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Energy Storage Optimization for Grid Reliability

Large scale renewable energy integration is being planned for multiple power grids around the world. To achieve secure and stable grid operations, additional resources/reserves are needed to mitigate the inherent intermittency of renewable energy sources (RES). In this paper, we present formulations to understand the effect of fast storage reserves in improving grid reliability under different cost functions. Our formulations and solution schemes not only aim to minimize imbalance but also maintain state-of-charge (SoC) of storage. In particular, we show that accounting for system response due to inertia and local governor response enables a more realistic quantification of storage requirements for damping net load fluctuations. The storage requirement is significantly lower than values determined when such traditional response are not accounted for. We demonstrate the performance of our designed policies through studies using real data from the Elia TSO in Belgium and BPA agency in the USA. The numerical results enable us to benchmark the marginal effect on reliability due to increasing storage size under different system responses and associated cost functions.


Introduction
Massive installation of solar and wind resources in power grids is slated to replace conventional sources of power. As renewable generation is a function of weather parameters such as solar irradiance and wind speed, such sources, unlike conventional resources, are inherently uncertain/stochastic in nature. For instance, solar generation could fluctuate more than 70% due to passing clouds during daytime and wind generation could ramp down 100% due to loss of wind [1]. As the share of renewable generation has increased, the amount curtailment has also proportionally increased, with a total cumulative curtailed energy of 1817 GWh from May 2014 to April 2019 in CAISO, see Traditional resources, due to the presence of rotation mass, provide system inertia to counter fluctuations [2]. The absence of inertia in several inverter corrected renewable resources compounds the problem of managing variability, when penetration levels increase further [3]. To ensure power system reliability, utilities have to hold ramp-up and ramp-down reserves in order to compensate for sudden loss of renewable generation. The authors in [4] observe that, to accommodate 15% of wind generation, traditional reserves have to be increased upto 9%. In order to compensate the renewable volatility and avoid massive reserve procurement, additional fast ramping resources, with associated performance based payments [5] are being incorporated. The authors in [6] observe that energy storage systems can mitigate issues with large scale renewable integration due to their fast ramping. Falling cost of Li-Ion batteries (and other energy storage technologies) has encouraged the bulk installation of batteries for this purpose 2 . While it is true that increasing energy storage can in theory lead to improvement in reliability, their true performance depends on available conventional responsive resources. Following a net imbalance in injection, the complete system response, due to conventional generation and fast ramping batteries, determines the dynamics in operational frequency, including the maximum deviation in frequency (termed 'nadir') and the time to reach there [2]. This is utilized by grid operators to determine reserves necessary to ensure secure operations [7]. The overarching goal of this work is to quantify the effect of storage, including its marginal value, on reliability by accounting for the system response in the presence of system inertia and generator governors.

Prior Work
Optimization of storage operations is a growing field of research. Storage usage for arbitrage and peak shaving operates at a slower time scale (minutes-hours to weeks) and has been analyzed in [8,9,10,11,12,13]. In work associated with storage usage for reliability, the authors in [14] consider investor-owned reserves that perform bidding to profitably provide balancing services [15]. In [16], storage along with energy dissipating resistors are used for primary frequency control. The authors of [17] use energy storage for providing inertial response along with primary frequency regulation and show that response similar to conventional power plants can be derived. [18] looks at the impact of energy storage parameters such as capacity, ramping parameters and conversion efficiency on the impact of renewable integration. [19] observes that myopic control algorithms for storage operation can approximate deterministic solution for cases with small time-difference between decisions. [20] looks at the utility's problem of minimizing power imbalance by using storage devices and presents threshold based control rules, however they ignore system response in their analysis. [21] shows that reserve sizing based on worst-case imbalance would not be financially plausible. Bringing the reliability requirement and system response into the picture can help in understanding the marginal increase in reliability due to integrating energy storage as reserves.

Contributions of the paper
We consider centralized optimization of utility owned/operated storage for improving grid reliability. While the profitability of battery is crucial, we assume payment for capacity to owners and do not include electricity prices while optimizing storage actions in real-time [5]. Rather, we are interested in identifying marginal system-wide reliability improvement due to available storage. We first justify a relaxed system reliability index measured in terms of net power balance that is principally aware of the conventional bulk system response due to inverters and governors. For both linear and quadratic cost functions of net imbalance, we present convex but non-smooth optimization formulations for real-time storage operation. The storage optimization problem can also incorporate state of charge (SoC) based constraints to ensure ramping availability during unexpected large imbalances. We present McCormick relaxation and threshold based exact schemes for our formulations, and benchmark the performance with respect to different reliability metrics, using real imbalance information from two regions: (a) Elia TSO in Belgium and (b) BPA (Bonneville Power Administration) in USA. Our work demonstrates that while increase in storage improves reliability, the marginal value of integrating storage deteriorates with the storage size. More importantly, the computed decrease in marginal benefit is more severe for systems with higher system response. As our optimal solutions are deterministic and require information of all fluctuations, we compare them with a myopic storage algorithm that uses only current information. We observe that due to significantly fast operation of reserves, the myopic storage operation approaches the reliability improvements in the deterministic solutions, and hence can be used for real-time operation.
The rest of the paper is organized as follows. Section II introduces the reliability indices used in the paper and formulation of the response aware storage operation to reduce imbalance. Section III provides optimization formulation for storage as reserve for different cost function. Section IV provides case studies for Elia in Belgium and BPA in the USA respectively. Section V concludes the paper.

