To decarbonize the electric power grid, there have been growing efforts to utilize clean, renewable energy sources. The utilization of wind and solar power generation is challenging because these energy sources are uncertain, intermittent, and nondispatchable. In particular, as the penetration of wind power increases, fast-ramping generators must be called upon more frequently to balance supply and demand, or wind power production must be curtailed [1
]. Such ancillary services and wind power curtailments will offset the economic and environmental benefits of wind energy.
One possible way to alleviate the negative impact of a growing wind power ramp rate is to utilize the flexibility that energy storage can offer. Energy storage devices are capable of shifting wind generation to reduce the ramp rate of wind generation [3
]. For an efficient charging/discharging operation of battery energy storage systems, a model predictive control approach was proposed in [5
]. However, a certain amount of wind generation must be curtailed when using this method. In [6
], the wind power ramp control problem using energy storage was formulated as a social welfare maximization problem. As the optimal solution to the problem requires information about the future wind generation and demand, a suboptimal online algorithm is presented; however, the suboptimal approach suffers a performance loss. In [7
], a storage control approach using a two-stage stochastic optimization was proposed. This operational strategy utilizes the forecast of wind energy obtained by a Gaussian process. Another optimization-based method was developed by using ramp scenario forecasts [8
]. The performance of both methods depends on the accuracy of forecasted information because the optimization problems in [7
] directly use wind forecasts. Arguably, the most popular method for efficient energy storage operation is stochastic optimal control [9
]. The associated stochastic optimal control problems are solved by dynamic programming or its approximate version, which often allows important structural properties of optimal strategies. Unfortunately, this method requires knowledge about the probability distribution of all the uncertainties such as future wind power generation. However, accurate distribution models are difficult to obtain in practice. Thus, the effectiveness of stochastic optimal control methods is limited as the wind power distribution at any given time deviates from the distribution estimated using historical data.
The methods mentioned above either require reliable information about future wind power generation or compromise the control performance. To account for these limitations, we seek an efficient storage operation strategy for wind power ramp management when only an inaccurate probability distribution of wind power generation is available. This method is based on distributionally robust stochastic control
, which minimizes the expected value of a given cost function in the face of the worst-case distribution drawn from a known set, called the ambiguity set
]. In this work, the ambiguity set is chosen as the set of all probability distribution whose Wasserstein distance
from an empirical distribution constructed from data is no greater than a certain threshold [20
]. The proposed storage control strategy is robust against wind ramp distribution errors characterized by the Wasserstein ambiguity set. It is worth mentioning that some storage control techniques do not require the exact distribution of uncertainties [23
]. However, these approaches do not aim to design a controller that is robust against distribution errors, unlike our method.
The contributions of this work can be summarized as follows. First, a novel storage operation strategy is proposed to provide a robust ramp management performance even when future wind power ramp distribution deviates from the empirical distribution obtained by historical data. Second, we develop a computationally tractable dynamic programming (DP) algorithm by using a piecewise linear approximation of the optimal value function with a uniform convergence property and Kantorovich duality. Thus, in each DP iteration, it suffices to solve linear programs for all grid points that discretize the state space. Third, the performance of the distributionally robust method is evaluated using the wind power generation data in the Bonneville Power Administration (BPA) control area and is compared with that of the standard stochastic optimal control method. Our simulation studies indicate that the proposed method reduces the ramp penalty by 4.82% on average compared to the standard stochastic optimal control method. We also examine how the ambiguity set size and the storage size affect the ramp management performance of the distributionally robust control method. This paper is significantly expanded from a preliminary conference version in many aspects [25
]. The problem studied in this paper is wind power ramp management, while [25
] considers a wind energy balancing problem. In addition, we use the Wasserstein ambiguity set and examine the effect of the set size, unlike [25
], which employs the moment-based ambiguity set (The performance of the moment-based approach in [25
] depends on the reliability of moments estimated from wind power generation data. However, the proposed Wasserstein approach does not have such an issue because it does not use information about moments.) Furthermore, a tractable dynamic programming solution is carefully developed in this work, using a linear programming approximation with a uniform convergence property.
The remainder of this paper is organized as follows. In Section 2
, the Wasserstein distributionally robust storage control problem is formulated for wind power ramp management using historical data. Its dynamic programming solution with a piecewise linear approximation is proposed in Section 3
. In Section 4
, the performance and utility of the proposed method are demonstrated and analyzed using the wind power generation data in BPA for the year 2018.
