Storage allows us to compensate for generation variability, especially from renewables. However, storage is expensive and complex to manage. Our work studies *sizing* of storage, drawing on its analogy to buffers in computer networks, and *operation* of storage, that is, when to charge and when to discharge storage.

**Battery sizing:**Although solar energy is varying throughout the day, if we could spread out the total daily solar energy throughout the day, we can have a useful constant power source. We aim to minimally size the batteries to guarantee a certain fixed power from the sun either for the whole day (firm-up solar PV power) or for a time interval during the day (day-ahead dispatch problem).**Battery operation:**The*application*of the renewable energy and battery system plays a role on how the battery should be operated. For example, a home owner might want to use the system to minimize payments to their electricity provider (utility). The optimal operation would differ substantially depending on the pricing scheme (eg. time-of-use pricing) that is in effect. We aim to find solutions to the question of how to optimally operate the system in a given environment, and propose practical online operating strategies that result in near-optimal system performance.

## Recent Projects

#### Lithium-Ion Storage Models for Energy System Optimization

This work looks at mathematical models for Lithium-ion batteries that are tractable enough to be included as part of mathematical optimization programs. We compare the state-of-the-art model with real Lithium-ion batteries, using experiments conducted at the Technology Center for Energy in Ruhstorf, Germany. Our experiments show that the accuracy of the state-of-the-art model is very limited with respect to a wide range of charging and discharging rates. Using the insights from our experiments, we derive and validate two new models that represent different trade-offs between accuracy and tractability. We evaluate the error of our models compared to a real battery, and make a favourable comparison with the state-of-the-art model. This work has been published in the proceedings of ACM eEnergy 2016, and was featured in a poster presented at the 2016 Cheriton Symposium and was a runner-up for the Best Poster award.

#### Practical Strategies for Storage Operation in Energy Systems

This work looks at grid-tied PV and storage systems that are deployed for self-consumption of renewable energy. These systems are starting to be deployed in homes and small businesses at an increasing rate, but existing research has not focused much on the efficient operation of the system. Moreover, existing ad-hoc operating strategies have been proposed but have not been properly evaluated. We model the system mathematically and solve a control optimization problem to get a benchmark for system performance, and create practical operating algorithms that are inspired by the optimal control. The contribution of our work is an approach to evaluate existing strategies and gain insight into designing new efficient and practical strategies, making it a valuable tool for manufacturers and users. One of our contributions is a novel operation strategy for peak-demand grid pricing schemes that has shown near-optimal performance. This work has been published in IEEE Transactions on Sustainable Energy, vol. 7, issue 4, May 2016.

#### Optimal Design of Solar PV Farms with Storage

We consider the problem of allocating a capital budget to solar panels and storage to maximize the expected revenue in the context of a large-scale solar farm participating in an energy market. This problem is complex due to many factors. To begin with, solar energy production is stochastic, with a high peak-to-average ratio, thus the access link is typically provisioned at less than peak capacity, leading to the potential waste of energy due to curtailment. The use of storage prevents power curtailment, but the allocation of capital to storage reduces the amount of energy produced. Moreover, energy storage devices are imperfect. A solar farm owner is thus faced with two problems: 1)deciding the level of power commitment and 2) the operation of storage to meet this commitment. We formulate two problems corresponding to two different power commitment approaches, an optimal one and a practical one, and show that the two problems are convex, allowing efficient solutions. Numerical examples show that our practical power commitment approach is close to optimal and also provide several other engineering insights. This work has been published in IEEE Transactions on Sustainable Energy, vol. 6, issue 4, October 2015.

## Past Projects:

- Y. Ghiassi-Farrokhfal, S. Keshav, and C. Rosenberg, Towards a Realistic Performance Analysis of Storage Systems in Smart Grids, IEEE Transactions on Smart Grids, Vol 6, No. 1, January 2015, pp. 402-410. Energy storage devices (ESDs) have the potential to revolutionize the electricity grid by allowing the smoothing of variable-energy generator output and the time-shifting of demand away from peak times. A common approach to study the impact of ESDs on energy systems is by modeling them as electric circuits in simulations. Although recent circuit models are becoming more accurate, to obtain statistically valid results, extensive simulations need to be run. In some cases, existing datasets are not large enough to obtain statistically significant results. The impact of ESDs on energy systems has also been recently studied using analytical methods, but usually by assuming ideal ESD behavior, such as infinite ESD charging and discharging rates, and zero self-discharge. However, real-life ESDs are far from ideal. We investigate the effect of nonideal ESD behavior on system performance, presenting an analytical ESD model that retains much of the simplicity of an ideal ESD, yet captures many (though not all) nonideal behaviors for a class of ESDs that includes all battery technologies and compressed air energy storage systems. This allows us to compute performance bounds for systems with nonideal ESDs using standard teletraffic techniques. We provide performance results for five widely used ESD technologies and show that our models can closely approximate numerically computed performance bounds.

