# Stochastic coordinated operation of wind and battery energy storage system considering battery degradation

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## Abstract

Grid-scale battery energy storage systems (BESSs) are promising to solve multiple problems for future power systems. Due to the limited lifespan and high cost of BESS, there is a cost-benefit trade-off between battery effort and operational performance. Thus, we develop a battery degradation model to accurately represent the battery degradation and related cost during battery operation and cycling. A linearization method is proposed to transform the developed battery degradation model into the mixed integer linear programming (MILP) optimization problems. The battery degradation model is incorporated with a hybrid deterministic/stochastic look-ahead rolling optimization model of wind-BESS bidding and operation in the real-time electricity market. Simulation results show that the developed battery degradation model is able to effectively help to extend the battery cycle life and make more profits for wind-BESS. Moreover, the proposed rolling look-ahead operational optimization strategy can utilize the updated wind power forecast, thereby also increase the wind-BESS profit.

## Keywords

Wind power Battery energy storage system (BESS) Battery degradation Stochastic programming Rolling optimization## 1 Introduction

With increasing penetration of wind generation, power systems are faced with great challenges in dealing with the variability and uncertainty of renewable resources. With very flexible charging-discharging characteristics, grid-scale energy storage is one of many potential sources of grid flexibility which can aid variable renewable integration. An energy storage system can provide multiple functions in coordinating wind power in the power system. For example, energy storage can be used for load time shifting, wind power smoothing or energy arbitrage. The idea is to store energy during the periods of low demand or low energy prices, and stands ready to dispatch energy to the grid during the periods of high demand or high energy prices. Energy storage can also provide ancillary services to the system. Moreover, energy storage can be used to reduce or eliminate the deviations from generation schedules, making wind farms more like conventional generators. Meanwhile, energy storage can provide some other services, like flexible ramping capacity [1] or increased do-not-exceed limits for wind power [2], in order to satisfy the integration requirements from the power grid.

Among the energy storage systems available, the battery energy storage system (BESS) is the most widely utilized technology for wind farms because this technology has been rapidly improved [3]. In recent years, there have been many BESS announced, contracted and built for wind farms around the world. For example, a 10 MW advanced lead-acid battery and a 31.5 MW lithium-ion phosphate battery are integrated with the 21 MW Kaheawa Wind Power II project in Hawaii and the 100.5 MW wind farm in West Virginia, respectively. In China, a 14 MW lithium-ion phosphate battery and a 2 MW vanadium redox flow battery have been built in Zhangbei National Wind and Solar Energy Storage and Transmission Demonstration Project. Recent research efforts have focused on how to coordinate the wind power with BESS for multiple objectives, such as smoothing power fluctuations [4, 5], energy arbitrage [6], regulation [7], and other ancillary services [8, 9].

It is well known that one of the biggest barriers for application of the grid-scale BESS is its high cost [10]. The average lithium-ion battery cost in 2015 is 350000~700000 $/MWh [11], and the realized investment costs of grid-scale BESS are substantially higher. Meanwhile, energy storages have limited lifespan. For example, the average full cycle life of the lithium-ion battery is only 4500 cycles [12]. Considering that the life of a battery is much shorter than that of a conventional generator, and the battery station investment cost and the battery cell replacement cost are high, it becomes important to consider the degradation cost of BESS. Furthermore, battery lifetime is closely related with the operational performances, for instance, frequent and deep cycles accelerate cyclic aging and reduce the cycle life [13]. Thus, the short-term operation of a battery affects the overall lifetime of the battery. The interplay between short-term operation and long-term battery lifetime are therefore necessary to be considered. There is a cost-benefit trade-off between two desirable goals, i.e. 1) better battery health status and longevity, and 2) better performance of a wind-BESS. Generally, if one is willing to accept making less profit in the energy market (such as curtailing more wind power sometimes), or a less smooth wind power output, the battery can spare some effort and degradation. Thus, it is important to consider the battery degradation in order to best take advantage of BESS. In many cases, it may be profitable to extend the BESS lifespan by adjusting its operation to some degree.

