Abstract
One battery energy storage system (BESS) can be used to provide different services, such as energy arbitrage (EA) and frequency regulation (FR) support, etc., which have different revenues and lead to different battery degradation profiles. This paper proposes a whole-lifetime coordinated service strategy to maximize the total operation profit of BESS. A multi-stage battery aging model is developed to characterize the battery aging rates during the whole lifetime. Considering the uncertainty of electricity price in EA service and frequency deviation in FR service, the whole problem is formulated as a two-stage stochastic programming problem. At the first stage, the optimal service switching scheme between the EA and FR services are formulated to maximize the expected value of the whole-lifetime operation profit. At the second stage, the output power of BESS in EA service is optimized according to the electricity price in the hourly timescale, whereas the output power of BESS in FR service is directly determined according to the frequency deviation in the second timescale. The above optimization problem is then converted as a deterministic mixed-integer nonlinear programming (MINLP) model with bilinear items. McCormick envelopes and a bound tightening algorithm are used to solve it. Numerical simulation is carried out to validate the effectiveness and advantages of the proposed strategy.
, Indices of typical service period and life stage
, The maximum indices of typical service period and life stage
, Indices of time intervals in energy arbitrage (EA) and frequency regulation (FR) services
, The maximum indices of time intervals in EA and FR services
, Charging and discharging efficiencies of battery energy storage system (BESS)
Linearized cyclic aging rate corresponding to unit cycle depth
Linearized calendar aging rate corresponding to unit operation time
Cyclic aging rate corresponding to unit cycle depth at life stage 1
Calendar aging rate corresponding to unit operation time at life stage 1
Calendar accelerating factor at life stage l
Possibility of the second-stage scenario
, The minimum and maximum values of variable
, The minimum and maximum values of variable
, Bound of dead band and the maximum tolerant frequency deviation in FR service
Frequency deviation
, Time spans of intervals in EA and FR services
Length of a time interval
, Parameters in piecewise linear calculation function for cyclic accelerating factor due to charging-discharging rate at segment and life stage l
Maintenance cost of BESS in unit operation time
The maximum rated energy capacity
, The minimum and maximum values for stored energy of BESS in EA service
, Life loss limitation for BESS at life stage l and whole lifetime
, The maximum and minimum power in FR ser- vice
, The maximum charging and discharging pow- er of BESS in EA service
Rated charging-discharging power
, Reserve price and reserve power in FR service
Hourly electricity price in EA service
Discount rate
Time length of whole service period
Binary variable that decides time span of whole service period
, Binary variables that limit time spans of EA and FR services at life stage l
Binary variable that limits life loss for BESS at life stage l
, Variables in McCormick envelopes
Stored energy of BESS at life stage l in EA service
, Revenues of BESS in EA and FR services
The second-stage optimization problem
Linearized cyclic aging rate
, , Linearized cyclic aging coefficients due to cycle depth, mean state-of-charge (SOC), and temperature
Linearized calendar aging rate
, , Linearized calendar aging coefficients due to operation time, initial SOC, and temperature
Comprehensive linearized aging rate
Multi-stage cyclic aging rate at life stage l
Cyclic accelerating factor due to charging-discharging rate at life stage l
Multi-stage calendar aging rate at life stage l
Comprehensive multi-stage aging rate at life stage l
, , Aging rates during charging and discharging , stages in EA and FR services at life stage l
Output power of BESS in EA service
, Charging and discharging power for battery (, )
, , Charging and discharging power of BESS in , EA and FR services at life stage l
Optimal value of the second-stage problem
Time of service switching point
Time span of whole operation period
, Time spans of EA and FR services at life stage l
, Random vectors in the second-stage optimization and scenario k
The second-stage constraint set
, Constraint set and optimization problem
Vector for variables in the first-stage decision
, Vectors for variables in the second-stage decision and scenario k
ENERGY storages are promising facilities to improve the robustness, resiliency, and efficiency of modern power systems and cope with the challenges brought by growing penetration of uncertain and intermittent renewable resources [
EA and FR are two main services for the BESSs. In EA service, BESSs operate according to the electricity price in hourly timescale [
Co-optimization strategies can be classified into two main categories: short-term joint-service optimization and long-term service-switching optimization. The short-term joint-service optimization is usually used in the daily operation of a BESS, whereas the long-term service-switching optimization is mostly considered in the whole-lifetime service. A robust optimization model is formulated in [
Aging characteristics significantly impact the economic benefit of a BESS [
This paper focuses on the whole-lifetime coordinated service over the long-term horizon. The contributions of this paper are as follows.
1) A whole-lifetime coordinated service strategy is proposed to maximize the whole-lifetime operation profit for a BESS by switching the service between EA and FR at proper SOHs.
2) A comprehensive multi-stage aging model is established to calculate the aging rate of a BESS considering both external impacts and internal chemical reaction process.
3) A two-stage stochastic programming model is formulated to model the optimization problem. At the first stage, the optimal service switching scheme between the EA and FR services are formulated to maximize the expected value of the whole-lifetime operation profit. At the second stage, the output power of BESS in EA service is optimized according to the electricity price in the hourly timescale, whereas the output power of BESS in FR service is directly determined according to the frequency deviation in the second timescale. McCormick envelopes and a bound tightening algorithm are used to solve the model.
The remainder of this paper is organized as follows. Section II introduces the framework of whole-lifetime coordinated service strategy. Section III illustrates the multi-stage battery aging model. In Section IV, the mathematical formulation is presented. The solution method is developed in Section V. The effectiveness of the proposed strategy is validated in Section VI. Finally, Section VII provides the conclusion.
The proposed whole-lifetime coordinated service strategy aims to maximize the total operation profit of a BESS through switching the service at proper SOHs. The operation revenues and costs are settled at the end of whole service period. An overview of the proposed strategy is shown in