- D. Fooladivanda, C. Rosenberg, and S. Garg, “Energy Storage and Regulation: An Analysis,” in
*Smart Grid, IEEE Transactions on*, pp.1-11. Electric system operators rely on regulation services to match the total system supply to the total system load in quasi real-time. The regulation contractual framework requires that a regulation unit declares its regulation parameters at the beginning of the contract, the operator guarantees that the regulation signals will be within the range of these parameters, and the regulation unit is rewarded proportionally to what it declares and what it supplies. We study how this service can be provided by a unit with a non-ideal storage. We consider two broad classes of storage technologies characterized by different state of charge evolution equations, namely batteries and flywheels. We first focus on a single contract, and obtain formulas for the upward and downward regulation parameters that a unit with either a battery or a flywheel should declare to the operator to maximize its reward. We then focus on a multiple contract setting and show how to analytically quantify the reward that such a unit could obtain in successive contracts. We quantify this reward using bounds and expectation, and compare our analytical results with those obtained from a dataset of real-world regulation signals. Finally, we provide engineering insights by comparing different storage technologies in terms of potential rewards for different contract durations and parameters.

- D. Fooladivanda, C. Rosenberg, and S. Garg, Analysis of Energy- and SoC-Neutral Contracts for Frequency Regulation with Energy Storage, Proc. IEEE Smart Grid Communications, November 2015.

- S. Singla, Y. Ghiassi-Farrokhfal, and S. Keshav. Using Storage to Minimize Carbon Footprint of Diesel Generators for Unreliable Grids, IEEE Transactions on Sustainable Energy, Vol 5, No. 4, pp. 1270-1277. 2014. Although modern society is critically reliant on power grids, modern power grids are subject tounavoidable outages. The situation in developing countries is even worse, with frequent load shedding lasting several hours a day due to a large power supply-demand gap. A common solution for residences is, therefore, to back up grid power with local generation from a diesel generator (genset).To reduce carbon emissions, a hybrid battery-genset is preferable to a genset-only system. Designing such a hybrid system is complicated by the tradeoff between cost and carbon emission. Toward the analysis of such a hybrid system, we first compute the minimum battery size required for eliminating theuse of a genset, while guaranteeing a target loss of power probability for an unreliable grid. We then compute the minimum required battery for a given genset and a target-allowable carbon footprint. Drawing on recent results, we model both problems as buffer sizing problems that can be addressedusing stochastic network calculus. Specifically, a numerical study shows that, for a neighborhood of 100 homes, we are able to estimate the storage required for both the problems with a fairly small margin of error compared to the empirically computed optimal value.

- D. Fooladivanda, C. Rosenberg, and S. Garg. An Analysis of Energy Storage and Regulation, Proc. IEEE Smart Grid Communications, November 2014. We focus on a region whose power system is controlled by an operator that relies on a regulationservice to balance the total system supply to the total system load in quasi real-time. We consider the existing contractual framework in which a regulation unit declares its regulation parameters at the beginning of the contract, the operator guarantees that the regulation signals will be within the range of these parameters, and the regulation unit is rewarded proportionally to what it declares. Our purpose is twofold. We first want to obtain formulas for the regulation parameters that a unit with non-ideal storageshould declare to the operator given its state of charge at the beginning of a contract. Second, we want to analytically quantify, ahead of time, the reward that such a unit could obtain in successive contracts by performing this regulation service. Since the state of charge at the beginning of a contract depends on what happened in the previous contract and, hence, is a random variable, we quantify this reward analytically using bounds and expectation. We then provide engineering insights by applying our results to three specific energy storage technologies that are often considered as candidates for regulation. In particular, we show the impact of the storage parameters and the length of one contract on the potential reward over a given period.

- Y. Ghiassi-Farrokhfal, S. Keshav and C. Rosenberg. An EROI-Based Analysis of Renewable Energy Farms with Storage, Proc. ACM e-Energy, June 2014. Large renewable energy (RE) farms, such as wind or solar farms are usually sited in remote areas, far from the transmission grid which typically interconnects population centers. Thus, they need to be connected on expensive access lines (distribution feeders) with limited capacity. The excess of RE generation over line capacity is wasted; this is called curtailment. We study curtailment using the metric of energy return on investment (EROI), defined as the ratio of useful energy extracted from each unit of energy invested in creating the renewable energy generation system. Curtailment reduces EROI. It may appear that we can extract more energy from an RE farm and increase EROI by adding storage to the system, where this storage is charged during generation peaks and discharged during off-peak times. However, manufacturing the storage requires an energy investment, and, after a certain number of cycles of usage, the storage becomes non-functional. Thus, adding storage may actually decrease the EROI. In this work, we study the EROI for RE farms when used with several types of storage technologies. Unlike prior work that makes numerous simplifying assumptions, our work accounts for storage size and storage imperfections and uses actual traces of renewable power generation. We find that lithium-ion batteries increase the EROI of both wind and solar farms, unlike lead-acid batteries which generally decrease their EROI. We also show that increasing access line capacity to achieve a target EROI is much more expensive for solar farms than for wind farms.

