods 1 and 2 and periods 3 and 4 are smaller, with the largest difference of 7.625 cents/kW (21.205 cents/kW - 13.58 cents/kW.) The possible increased profit margin of 7.625 cents/kW is not large enough to compensate for the lost revenue of storage losses. Therefore, five battery cells of both cell types are not excited for load shifting.

Table 4. Change in average marginal cost/market price.

Periods,	Average marginal cost/market price, cents/kW
hour	before dispatching	after dispatching
1	6.395	6.485
2	6.575	6.665
3	31.86	31.62
4	34.02	33.54
5	6.755	6.755
6	7.175	7.175

Table 5. Change in profit margin for increased load

Periods,	Profit margin for increased load, cents/kW
hour	before dispatching	after dispatching
1	21.205	21.115
2	21.055	20.965
3	15.85	16.09
4	13.58	14.06
5	20.775	20.775
6	20.265	20.265

VI. CONCLUSION

BESS dispatching algorithm was successfully formulated for a price-based operation. Significant improvements have been made over previous work to include storage losses and full charge/discharge characteristics of batteries. Since the algorithm decides the number of battery cells to be charged or discharged during observed periods, physical feel of actual system is enhanced. Besides providing a physical feel to actual systems developed in linear programming, the BESS algorithm considers nonlinear cost/market price function through successive approximation. Finally, results show that maximum energy efficiency of BESS may not desirable when system cost is considered.

VII. ACKNOWLEDGMENT

The first author would like to acknowledge the contributions of Dr. V. Vittal, Dr. J. McCalley, Dr. M. Khammash, and Dr. D. Starleaf for reviewing his thesis, from where the formulated concepts throughout this paper have been extracted.

VIII. REFERENCES

[1] T. Y. Lee, and N. Chen, “The Effect of Pumped Storage and Battery Energy Storage Systems on Hydrothermal Generation Coordination,” IEEE .Trans. on Energy Conversion, Vol. 7, No. 4, pp. 631-637, December 1992.

[2] T. Y. Lee, and N. Chen, “Optimal Capacity of the Battery Storage System in a Power System,” IEEE Trans. on Energy Conversion, Vol. 8, No. 4, pp. 667-673, December 1993.

[3] T. Y. Lee, and N. Chen, “Effect of Battery Energy Storage System on the Time-of-use Rates Industrial Customers,” IEE Proc. -Gener. Transm. Distirb., Vol. 141, No. 5, pp. 5521-528, September 1994.

[4] D. K. Maly. and K. S. Kwan, “Charge Scheduling with Dynamic Programming,” IEE Proc. -Sci. Meas. Technol., Vol. 142, No. 6, pp. 453-458, November 1995.

[5] J. T. Alt, M. D. Anderson, and R. G. Jungst, “Assessment of Utility Side Cost Savings from Battery Storage-type Customers,” 96 SM 474-7 PWRS, IEEE Summer Meeting, 1996.

[6] B. Daryanian, R. E. Bohn, and R. D. Tabors, “Optimal demand-side Response to Electricity Spot Prices for Storage-type Customers,” IEEE Trans. on Power Systems, Vol. 4, No. 3, pp. 897-903, August 1989.

[7] Electric Power Research Institute, Development of the Zinc Chloride Battery for Utility Applications, EPRI EM-3136, Research Project 226-5 Interim Report. Palo Alto, California: EPRI, 1983.

[8] Electric Power Research Institute, Development of the Zinc Chloride Battery for Utility Applications, EPRI SP-5018, Projects 226-5, -9 Final Report. Palo Alto, California: EPRI, 1987.

[9] G. B. Sheblé, “Price Based Operation in an Auction Market Structure”, presented at the IEEE PES winter meeting, 96 WM 191-7 PWRS, Baltimore, Maryland, 1996.

Kah-Hoe Ng received his BSEE with distinction (May, 1995) and MSEE (May, 1997) from Iowa State University. His research interests include power system economics and optimization.

Gerald B. Sheblé (M 71, SM 85) is a Professor of Electrical Engineering, Iowa State University, Ames, Iowa. Dr. Sheblé received his B.S. and M.S. degrees in Electrical Engineering from Purdue University and his Ph.D. in Electrical Engineering from Virginia Tech. His industrial experience includes over fifteen years with a public utility (Commonwealth Edison), a research and development firm (System Control), a computer vendor (Contrl Data Corporation - Empros), and a consulting firm (Energy and Control consultants). His research interests include power system economics and optimization, especially scheduling and control.

APPENDIX

Table 6. Initial system load, .

Period, j	Load level, kW
hour
1	7300	3500	3800
2	7600	3600	4000
3	12000	5500	6500
4	12900	6200	6700
5	7900	3900	4000
6	8600	4400	4200

Table 7. Rate structure.

Period, j	Rate, cents/kW
hour
1	27.60	25	30
2	27.63	25	30
3	48.79	45	50
4	47.60	45	50
5	27.53	25	30
6	27.44	25	30

Table 8. System cost function, .

Period, j	System cost function,
1,2,5,6	0.02 + 0.000003
3,4	0.03 + 0.000012

Table 9. Data pertinent to BESS.

Parameter	Description
	35
	20
Maximum energy efficiency of battery
type 1	90.00%
type 2	90.00%
Minimum energy efficiency of battery
type 1	84.21%
type 2	84.52%

Table 10. Charging and discharging characteristic of battery.

Energy required	Duration of charging/discharging period
during each period	1	2	3
Battery type 1, kW
Charging	7.60	3.70	2.40
Discharging	6.40	3.22	2.16
Battery type 2, kW
Charging	8.40	4.10	2.70
Discharging	7.10	3.60	2.43