Fig. 1 Overview of whole-lifetime coordinated service strategy.
A BESS can provide EA service either as a seller or a buyer according to the hourly electricity price [
(1) |
A BESS can be used to provide FR services in the short timescale. The operation rule in FR service [
(2) |
According to (2), a dead-band of is set for the BESS to avoid unnecessary frequent usage. When the frequency deviation is in , the output of BESS is increased linearly and limited by the reserve power and the maximum and minimum power values. When the power is positive, the BESS is charged. When the power is negative, the BESS is discharged. During each period in FR service, the BESS operates according to the operation rules and real-time power demand for 900 s, after which a break of 900 s is permitted to recover the SOC to the initial value.
The FR service considered in this paper is a reliable FR service with fixed standby capacity. According to the European FR market, the BESS in reliable FR service is paid by the system operator (SO) with a fixed price for reserve power [
(3) |
This paper considers the lithium-ion battery which is most widely adopted in practice. The formulation and growth of SEI cause the capacity loss of the lithium-ion batteries [
In this work, lifetime loss is used to evaluate the aging rate L. When , the lithium-ion battery has no lifetime loss. When , the lithium-ion battery degrades from the beginning to the end of life. The end of life is typically defined as the point at which the battery only provides 80% of its rated maximum capacity.
A linearized aging model depending on the external stress factors is widely used to describe the aging characteristics of the battery. The linearized aging model considers the cyclic and calendar aging functions, which are described as follows.
1) Linearized Cyclic Aging Function
A linearized cyclic aging function [
(4) |
In (4), the linearized cyclic aging rate is the product of aging coefficients corresponding to the cycle depth, mean SOC, and temperature, i.e., , , and , respectively. In the application of above linearized aging function, the aging coefficients due to mean SOC and temperature are usually regarded as constant parameters. The cyclic aging rate is commonly fitted as a linear function of cycle depth. As the cycle depth is a linear function of charging-discharging power, the linearized cyclic aging rate can be calculated as a linear function of charging-discharging power [
(5) |
2) Linearized Calendar Aging Function
A linearized calendar aging function [
(6) |
In (6), the calendar aging rate is the product of aging coefficients corresponding to operation time, initial SOC, and temperature, i.e., , , and , respectively. As the temperature of the BESS can be well controlled by the cooling system and the initial and final SOCs in this paper are all 50%, the aging coefficients due to the impacts of initial SOCs and temperature are regarded as constant parameters. Then, the calendar aging rate can be calculated as a linear function of stored time [
(7) |
3) Comprehensive Linearized Aging Function
According to the functions in (5) and (7), the comprehensive linearized aging function is expressed as:
(8) |
The aging rate of a BESS depends on not only the external stress factors but also on its internal chemical reaction process. A multi-stage aging model is formulated to show the comprehensive aging characteristics.
1) Multi-stage Cyclic Aging Function
Multi-stage cyclic aging rates are calculated according to the accelerating factors corresponding to the charging-discharging rates at different life stages, which is expressed as:
(9) |
where is calculated in (5) by replacing with ; and is calculated by:
(10) |
The charging-discharging rate is defined as the ratio of the charging-discharging power to rated power. Then, the multi-stage cyclic aging rate can be calculated as:
(11) |
2) Multi-stage Calendar Aging Function
Multi-stage calendar aging rates are calculated according to the accelerating factors corresponding to different life stages in (12).
(12) |
where is calculated in (7) by replacing with .
Then, the multi-stage calendar aging function is shown as:
(13) |
3) Comprehensive Multi-stage Aging Function
Based on the multi-stage cyclic and calendar aging functions, the comprehensive multi-stage aging function is formulated as:
(14) |