Reliability of contemporary power grids
In this section, we provide the definition of reliability index and a framework for our analysis. The system average interruption duration index, SAIDI [22], is commonly used as a reliability indicator by electric power utilities. SAIDI is the average outage duration for each customer served and is 3 given as:

SAIDI =
Sum of customer interruption duration Total number of customers served where N i is the number of consumers for the outage time U i for incident i and C T is the total number of customers served. We define reliability index (RI) as RI = 1 − SAIDI in units of time Time horizon for calculating SAIDI (2) Almost all developed countries have a power system reliability higher than 99.9% (see Table 1), that is expected to be ensured in the presence of renewables [22]. For research purposes, the detailed real-world information for faults and consumers affected by each fault as needed for calculating SAIDI may be hard to get. Therefore, we redefine SAIDI in terms of the power imbalance in demand and supply relative to the aggregate load. We define residual R(i) = ∆ i + s i , where ∆ i and s i denote net imbalance (without storage) and storage power output at time i, respectively. For our system, modified SAIDI is defined as where N T is the total number of samples in the time horizon for SAIDI computation.P g is the mean of active power and is given asP Note that the sample based SAIDI definition, similar to the cost function in [20], intuitively assumes that the number of customers interrupted is captured in the size of power imbalance in the system.
While our reliability measure increases with decreasing net imbalance, it does not account for the system response following an imbalance.
Including system response in reliability: In power grids, demand and supply are matched approximately at every time instant to maintain frequency within a narrow band as listed in Table 2. Rotational generators such as synchronous and induction machines in the grid provide an inherent Table 2: Continuous operating frequency range Country COFR Germany [23], China [24] 49.5 to 50.5 Hz Australia [24] 47 to 52 Hz A-zone: 59.95 to 60.05 Hz, USA [23] B-zone:59. rotational inertia as well as governor feedback (called Primary Frequency Control) that act to correct the operating frequency f (t), following imbalance. This dynamics is modelled by the Swing equation [2,7]: where P m and P e are the system's mechanical power and electrical load. M H is the inertia in the system. Considering a system-wide ramp rate of C MW/s that provides frequency damping services, the frequency nadir f min reached due to a net imbalance/residual R(i) in the system is given by (see [7] for the derivation): Here f 0 is the normal operating frequency, while f db is the dead-band frequency for governor response. This is derived in [7] by first determining the time to reach the frequency nadir from the event beginning and then using that to determine the system frequency. The operation of the primary response takes places within the first 30 seconds following an imbalance, as shown in Fig. 2.
Note that a similar analysis can be conducted for frequency incursion above the rated frequency.
If system inertia and ramp rates of different conventional generators are comparable (else consider the minimum per mW), then the total system inertia M H and ramp rates C can each be considered proportional to the total scheduled generation P g in the system. Going forward to regimes with similarly sized local generation (Eg. networked micro-grids), one can approximate M H C with a constant times P 2 g , the square of the total system load. Consider a pre-fixed maximum frequency deviation f 0 −f min for system safety. As dead-band f db is pre-determined, it follows from Eq. (6) that the maximum imbalance R(i) that the grid can safely sustain, is proportion to the scheduled system load or generation, as noted below.
Here is a constant that depends on system inertia and ramp rate. Fig. 3 represents this permissible imbalance range, which in agreement with the observation that larger disturbances can be tolerated 5 Figure 2: Reserves activation during a contingency [25] Permissible active power unbalance  in system with larger load or online generation. Note that conservative operators can determine a lower to be on the safe side. To account for the system safety for small imbalances below P g , we modify the SAIDI mod formula in Eq. (3) and define response-aware SAIDI mod below.
Using Eqs. (2,3,8) the reliability for our system, with and without response awareness, are respectively given by .
Time-scale of battery operation: Note that we assume storage operation to be without delay after an imbalance is observed. In practice, such seamless storage operations can be conducted through rate of change of frequency (RoCoF) measurements [26] directly or through the use of phasor measurement devices. In the next section, we describe the storage optimization problems that consider the defined reliability functions RI mod and RI mod (with response awareness).