The battery lifetime and degradation problems have attracted attentions in recent literatures. For microgrid operation, [14] proposes a detailed physical-based degradation model solved by dynamic programming. References [15, 16] propose degradation models to extend battery life in electric vehicles. Reference [17] builds a battery degradation cost model in a short-term scheduling problem, and proposes a big-M linearization method for degradation calculation. Reference [8] considers degradation cost for battery bidding in the ancillary service market, and solves it by nonlinear solvers.

- 1)
We propose a hybrid deterministic/stochastic look-ahead rolling optimization model for coordinated wind-BESS operation in the real-time market. Inspired by the rolling optimization proposed in [18], we derive a deterministic/stochastic model based on the timelines of a rolling wind power forecast and the real-time market, which is suitable for wind-BESS real-time operation and bidding.

- 2)
We build a battery degradation model for the wind-BESS operation. This is accomplished by introducing degradation percentage constraints as well as a degradation cost in the objective function. Compared to the literature which ignores or simplifies the battery cost, such as [4, 18], this paper introduces a more accurate battery degradation cost model representing degradation under different states of charge (SoCs) during charging and discharging processes. Compared to the existing battery degradation studies, such as [16, 17], we propose a different linearization method to transform the degradation model into a mixed integer linear programming (MILP) problem, which can be solved by commercial solvers, and therefore have great potential to be used in multiple applications.

This paper is organized as follows. Section 2 describes the overall wind-BESS look-ahead rolling optimization method and its timeline. Section 3 introduces the background, mathematical formulation and linearization method of battery degradation. Section 4 presents the wind-BESS model formulation. Section 5 provides results from a case study. Section 6 concludes this paper.

## 2 Wind-BESS coordinated operation timeline

*H*, the generation plan of wind-BESS is submitted to Independent System Operator(ISO) at the beginning of hour

*H*−1. The wind-BESS submits their bids every one hour. It should be noted that this timeline is flexible and adjustable, which can be changed according to the detailed timeline in different electricity markets.

- 1)
Wind-BESS bids will always be accepted. The competition and interaction among different generators are not considered.

- 2)
The wind-BESS is a price-taker. For consideration of price-maker impacts, we refer to [22].

*H*−1 and can be split into three time segments, as illustrated in Fig. 1 and explained below.

- 1)
For hour

*H*−1

*H*−1 has previously been submitted at the beginning of hour

*H*−2. The submitted generation schedules cannot be revised. Here we extend the definition of uninstructed deviation penalty (UDP) in CAISO for conventional generation to wind generation, and penalize the deviation of power between the real-time market offer (which is equal to the generation schedule) and the realized delivery if the deviation exceeds a defined tolerance band [23]. Considering that the previous bid is based on previous wind power forecasts, forecast errors are inevitable. Thus, the operation of the wind-BESS needs to be further optimized and adjusted to avoid deviation penalties between the real-time schedule and the realized delivery. Meanwhile, the wind power forecast for hour

*H*−1 is updated at the beginning of hour

*H*−1, which is assumed to be accurate without errors due to the short forecast horizon for simplicity. Thus for hour

*H*−1, we build a deterministic optimization model with the updated wind power forecast, which is assumed perfect for this hour.

- 2)
For hour

*H*

*H*is less accurate than the one for hour

*H*−1. In order to address the impacts of wind power forecast uncertainty, we make use of stochastic scenarios to fully account for the probability distribution of the wind power forecast errors [24, 25].

- 3)
For hours

*H*+1 and*H*+2

The generation schedules for hours *H*+1 and *H*+2 are not offered to the market at the beginning of hour *H*−1. However, BESS needs to keep an appropriate energy storage level for the next few hours because of the inter-temporal constraints of BESS. Therefore, a two-hour look-ahead operation is proposed, rather than optimizing bids for only one individual time period. Overall, the time horizon for the optimization problem is therefore 4 hours, i.e., from *H*−1 to *H*+2, as indicated in Fig. 1.