The optimal whole-lifetime coordinated service strategy aims to maximize the whole-lifetime operation profit by switching the service at proper SOHs. The whole-lifetime operation profit is evaluated by the net present value (NPV) considering the revenues and costs during the whole service period. Decision variables are the selection for total service time, time spans of services at different life stages, and charging/discharging power for the BESS in EA and FR services. The mathematical model is formulated as:
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
(29) |
(30) |
(31) |
(32) |
(33) |
The objective function in (15) consists of two parts: maintenance cost and operation revenue.
Considering the uncertainty of the electricity price in (1) and the frequency deviation in (2), a two-stage stochastic programming model is compactly formulated as:
(34) |
where is the first-stage problem, i.e., maintenance cost, and the first-stage decision vector consists of the selection for total service time and time spans of services at all life stages; is the optimal value of the second-stage problem, i.e., the operation revenue , is the second-stage decision vector, consisting of the charging/discharging power for the BESS in EA and FR services, and is the random vector, which consists of the uncertain electricity price in EA service and uncertain frequency deviation in FR service; is the expected value of the second-stage problem.
Assuming has a finite number of possible scenarios, denoted as with respective possibilities of , then the expectation form in (34) can be written as:
(35) |
Then, the original two-stage stochastic model can be formulated as the following deterministic equivalence:
(36) |
s.t. | (37) |
(38) |
It can be seen from the formulation of optimization model that, at the first stage, a “here-and-now” decision is made before the realization of the uncertain data is known. Then, at the second stage, after a realization of uncertain scenario , a “wait-and-see” decision is made to compensate for a possible inconsistency between the prediction and the reality at the first stage. In practice, the uncertain scenarios should be sampled from the probability distribution function of the uncertain variables, and the number needs to be reduced to a tractable number for solution.
Variables in the first-stage decision vector are , , and , which correspond to the selection for total service time and time spans of EA and FR services at the life stage l, respectively. The objective function in the first-stage problem is given as:
(39) |
Constraint set corresponds to the constraints in (16)-(20) and (30)-(33).
In the second-stage decision, battery charging and discharging power in FR service is formulated according to the frequency deviations in each scenario. Variables in the second-stage decision vector are and , which are the charging and discharging power of the BESS in EA service of each scenario, respectively. The objective function in the second-stage problem is given as:
(40) |
where and are calculated in (1) and (3), respectively.
Constraint set corresponds to the constraints shown in (21)-(33). Based on the above analysis, the deterministic equivalence (36)-(38) is a mixed-integer nonlinear programming (MINLP) problem, which cannot be solved directly. The solution method used in this paper is shown in the next section.
As and are calculated by the piecewise quadratic function in (14), and in (27)-(29) are highly nonlinear and make the optimization model difficult to be solved. A piecewise linear function is used to simplify the function in (14). In the piecewise linear function, battery power is divided into several intervals with a series of breaking points. Then, a slope for each linear equation between the adjacent breaking points is calculated. Finally, the aging rate in (14) can be calculated by summing all the separated linear equations. Based on this linearization method, the highly nonlinear deterministic equivalent model is converted to a bilinear programming problem.
McCormick envelopes and a bound tightening algorithm are used to solve the bilinear programming problem. Variable and are used to present the bilinear items, i.e., in (15), and and in (27)-(29). The auxiliary variable is used to design the McCormick envelopes for the bilinear items:
(41) |
McCormick envelopes are presented as:
(42) |
(43) |
(44) |
(45) |
A bound tightening algorithm is developed in
The proposed whole-lifetime coordinated service strategy is tested by a multifunctional BESS in EA and FR services. The NPV is used to evaluate the profit of BESS. The discount rate is 5% per year. The parameters of the BESS [