Optimization of Storage
In this section, we outline optimization formulations for battery performing supply-demand balancing considering (a) linear or quadratic cost function for minimizing imbalance, (b) response of the power network, (c) maintaining the SoC of the battery. While we first consider optimal deterministic solutions schemes over a time-horizon with perfect information of fluctuations, in real-world future information will not be available. Thus, we also provide myopic rule-based real-time algorithms and benchmark them against the optimal deterministic formulations. For normalized time-instance i, the battery energy b i and power output s i needs to satisfy the following constraints: where Eq.(11) reflects the bounds on b i and s i , and Eq.(12) describes the linear dynamics in b i . η ch and η dis are the efficiencies of battery charging and discharging. The state-of-charge of battery at time i is where b rated denotes the rated battery capacity.

Linear Cost Function
The linear cost functions are depicted in Fig. 4 (a),(b) and follow from the definition of RI mod and RI mod in Eqs. (9) and (10) respectively. Under response awareness, the cost for imbalance below a threshold is 0. We first describe the case where system response is not considered in making storage decisions.

Reliability without response awareness
The optimization problem (P L ) is given as (11,12) Note that for a linear cost function, the non-zero net/marginal improvement made in reducing imbalance is the same irrespective of time-instant or overall imbalance magnitude. Thus, storage can be operated myopically considering only the current imbalance, using thresholds, as described in Algorithm 1.

Reliability with response awareness
Based on RI (see Eq. (10)), the optimization problem here is given by: , 0)}, subject to (11,12) Problem (P L ) can also be solved by a rule-based policy given in Algorithm 2 that differs from Algorithm 1 due to -induced thresholding in the cost-function. 8

Quadratic Cost Function
We now describe optimal storage actions where the cost for imbalance is quadratic. In this setting, imbalance minimization at larger imbalances are prioritize for storage operation.

Reliability without response awareness
The optimization problem for storage without response awareness uses the cost function in Fig. 4  (c), and is formulated as: (11,12) Clearly, look-ahead is essential for solving (P Q ), unlike (P L ). However, standard convex optimization is sufficient to solve it as the cost function is smooth.

Reliability with response awareness
Under knowledge of system response, we use a quadratic cost that ignores imbalances below P g (i), similar to the linear setting. The optimization problem is given by: (11,12) The cost function in (P Q ) is not smooth due to the absolute value operator. We use θ i to denote max(|∆ i + s i | − P g (i), 0), and then derive a McCormick relaxation [27] scheme for the absolute value operator to solve (P Q ), as described below.
where ∆ lb i = ∆ i + S min , ∆ ub i = ∆ i + S max , z i denotes a binary variable which is equal to 1 when net imbalance with storage, (∆ i + s i ) is positive. Note that the McCormick relaxation is used to approximate the values of a bilinear variable by creating a quadrilateral feasible space bounded by 4 constraints derived using the upper and lower limits of the individual variables in the bilinear variable. This form is exact when one of the variables in the bilinear form is binary [27]. Here y i = z i (∆ i + s i ) has a binary component z i .
Note that the storage SoC is not included in the optimization problems discussed till now. An operator may be interested in keeping SoC within a certain band to ensure available storage for future unforecasted large fluctuations. Next we discuss formulations where SoC targets are promoted through penalized SoC deviations.

Reliability with response awareness and SoC management
Consider the setting where storage SoC needs to be maintained within a band, denoted as [SoC l , SoC u ], where SoC l , SoC u denote the lower and upper boundaries, and mean SoC level is denoted as SoC = 0.5 × (SoC l + SoC u ). We define γ as γ =S oC − SoC l = SoC u −S oC.
On the other hand, the objective with quadratic cost for imbalance is given as (11,12,13).
Note that with SoC management, the optimal solutions for both linear and quadratic cost formulations do not have an optimal myopic form. We, thus, revert to two Mccormick relaxation schemes to overcome the non-smooth parts of the cost function (one for reliability, another for SoC). The additional associated constraints for both P LS and P QS are given by: (e) Binary variable: where ∆ lb i = ∆ i + S min , ∆ ub i = ∆ i + S max , z 1 i denotes a binary variable which is equal to 1 when (∆ i + s i ) is positive. z 2 i denotes another binary variable which is equal to 1 when SoC i −S oC ≥ 0 is positive.