We use the forecast data from the high-resolution rapid refresh (HRRR) model of National Oceanic and Atmospheric Administration (NOAA) in the U.S. as a typical high resolution wind forecast. The forecast is updated hourly with a 4~15 hours ahead the horizon at 15 minutes resolution [26]. Thus, for the *H*−1 time period, the deterministic optimization is at 15 minutes time resolution. As for the *H* to *H*+2 time periods, the stochastic optimization is done at one hour time resolution. The overall look-ahead rolling optimization runs every one hour with an updated wind power forecast.

## 3 Modeling and linearization of battery degradation

### 3.1 Background

_{0.3}MnO

_{2}) is DoD due to reactions with deposited lithium [27]. Cycling the battery at a high DoD can therefore dramatically reduce the number of lifetime cycles for lithium-ion batteries [28]. Reference [29] presents a cycle life versus DoD curve of a lithium nickel manganese cobalt oxide (NMC) battery by using a least-square fitting method based on test bed experiment data, as shown in Fig. 2. In some literatures the curve is assumed to be an exponential function. For example, the function is 1591.1×

*DoD*

^{−2.089}[17], or 10000×

*DoD*

^{−k }(

*k*is a constant ranging from 0.8 to 2.1) [8]. Reference [30] presents a degradation map in terms of DoD and current rate.

Typically, the cycle life versus DoD curve is acquired by repeatedly discharging the battery to a specified DoD level, and recharging it to the full capacity, which called “regular cycles”. However, grid-scale BESS are usually operated at irregular cycles. For example, BESS will response to energy price signals in the energy market and automatic generation control (AGC) signals if it participates in the ancillary service market. Grid-scale BESS can hardly run regular cycles from 100% to a specific DoD. Thus, it is necessary to build a battery degradation model to reflect irregular cycle impacts. Considering that lithium-ion batteries are mostly applied in practice today, we mainly focus on lithium-ion batteries in this paper.

### 3.2 Modeling

*DP*is the degradation percentage;

*DP*

^{ cy }is the cycle degradation percentage;

*DP*

^{ shelf }is the shelf degradation percentage.

*L*

^{ shelf }is the shelf life.

*N*

^{ cy }is the total number of the battery cycles.

*deg*

_{ t }can be obtained by the degradation-SoC curve, where

*SoC*= 1−

*DoD*, and the value is the inverse function of the cycle life versus DoD curve, as shown in Fig. 3. For example, if the battery is discharged from 20%

*DoD*to 80%

*DoD*in one period, the degradation percentage during this process is |1/34957 − 1/3221|×0.5 = 0.014%.

*deg*

_{ t }is the value in degradation-SoC curve at time point

*t.*

*α*is the allowance degradation percentage;

*C*

^{ BD }is the cost of battery degradation; \(\pi_{B}^{rpl}\) and \(\pi_{B}^{res}\) are the replacement and residual value, respectively.

### 3.3 Linearization

The battery degradation model needs to reflect a degradation-SoC curve, which consists of nonlinear functions. Considering that (6) is an inequality constraint, and (7) will be minimized in the objective function, we can linearize the non-linear parts by adding binary variables and dummy continuous variables.