- S. Singla, Y. Ghiassi-Farrokhfal, and S. Keshav. Battery Provisioning and Scheduling For a Hybrid Battery-Diesel Generator System, ACM SIGMETRICS Performance Evaluation Review, December 2013. Diesel generators (gensets) are commonly used to provide a reliable source of electricity in off-grid locations. Operating a genset is expensive both in terms of fuel and carbon footprint. Because genset efficiency increases with offered load, this expense can be reduced by using a storage battery to ensure that a genset always runs at full load, charging and discharging the battery as necessary. However, the cost of batteries requires us to size them parsimoniously and operate them efficiently. We, therefore, study the problem of provisioning and optimally managing a battery in a hybrid batterygenset system. To aid in sizing a battery, we analytically study the trade-off between battery size and carbon footprint. We also formulate the optimal scheduling of battery charging and discharging as a mixed-integer program, proving that it is NP-hard. We then propose a heuristic online battery scheduling scheme that we call alternate scheduling and prove that it has a competitive ratio of
*k*_{1}*G*/*C*+*k*^{2}*T*/_{u}*k*_{1}+*k*_{2}*T*with respect to the offline optimal scheduling, where G is the genset capacity, C is the battery charging rate, k1, k2 are genset-specific constants, and Tu is the duration of a time step. We numerically demonstrate that alternate scheduling is near-optimal for four selected industrial loads._{u}

- Y. Ghiassi-Farrokhfal, S. Keshav, and C. Rosenberg. Firming Solar Power, Extended Abstract/Poster, Proc. ACM SIGMETRICS, June 2013. The high variability of solar power due to intrinsic diurnal variability, as well as additional stochastic variations due to cloud cover, have made it difficult for solar farms to participate in electricity markets that require pre-committed constant power generation. We study the use of battery storage to ‘firm’ solar power, that is, to remove variability so that such a pre-commitment can be made. Due to the high cost of storage, it is necessary to size the battery parsimoniously, choosing the minimum size to meet a certain reliability guarantee. Inspired by recent work that identifies an isomorphism between batteries and network buffers, we introduce a new model for solar power generation that models it as a stochastic traffic source. This permits us to use techniques from the stochastic network calculus to both size storage and to maximize the revenue that a solar farm owner can make from the day-ahead power market. Using a 10-year of recorded solar irradiance, we show that our approach attains 93% of the maximum revenue in a summer day that would have been achieved in daily market had the entire solar irradiance trace been known ahead of time.

- O. Ardakanian, C. Rosenberg, and S. Keshav. On the Impact of Storage in Residential Power Distribution Systems, ACM SIGMETRICS Performance Evaluation Review 40.3 (2012): 43-47. It is anticipated that energy storage will be incorporated into the distribution network component of the future smart grid to allow desirable features such as distributed generation integration and reduction in the peak demand. There is, therefore, an urgent need to understand the impact of storage on distribution system planning. In this paper, we focus on the effect of storage on the loading of neighbourhood pole-top transformers. We apply a probabilistic sizing technique originally developed for sizing buffers and communication links in telecommunications networks to jointly size storage and transformers in the distribution network. This allows us to compute the potential gains from transformer upgrade de- ferral due to the addition of storage. We validate our results through numerical simulation using measurements of home load in a testbed of 20 homes and demonstrate that our guidelines allow local distribution companies to defer trans- former upgrades without reducing reliability.

- T. Carpenter, S. Singla, P. Azimzadeh, and S. Keshav. The Impact of Electricity Pricing Schemes on Storage Integration In Ontario, Proc. ACM e-Energy, May 2012. The Ontario electrical grid is sized to meet peak electricity load. If this worst-case load were reduced, the government and Ontario tax-payers could defer large infrastructural costs, reducing the cost of generation and electricity prices.
*Storage*, batteries that can store energy during times of low load and be discharged during times of peak load, is one proposed solution to reducing peak load. We evaluate the effect of storage on the electrical grid under different customer electricity pricing schemes. We find that for existing pricing schemes, adopting storage is not profitable. Furthermore, as the level of storage adoption in the population increases, pricing schemes that incentivize charging at times known to all homeowners will eventually increase the peak load rather than decrease it. However, in some circumstances particular levels of homeowner storage adoption helps the grid reduce peak load, and thus the grid may choose to subsidize the cost of storage. We discuss hypothetical pricing schemes under which storage adoption is profitable for homeowners.