Myopic control of reserves considering SoC management and network inertia
Storage control in the real-world will not have access to accurate information of future imbalances. In those settings, optimal solutions for problems (P LS ) and (P QS ) that require perfect information will not be practical. Instead, we propose a myopic Algorithm 3 in this section, for linear cost on reliability with response awareness and SoC management. Algorithm 3 is thus an extension of Algorithm 2. It uses the current information (SoC and imbalance in the power network) and network response to make charge/discharge decisions to minimize the imbalance. When the imbalance is within bounds (see Eq. (7)), it also attempts to keeps the SoC within the desired SoC band. Lines 3 to 6 decides whether the SoC is outside the target band. The SoC target band is decided based on battery type. For example, LiIon battery cannot be over-charged above an SoC level or over-discharged below a certain level [28]. Similarly, the zones for imbalance is identified in Algorithm 3's lines 7 to 11.
The storage operation is further constrained by capacity and ramping constraint. Based on the SoC and imbalance levels designated by Flag SoC and Flag ∆ respectively, different combinations are possible. The respective actions under each case are described in lines 12-34 of the pseudo code.
The algorithm can be similarly extended to derive a sub-optimal myopic policy for quadratic costs. In the next section, we provide simulation results on benefits from storage usage in reliability using real power grid imbalance data.

Numerical simulations for storage optimization
To compare benefits from our optimization algorithms for different cost functions and constraints, we use the following performance indices: • SAIDI mod and Reliability index RI mod

• Mean SoC
First we discuss results for data from the Elia TSO in Belgium.

Imbalance minimization in Elia, Belgium
The data considered in this case study is from the month of January 2019. Fig. 5 shows the aggregate load, demand and supply imbalance and the imbalance in percentage with respect to the aggregate load. Without storage, the reliability RI mod is equal to 98.845%. SAIDI mod for this month is 515.5 minutes. Observe that at hour index 253, an imbalance of the order of 17% with respect to the total load occurs, due to a sudden loss of generation of approximately 2000 MW. The reserve sizing necessary to completely mitigate this unbalance will require an astounding ramping capability of 2000 MW or more.
The objective of study for the Elia data is to identify the marginal value of adding storage as reserves, for different values of system response, that is measured in terms of (see Eq. (7,8)). We vary from 0 to 5% and implement the following 5 storage settings 3 : (i) No storage (nominal case), (ii) with 100 MW 1C-1C 4 , (iii) with 200 MW 1C-1C, (iv) with 500 MW 1C-1C, (v) with 1000 MW 1C-1C. Fig. 6 belabors the fact that the benefit of storage sizing for reliability is higher at lower (less conventional reserves), which is the regime of operation for grids with high renewable penetration. From Fig. 7, it is clear that the marginal benefit in reliability due to storage decreases with increasing storage sizes, as expected. However, note that the decay in marginal benefit due to increasing storage is much sharper at higher system response . This suggests that analysis on greater installation of storage in a grid should involve thorough studies of current and future trends in conventional reserve availability. Fig. 8 shows the battery control signal for the imbalance data in Fig. 5, using a 500 MWh (1C-1C) battery. The myopic control Algorithm 3 is used for the optimization with linear cost, but takes into account system response ( = .5%) and SoC maintenance within 40% − 80% band. Whenever the SoC dips below the minimum level or rises above the maximum, the controller follows by re-adjusting the SoC, during time instants when the imbalance R i is within the response-aware bound in Eq. (7). 3 One of the largest installed battery project is located in Hornsdale Power Reserve in Australia. The installed capacity of this plant is 100 MW in power and 129 MWh in energy. Considering the storage installations are going to grow in future we consider significantly large battery in our numerical simulation. 4 Battery model xC-yC means that the battery takes 1/x hours to charge from completely discharged state at the maximum charging rate and 1/y hours to discharge from completely charged state at the maximum discharging rate

Imbalance minimization in BPA, USA
We now consider the aggregated load and generation variation in BPA, for 6 days starting from May 10 2019, collected from their website 5 . Using this data, we compare the performance of our optimization schemes for linear and quadratic cost functions (with differing system responses), for a 1C-1C battery of capacity 100 MW. The results are provided in Table. [19] where myopic stochastic control has an optimality gap of less than 4% compared to the ground truth. For illustration, the tracking of imbalance signal and corresponding change in storage charge levels for response-aware linear (P L ) and quadratic (P Q ) optimization problems (no SoC) are shown in Fig. 9 and Fig. 10, respectively.

Conclusion and Perspectives
The paper presents algorithms for control of energy storage for minimizing demand and supply imbalance. Through theoretical motivation corroborated with numerical simulations, we show that the system dynamic response due to inertia and governor control, impacts the effect of storage in improving grid reliability. In particular, the marginal reliability benefit due to increasing storage decays rapidly for systems with higher conventional reserves. For real-time optimization of storage, we present myopic alternates to deterministic storage algorithms requiring full information, and show their comparable performance using real data from Elia, Belgium and BPA, USA. Furthermore, we demonstrate that storage control algorithms can maintain SoC without significant loss in reliability performance.
In future work, we plan to theoretically study the relationship between variance of stochastic imbalances, and response aware storage operation. Moreover, we plan to extend our numerical analysis to smaller grids/micro-grids with greater fraction of inverter-connected generation, and study financial incentives to maximize reliability performance from storage.