*m*is the index of piecewise segment of battery SoC;

*∆*

_{ m,t }and

*δ*

_{ m,t }are SOSs of linearization degradation-SoC curve at time point

*t.*

## 4 Wind-BESS model formulation

The detailed formulation is described as follows. *j, t, k, s* are the indices of wind farms, time points, battery energy storage, scenarios, respectively. *W* is the set of wind farms. *BAT* is the set of battery storage plants. *T* _{1} and *T* _{2} are the sets of deterministic/stochastic optimization time periods, respectively. *S* _{1} and *S* _{2} are the sets of deterministic and stochastic scenarios, respectively. If there is no specific explanation, \(k \in BAT\), \(j \in W\), \(t \in T_{1} \cup T_{2}\), \(s \in S_{1} \cup S_{2}\). Equation (17) defines the objective to maximize the profit of the wind-BESS. Equation (17) includes the revenue from the energy market, the cost of battery degradation, the cost of wind curtailment (if any), and the cost due to power deviation penalty. For the deterministic optimization period, \(\rho_{s} = 1\). Equation (18) defines the revenue from net output/input to the grid. Equations (19) and (20) define the costs of BESS degradation and power deviation penalties, respectively. Equation (21) defines the wind power curtailment cost. Equation (22) represents the net power output/input of the wind-BESS. Equation (23) is the wind power curtailment constraint. Equations (24)–(26) set power limits for BESS charging/discharging levels, indicating the BESS cannot charge and discharge at the same time. Equations (27) and (28) represent the SoC limits, and the relationship between SoC and the BESS charging/discharging behavior. Note that the SoC at the end of *H*−1 in *S* _{1} must be the same as the SoC at the beginning of *H* in *S* _{2}. Equation (29) sets the degradation percentage limit for BESS operation, as explained in Section 3.

*s*; \(R_{s}^{P}\) is the revenue in the energy market of scenario

*s*; \(C_{s}^{PN}\) is the cost of deviation penalty between the real-time market offer and realized delivery; \(C_{s}^{BD}\) is the cost of battery degradation in scenario

*s*; \(C_{s}^{WC}\) is the cost of wind curtailment in scenario

*s.*

*t*; \(P_{G,t,s}\) is the net power input/output of wind-BESS at time point

*t*in scenario

*s.*

*t*in scenario

*s*; “+” is positive deviation, “−” is negative deviation; \(\pi_{t}^{{PN{ + }}}\) and \(\pi_{t}^{PN - }\) are penalty prices of deviation between the real-time market offer and the realized delivery exceeding the tolerance band, “+” is positive deviation, “−” is negative deviation.

*t*; \(P_{W,jt,s}^{WC}\) is the wind curtailment at wind farm

*j*at time point

*t*in scenario

*s*.

*j*at time point

*t*in scenario

*s*; \(P_{B,kt,s}^{ch,G}\) and \(P_{B,kt,s}^{dis,G}\) are charging and discharging power provided by battery

*k*at time point

*t*in scenario

*s*, respectively .

*k*, respectively; \(I_{kt}\) is the discharging status of battery

*k*at time point

*t*in scenario

*s*(\(I_{kt}\) = 1: discharging, \(I_{kt}\) = 0: not discharging).

*k*at time point

*t*in scenario

*s*; \(SoC_{k}^{\hbox{min} }\) and \(SoC_{k}^{\hbox{max} }\) are the minimum and maximum states of charge of battery

*k*, respectively; \(\varepsilon_{k}\) is the self-discharge rate of battery

*k*; \(\eta_{k}^{{}}\) is the charging/discharging efficiency of battery

*k*; \(E_{k}\) is the installed energy of battery

*k*.

As to the power deviation between the real-time market offer and the realized power, we refer to the power penalty calculation method of UDP in CAISO, as shown in (30), (31). Only the power deviation exceeding the tolerance band is penalized.

*t*in scenario

*s*; \(P_{TD}\) is the tolerance band of the deviation between the real-time market offer and the realized delivery.

