Shared Economy for Battery Storage
Introduction
With growing dependence on technology, there is an ever increasing demand for electricity. Not only has this burdened generation units in terms of large power demand but also in terms of its highly irregular demand profile. This has ensued various pricing policies by utilities to mitigate the run time cost of power generating units. In developed countries, one such scheme utilizes the concept of dynamic tariff. During peak hours, cost per unit of electricity is kept high, and during off-peak hours tarrifs are brought down to moderate the energy demand profile.
One way to minimize loss is to decrease individual's exposure to real time market. This can be easily achieved by an investment on battery storage. Size of battery is identified by solving an optimization problem to minimize expected cost in long run. It turns out that investing on a shared battery resource is better than investing on individual battery storage, therefore, highlighting the benefit of coalition formation.
This project is a literature survey of several game theoretic models to study the benefit of coalition for electricity consumers under different pricing models. Moreover, focus was restricted to 4 published articles by a research group at University of California, Berkley.
Shared Economy for Battery Storage: Risk Avoidance
Consider an electricity market with dynamic pricing scheme. There exists a slot ahead market to purchase \(\hat{x}_i\) amount of electricity at nominal cost \(\pi_l\). Any excess real time purchase is at higher cost \(\pi_h>\pi_l\). Any rational user in this market would want to minimize exposure to the real time market. One way to achieve this is by investing on a battery storage system, The battery provides electricity backup to avoid or minimize purchase of electricity from the real time market at higher cost.
Example
We will illustrate an example to motivate the study of battery sizing problem.
When the battery storage system fails to meet the excess dynamic electricity demand, the player has to purchase electricity from the real time market.
There is a trade-off between exposure to real time market and overhead cost associated with a battery storage. In the battery sizing problem, the aim is to minimize the overall cost by identifying the optimal size of battery storage.
Notation
\(\mathcal{N}=\{1,2,\dots,N\}\) denotes consumers.
\(\mathcal{T}=\{1,2,\dots,T\}\) denotes partition of time slots each of duration \(\Delta T\).
\(\textbf{x}_i(t)=\hat{x}_i(t)+x_i(t)\) denotes average + deviated electricity consumption, where \(x_i(t)\)'s are i.i.d. random variable \(\forall i,t\).
\(s_i(t)\) denotes battery state of charge at time \(t\forall i\in\mathcal{N}\).
\(s_i=\;s_i^{max}\) denotes capacity of battery storage \(\forall i\in\mathcal{N}\)
\(\mathbb{E}[x_i(t)]=0;\;\forall\;i\in\mathcal{N},\;\forall t\)
\(\Sigma_{ij}=\mathbb{E}[x_i(t)x_j(t)]=\rho_{ij}\sigma_i\sigma_j\)
\(\Sigma=\{\Sigma_{ij}\}_{1\leq i,j\leq N}\)
Battery Storage Sizing (Single Player)
Battery storage sizing refers to deciding the capacity of a battery backup to minimize purchase of electricity from real time market while keeping in mind the investment overhead associated with a battery. In the single player version, each player buys their own battery storage and aims to minimize their overall cost. This can be formulated as the following optimization problem.\(\displaystyle \min_{s}(ks+c_0)\)
s.t. \(p^c=\mathbb{P}\left(\max_{t\in\mathcal{T}}||{s(t-1)-x(t)-s(0)}||\geq\frac{s^{max}}{2}\right)\leq\theta\)
where, \(s^{max}\) is the battery storage size,
\(k\) is the unit price for storage capacity, and
\(c_0\) is the cost of installation of inverter.
For the above problem, bound on battery size is given by
\(s^{max}\geq\frac{2\sqrt{T}\sigma}{\sqrt{\theta}}\)
Battery Storage Sizing (Multiple Players)
In the multiple players version, players collectively invest on a single battery to minimize their purchase of electricity from the real time market; the battery storage is shared amonst the players. The idea is to exploit that fact that each player's electricity consumption is not completely correlated with others. The problem of battery sizing for multiple players can be formulated as follows.\(\displaystyle \min_{s_{\mathcal{N}}}(ks_{\mathcal{N}}+c_0)\)
s.t. \(p_i^c\leq\theta\;\forall\;i\)
where,
\(s^{max}_{\mathcal{N}}\) is the shared battery storage size
\(k\) is the unit price for storage capacity
\(c_0\) is the cost of installation of inverter
For the above problem, bound on the battery size is given by
\(s^{max}_{\mathcal{N}}\geq\frac{2\sqrt{T\mathbb{1}^{'}\Sigma\mathbb{1}}}{\sqrt{\theta}}\)
Results
- Sharing of battery minimizes battery cost.
- Cost allocation scheme is given by \(\pi_c(i)=\frac{\sum_{j\in\mathcal{N}}\rho_{ij}\sigma_{i}\sigma_{j}}{\mathbb{1}^{'}\Sigma\mathbb{1}}\times Total\;cost\)
- The abovementioned cost allocation is in the core of the game.
Documentation
To learn about other models used to study cooperation between multiple players in electricity market, refer to the presentation file.
Author
Anurag Gupta is an M.S. graduate in Electrical and Computer Engineering from Cornell University. He also holds an M.Tech degree in Systems and Control Engineering and a B.Tech degree in Electrical Engineering from the Indian Institute of Technology, Bombay.
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