4. Case Studies
To demonstrate the performance and utility of the proposed distributionally robust ramp management method, we perform simulation studies using the wind power generation data in the BPA control area for the year of 2018 [35
]. The storage size is chosen as
MWh with power rating
MW. The initial SOC of storage is chosen to be the half of its capacity, i.e.,
. The storage operation interval is
min, and the number of time steps T
is 288. Thus, the storage device manages the ramp rate for 24 h. The ramp-rate limit is chosen as
MW/min by setting
MW. Additionally, the following parameters are used in the simulations:
. The sample of the ramp variable
is constructed for each month from the BPA wind generation data. Since outliers bias the trained controller, we clipped out data points lying outside
MW and replaced such outliers with
MW. The state space
was discretized with
grid points with grid spacing 1 for x
-axis and 12 for y
-axis. All the numerical experiments were performed on a Mac with 4.2 GHz Intel Core i7 and 64 GB RAM. The LP problem (9
) was solved using CPLEX for MATLAB.
4.1. Comparison with Stochastic Optimal Control
We first compare the performance of the proposed distributionally robust method and that of the standard stochastic optimal control method, which is the most popular approach to energy storage operation (e.g., [9
]). The stochastic optimal controller is designed by solving (3
) via dynamic programming. To evaluate the performance for all four seasons, the BPA wind data of January, April, July, and October are used. For each month, we split the data into the training and test data set: the training data are chosen as the ramp data for the first 15 days, i.e., from day 1 to day 15; and the test data are selected as those for the next 15 days, i.e., from day 16 to day 30. We use three different training sample sizes, 5, 10, and 15, in this comparison study. The training data are chosen from day 15, backward in time. For example, in the case of sample size 5, the training data are selected for days 11–15.
The performance of the two controllers are evaluated as the total ramp penalty for 24 h relative to the “no storage” case. In other words, the cost is evaluated as the ratio of the total ramp penalty with energy storage to that without storage. As shown in Figure 3
, the proposed distributionally robust controller outperforms the standard stochastic optimal controller for all months and for all sample sizes. Specifically, the distributionally robust method saves the ramp penalty by 4.82% on average compared to the stochastic optimal control method, as summarized in Table 1
. This result indicates that the distributionally robust method consistently resolves the issue coming from distribution errors for every season and sample size. The stochastic optimal controller sometimes performs worse than the “no storage” case. This is because the training set distribution is different from the test set distribution, i.e., the training set does not offer useful information about the behavior of wind power ramping in the near future. The stochastic optimal controller believes such a misleading or uninformative training set distribution, while the distributionally robust controller does not. The proposed method actively takes into account potential distribution mismatches and makes control decisions robust against the distribution errors. The effect of the distributionally robust method on net power production is shown in Figure 4
. The controlled storage smoothens wind power fluctuations and thus reduces the ramp penalty.
When the sample size is too small, the data may provide too little information that is useful in decision making. On the other hand, as the sample size increases, old data are used for designing controllers. This addition of old data, which may be different from the future ramping behavior, may distort the training set distribution in an undesirable way. Thus, in both control cases, the performance is improved when increasing the sample size from 5 to 10 and is almost unaffected when increasing the sample size from 10 to 15.
4.2. Effect of Ambiguity Set Size
A notable advantage of Wasserstein distributionally robust control is that it provides a nonasymptotic probabilistic guarantee on the out-of-sample performance
, which is the control performance evaluated with unseen test samples drawn from the true distribution [19
]. It is well known that the out-of-sample performance critically depends on the radius
that controls the size of ambiguity set (4
). In our simulation studies, the ramp penalty computed with test samples is the measure of out-of-sample performance. We now examine how the radius affects the ramp management performance of the distributionally robust method. Figure 5
displays the effect of
on the total ramp penalty relative to the “no storage” case, where the data of April are used for the test. As
increases from 0.05, the performance initially decreases and then increases for
. When the radius is too small, the resulting controller is not sufficiently robust to take into account the deviation of the test set distribution from the training set distribution. On the other hand, in the case of a large radius, decisions made by the distributionally robust storage controller are overly conservative. Thus, it is incapable of aggressively charging or discharging energy storage to minimize the ramp rate. According to the simulation result, the proposed controller with
presents the best out-of-sample performance for wind power ramp management in the setting used for these simulations.
4.3. Effect of Storage Size
To examine the impact of storage size, the total ramp penalties relative to the “no storage” case are computed for different sizes
of energy storage with radius
, using the data of April. As shown in Figure 6
, the ramp penalty decreases as the storage size increases up to 11 MWh in both standard stochastic optimal control and distributionally robust control cases. This is because a bigger storage device provides greater operational flexibility, which can be utilized to mitigate the ramp rate of wind power generation. However, the benefit of such flexibility is saturated around 11 MWh: the ramp penalty even slightly increases with storage size. This counterintuitive result is caused by the mismatch between the training and test sets. The controllers use the prior knowledge obtained from the training set about wind power ramp to fully utilize the flexibility provided by energy storage. However, such behaviors can be overly aggressive when the storage size is large. Thus, as the test set distribution deviates from the training set distribution, the aggressive storage operation produces undesirable ramp events.