## 5 Case study

### 5.1 Basic parameters

Parameters of BESS

\(P_{{}}^{ch,\hbox{max} }\) (MW) | \(P_{{}}^{dis,\hbox{max} }\) (MW) | \(E_{{}}\)(MWh) | \(\eta_{{}}^{{}}\) | \(SoC_{{}}^{\hbox{min} }\) | \(SoC_{{}}^{\hbox{max} }\) | \(\varepsilon_{{}}\) |
---|---|---|---|---|---|---|

6 | 6 | 18 | 0.9 | 0.2 | 0.9 | 0 |

The wind farm is assumed to be located in flat terrain in the State of Oklahoma in the U.S. The forecast and realized rolling wind speed data are from the HRRR odel of NOAA. Wind power is calculated for the wind speed at 80 m height by using a standard wind turbine power curve. The cut-in and cut-out wind speeds are 3.07 m/s and 25 m/s, respectively. The probabilistic wind power forecast scenarios are generated by using conditional kernel density estimation, following the procedure in [35]. The probabilistic forecast consists of 1000 discrete realizations according to the estimated probability mass functions for each hour of the next day, and reduced to 10 scenarios for computational tractability [36]. For the purpose of training and generating the wind power forecast, the data are split into a training period (January 1^{st} to July 30^{st}, 2012) and a testing period (August 1^{st}, 2012 to August 31^{st}, 2012). The energy prices for the real-time market are taken from the historical data in the PJM market from August 1^{st}, 2012 to August 31^{st}, 2012. For simplicity, the energy prices are assumed to be known in advance, i.e. price forecasting errors are not considered.

The tolerance band of UDP in CAISO is defined as the greater one of 5 MW or 3% of a unit maximum capacity. In this paper, recognizing the special characteristics of wind power, we define the tolerance band of wind-BESS power deviation from the real-time market offer and the realized delivery to be 10% of the wind farm capacity. In CAISO, the deviation penalty prices are set as 50% and 100% of the local energy price for negative and positive deviations, respectively. However, penalty prices could also be fixed. In this paper, we use the default assumption of a penalty price of 5 $/MWh (as is used for wind power deviations between the day-ahead and the real-time markets in [37]), but we also run several sensitivity cases, also with variable penalties similar to the CAISO rules (as shown in Section 5.5.1). The price of wind curtailment is set as 0, with sensitivity analysis presented in Section 5.5.2.

### 5.2 One-time optimization results

^{th}, forecast time: 00:00:00, look ahead time: 00:00:00−04:00:00) are presented in this section to show the effectiveness of the proposed algorithm. The energy prices from August 12

^{th}00:00:00−04:00:00 are $22.79, $22.64, $14.68, and $18.19, respectively. The deterministic (00:00:00 to 01:00:00) and stochastic (01:00:00 to 04:00:00, 10 scenarios) wind power forecast for the next four hours are shown as Fig. 4.

Results (expected values) of 4 hour optimization (00:00:00, August 12^{th})

Model | \(R_{{}}^{P}\)($) | \(C_{{}}^{BD}\) ($) | \(C_{{}}^{PN}\) ($) | \(P_{W}^{WC}\) (MW) | \(DP_{{}}\) | Total profit ($) |
---|---|---|---|---|---|---|

M1 | 3102.0 | 1430.9 | 0 | 0 | 1.77 × 10 | 1671.1 |

M2 | 3083.2 | 369.9 | 0 | 0 | 4.57 × 10 | 2713.3 |

M3 | 3040.4 | 122.9 | 0 | 0 | 1.52 × 10 | 2917.6 |

Results of the first hour (00:00:00, August 12^{th})

Model | \(R_{{}}^{P}\) ($) | \(C_{{}}^{BD}\) ($) | \(C_{{}}^{PN}\) ($) | \(P_{W}^{WC}\) (MW) | \(DP_{{}}\) | Total profit ($) |
---|---|---|---|---|---|---|