Fig. 2 Cyclic aging characteristics. (a) SOH variation when cycle depth is 100% and charging-discharging rate is 1. (b) Cyclic accelerating factor with different charging-discharging rates and life stages.
Calendar aging rate corresponding to the operation time at the life stage 1 is per day [
According to the calendar aging parameters, the maximum stored time of the BESS is 10.17 years. Hence, the maximum time span of whole service period is set as 10 years, i.e., . The mean value and standard deviation of the historical electricity price [

Fig. 3 Mean value and standard deviation of historical electricity price and frequency deviation. (a) Mean value of historical electricity price. (b) Standard deviation of historical electricity price. (c) Mean value of frequency deviation. (d) Standard deviation of frequency deviation.
The stochastic variations of predicted electricity price and frequency deviation are assumed to follow the normal distribution [
The simulation is conducted on a 64-bit PC with 2.50 GHz CPU and 8 GB RAM, using Anaconda platform with Python 3.7.6 and GUROBI solver. The bilinear programming problem is solved by McCormick envelopes and the bound tightening algorithm in
1) First-stage Decision Results
The first-stage decision results are shown in

Fig. 4 First-stage decision results.
From
Based on the above service strategy, the expected value of the operation profit and aging rate in each day at each life stage is shown in

Fig. 5 Per-degradation-rate operation profit.
Based on the per-degradation-rate operation profit in
2) Second-stage Decision Results
For illustration purpose, typical scenarios shown in

Fig. 6 Scenario information in second-stage decision. (a) Electricity price in EA service. (b) Frequency deviation in FR service at life stage 2.
The second-stage decision results in EA service are shown in

Fig. 7 Second-stage decision results in EA service. (a) Active power. (b) SOC.
The second-stage decision results in FR service at life stage 2 are shown in

Fig. 8 Second-stage decision results in FR service. (a) Active power. (b) SOC.
The comparison with single service strategies is shown in
From
The existing whole-lifetime coordinated service strategy divides the lifetime of a BESS into two stages [

Fig. 9 Results using existing whole-lifetime coordinated service strategy. (a) Estimated SOH. (b) Actual SOH.
From
The computation time of the proposed strategy is shown in
As can be observed from
Unlike previous works using the BESS to provide a single service or switch the service of the BESS only one time, this paper proposes a much more flexible whole-lifetime coordinated service strategy for BESSs. The proposed strategy switches the service of the BESS in EA and FR according to the battery aging characteristics at different life stages. Both external aging stress and internal reaction process are considered in the analysis on battery aging characteristics. A two-stage stochastic programming problem is formulated to optimize the proposed strategy. It is converted into an equivalent deterministic MINLP problem and solved using McCormick envelopes and a bound tightening algorithm. Testing results show that the proposed strategy can make full use of the battery aging characteristics. Compared with the existing single service strategies and whole-lifetime coordinated service strategies, the proposed strategy achieves the highest whole-lifetime operation profit.
With the development of battery modeling methods, the aging model in this paper can also be further improved to enhance the effectiveness of the proposed strategy. It should be noted that the proposed coordinated service strategy is not limited by the service modes or the number of life stages presented in this paper. For other service modes and other number of life stages, the proposed strategy can also formulate an optimal coordinated service scheme by including new service rules and new degradation properties. The research on other service modes and other numbers of life stages will be carried out in our future work.
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