M1 | 850.2 | 477.0 | 0 | 0 | 5.89 × 10 | 373.3 |

M2 | 831.7 | 369.9 | 0 | 0 | 4.57 × 10 | 461.8 |

M3 | 788.9 | 122.9 | 0 | 0 | 1.52 × 10 | 666.1 |

### 5.3 One-day simulation results

^{th}. The energy price, one hour-ahead forecast (average) and real-time realized wind power are shown in Fig. 6. The hourly total profit of M1, M2 and M3 are shown in Fig. 7. The profits are negative in some hours due to large wind power forecast errors (04:00:00) and low wind speed (17:00:00 to 18:00:00), when the revenues from the energy market are less than the total cost. In general, high wind speed and high energy prices will help the wind-BESS to make more profit. Meanwhile, the degradation constraint in M2 and cost in M3 help to increase the total profit, also over the full day, by reducing the degradation cost. A more detailed comparison of the aggregate results for the day is given in Table 5. The total profits of M2 and M3 are 7.7% and 9.2% higher than that of M1.

Results of one-day (August 12^{th})

Model | \(R_{{}}^{P}\)($) | \(C_{{}}^{BD}\) ($) | \(C_{{}}^{PN}\) ($) | \(P_{W}^{WC}\) (MW) | \(DP_{{}}\) | Total profit ($) |
---|---|---|---|---|---|---|

M1 | 19389.2 | 3002.3 | 654.3 | 61.8 | 3.71 × 10 | 15732.6 |

M2 | 19032.6 | 1409.9 | 676.2 | 65.4 | 1.74 × 10 | 16946.5 |

M3 | 18989.0 | 1132.2 | 675.0 | 65.9 | 1.40 × 10 | 17181.8 |

### 5.4 One-month results analysis

^{th}. Moreover, M3 consistently outperforms M2. The total profits of M2 and M3 are 4.2% and 6.0% higher than that of M1 over the month. As to the battery, the overall degradation of M2 and M3 is 46.8% and 32.0% compared with M1, which means the expected battery life is extended more than two and three times, respectively.

Results of one-month simulation (August, 2012)

Model | \(R_{{}}^{P}\)($) | \(C_{{}}^{BD}\) ($) | \(C_{{}}^{PN}\) ($) | \(P_{W}^{WC}\) (MW) | \(DP_{{}}\) | Total profit ($) |
---|---|---|---|---|---|---|

M1 | 789900 | 82600 | 36900 | 135.7 | 1.02 × 10 | 670400 |

M2 | 776400 | 38600 | 39300 | 139.3 | 0.48 × 10 | 698600 |

M3 | 776400 | 26400 | 39600 | 139.7 | 0.33 × 10 | 710400 |

### 5.5 Sensitivity analysis

Sensitivity analysis of deviation penalty prices, wind curtailment prices, BESS sizes and BESS replacement cost are provided for the full day of August 12^{th} in this section.

#### 5.5.1 Sensitivity analysis of deviation penalty prices

^{th}) simulations of different deviation penalty prices are compared in Table 7 and Table 8. The prices of four cases are shown as follows. The penalty prices in Case 1-2 are fixed. The prices of Case 3 are the same as the UDP prices in CAISO. Case 4 are similar to case 3 but have the same prices for negative and positive deviations. From the results we can find that the higher penalty prices lead to less penalized power deviations, although the differences are relatively small (Table 7). Higher deviation penalties also reduce the total profits (Table 8).

Penalized power deviations under different penalty prices (August 12^{th})

Model | Penalized power deviation (MW) | |||
---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 4 | |

M1 | 130.8 | 125.6 | 117.2 | 189.4 |

M2 | 135.2 | 134.0 | 130.2 | 200.2 |

M3 | 135.0 | 133.9 | 130.0 | 199.8 |

Total profit under different penalty prices (August 12^{th})

Model | Total profit ($) | |||
---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 4 | |

M1 | 15732.6 | 15132.2 | 14827.3 | 14796.0 |

M2 | 16946.5 | 16192.4 | 14918.1 | 15914.1 |

M3 | 17181.8 | 16463.1 | 15415.2 | 16170.7 |

#### 5.5.2 Sensitivity analysis of wind curtailment

^{th}) simulations with different wind power curtailment costs are compared in Table 9 and Table 10. The prices of three cases are shown as follows. The results show that the wind power curtailments are sensitive to the curtailment cost, and that a curtailment cost of 5 $/MWh brings the curtailment to zero for all models.

Wind power curtailment under different curtailment prices (August 12^{th})

Model | Wind power curtailment (MW) | |
---|---|---|

Case A | Case B | |

M1 | 61.8 | 0 |

M2 | 65.4 | 0 |

M3 | 65.9 | 0 |

Total profit under different wind power curtailment costs (August 12^{th})

Model | Total profit ($) | |
---|---|---|

Case A | Case B | |

M1 | 15732.6 | 15582.9 |

M2 | 16946.5 | 16425.8 |

M3 | 17181.8 | 17025.1 |

#### 5.5.3 Sensitivity analysis of BESS sizes

^{th}) simulations with three different BESS sizes are compared in Table 11. Comparing the results in Case a and Case b, it can be found that with the same energy capacity of 18 MWh, the higher power capability results in higher profit, but the increases are minor. Comparing the results in Case a and Case c, the results are interesting because the BESS with smaller energy capacity makes a higher profit. A plausible explanation for this finding is that the larger BESS has higher calendar degradation cost. This has implications for the optimal sizing of BESS, which is not considered in this paper.

Total profit under different BESS sizes (August 12^{th})

Model | Total profit ($) | ||
---|---|---|---|

Case a | Case b | Case c | |

M1 | 15732.6 | 15917.6 | 16955.8 |

M2 | 16946.5 | 16980.3 | 17384.9 |

M3 | 17181.8 | 17182.0 | 17519.4 |

#### 5.5.4 Sensitivity analysis of BESS replacement cost

Total profit under different battery replacement costs (August, 2012)

Battery replacement cost ($/MWh) | Total profit ($) | ||
---|---|---|---|

M1 | M2 | M3 | |

200000 | 725461.0 | 724294.4 | 729472.0 |

300000 | 707108.5 | 715714.6 | 722658.9 |

400000 | 688755.9 | 707134.8 | 716133.2 |

500000 | 670403.4 | 698555.0 | 710360.4 |

600000 | 652050.9 | 689975.2 | 704505.6 |

### 5.6 Computational performance

The experiments are performed on a PC with Intel(R) Core(TM) i5 2.60 GHz and 8 GB of memory. The algorithms are implemented in MATLAB and programmed using YALMIP [38]. The MILP solver is CPLEX 12.6 [39]. The optimality gap is set as 10^{−6}. The average solution time for one single optimization is 0.76 seconds.

## 6 Conclusions

- 1)
We propose a novel hybrid deterministic/stochastic look-ahead rolling optimization model of wind-BESS coordinated operation in the real-time market. This model helps to minimize the power deviation penalty for the real-time operation and maximize the profit for the real-time bidding at the same time. The proposed method utilizes updated probabilistic wind power forecasts, effectively mitigating the impacts from forecast uncertainties.

- 2)
A linearized battery degradation model including battery degradation percentage constraints and degradation cost is proposed. The proposed model can accurately represent the battery degradation at different SoC levels during irregular battery cycling. Case studies based on practical data show that, by incorporating the battery degradation model, the battery life can be extended two to three times with substantial increases in wind-BESS profits.

In future work, we plan to investigate battery degradation for a wider range of services in the power system, like the provision of regulation and spinning reserves. This entails shorter-time cycling with different impacts on battery degradation and lifetime. Other interesting directions for future research include the potential consideration of more complex battery degradation models, more detailed representation of battery electrochemistry and how it affects losses and power limits, and applications in large-scale optimization problems such as unit commitment and expansion planning.

## Notes

### Acknowledgment

This work was supported by National Natural Science Foundation of China (No. 51477157), State Grid Corporation of China (Research on Probabilistic Economic Dispatch and Security Correction with Large-scale Renewable Energy Integration) and China Scholarship Council, as well as the U.S. Department of Energy’s Wind Power Program.

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