Modeling and Optimal Operation of Distributed Battery ...

1 1. Modeling and Optimal Operation of Distributed Battery storage in low voltage Grids Philipp Fortenbacher, Johanna L. Mathieu, and G oran Andersson Abstract Due to high power in-feed from photovoltaics, it Br branch flow matrix can be expected that more Battery systems will be installed in Bv linearized active and reactive power to the distribution grid in near future to mitigate voltage violations voltage matrix and thermal line and transformer overloading. In this paper, we [ ] 16 Mar 2017. present a two-stage centralized model predictive control scheme Bq matrix to describe polygonal regions for for Distributed Battery storage that consists of a scheduling entity active and reactive power and a real-time control entity. To guarantee secure grid Operation , Cg controllable generator to bus mapping ma- we solve a robust multi-period Optimal power flow (OPF) for the trix scheduling stage that minimizes Battery degradation and max- C pv PV generator to bus mapping matrix imizes photovoltaic utilization subject to grid constraints.

2 The real-time controller solves a real-time OPF taking into account cE Battery capacity storage allocation profiles from the scheduler, a detailed Battery cE Battery capacity vector model, and real-time measurements. To reduce the computational cd Battery degradation cost in e/kWh complexity of the controllers, we present a linearized OPF that cn feeder energy cost in e/MWh approximates the nonlinear AC-OPF into a linear programming cp generator energy cost vector problem. Through a case study, we show, for two different Battery technologies, that we can substantially reduce Battery cr inverse charge recovery time in sec 1. degradation when we incorporate a Battery degradation model. cw parameter to describe the size of capacity A further finding is that we can reduce Battery losses by 30% by wells using the detailed Battery model in the real-time control stage.

3 E state of energy of one Battery system in Index Terms Optimal control, power systems, predictive con- MWh trol, energy storage e state of energy vector of ns Battery systems e(0) initial state of energy vector of ns Battery systems N OMENCLATURE. emin , emax storage allocation bounds from scheduler bat , in Battery stack and inverter efficiency E state of energy evolution vector of ns bat- dis ch bat , bat Battery stack discharging and charging ef- tery systems ficiency H discrete Battery control input matrix dis , ch total Battery system discharging and charg- i nodal current vector in ing efficiency ib branch current vector in SOS2 set imax b max branch current vector in PV standard deviation of the PV forecast error Ibat Battery current in A. discrete Battery system dynamics matrix l Luenberger observer gain a1 , a2 , a3 degradation plane parameter vectors of the m penalty factor for setpoint regions triangles from the convex hull n number of buses A continuous Battery system dynamics matrix ng number of controllable generators A1 , A2 , A3 , Az matrices to include Battery degradation for nl number of branches multiple Battery systems and time steps np number of triangles Aq matrix to describe polygonal regions for npv number of PV units active and reactive power ns number of Battery systems bbat Battery system control input vector N number of time steps in the scheduler hori- B Battery system control input matrix for zon multiple Battery systems Pbat active Battery stack power B 1i , B 2i.

4 Bl Matrices and vector specifying the PWA pagg aggregated Battery set point bat network loss approximation loss Pbat total active power losses of one Battery system This work was, in part, financed by the Swiss Commission for Technology and Innovation (CTI) 4/2013-10/2014 ploss bat total active power loss vector of ns Battery P. Fortenbacher and G. Andersson are with the power Systems Labora- systems tory, ETH Zurich, Switzerland (e-mail: anders- + . Pbat,r , Pbat,r active Battery power range for linear power J. L. Mathieu is with the Department of Electrical Engineering and loss approximation Computer Science, University of Michigan, USA (e-mail: Pcell active Battery cell power 2. pd , q d non-controllable nodal active and reactive communication infrastructure. However, addressing thermal power load vectors overloading requires coordination either via centralized or pgen , q gen controllable active and reactive generator Distributed control, since the power flows are not observable power vectors at a local level.))

5 In [4], a centralized predictive control scheme pv pv ppv gen , q gen , pgen active and reactive power PV measurement is developed to avoid thermal line overloading, but the authors and prediction vectors do not consider voltage constraints. Furthermore, the proposed ps,dis s gen pgen active discharging Battery system grid control strategies in [2] [4] do not consider Battery degradation power vector or use detailed Battery models. ps,ch s gen pgen active charging Battery system grid power Since investment costs for batteries are still high [5], a vector Battery 's expected lifetime greatly affects assessments of its ps,P. gen supporting point vector for nonconvex bat- economic viability. Each control action applied to a Battery tery system loss calculation leads to charge capacity loss ( , degradation) [6], reducing ppl , pql decision vectors of active network losses its lifetime.

6 Some recent papers propose methods to include ld pld , q ld , p active and reactive power load measure- Battery degradation costs in economic cost functions [7], [8]. ment and prediction vectors Another way to increase the economic viability of a Battery is pmin , pmax min and max active generator power vec- to exploit its full capacity, enabling Operation in low/high state tors of charge regimes when the benefits outweigh the degradation pnet active feeder generator power costs. However, simple Battery models do not capture the R internal Battery resistance dynamics that are present when batteries operate in these smax max apparent generator power vector regimes. For example, Operation in these regimes is only Sx, Su matrices to describe the storage evolution possible at low charging/ discharging powers.

7 It is possible to T1 , T2 sample times for scheduler and RT con- model these dynamics with detailed Battery models [9], [10]. troller The objective of this paper is to develop computationally- U decision vector for active Battery power tractable methods to control Distributed batteries in distri- v nodal RMS voltage vector in bution networks with high penetrations of PV to manage v min , v max min and max nodal RMS voltage vectors network constraints such as voltage and thermal constraints. vs slack bus voltage vector To exploit the full potential of the network and to dispatch Voc Battery stack open-circuit voltage the Battery systems in an economic and efficient way, we wk W box-constrained uncertainty set formulate multi-period AC Optimal power Flow (AC-OPF). xk X decision vector for grid variables problems.

8 We use a linear approximation of the nonlinear xE dynamic state vector of one Battery system nonconvex AC-OPF from our previous work [11] that exploits xE1 state of accessible energy well the radial structure of a LV network. The approximation xE2 state of non-accessible energy well captures line losses and is linear in active and reactive power xset vector of helper decision variables to pe- injections. Linear approximations have been developed for nalize deviations from the storage alloca- distribution networks [12], [13]; however, these neglect line tion band losses. The so-called DC power flow approximation and the X 0, X 1 Battery system state vectors (initial and approximation in [14] are not applicable for distribution grids after one time step) for ns Battery systems since, in LV grids, active power flow is dominated by voltage agg yset helper decision variable to penalize devia- magnitude differences rather than voltage angle differences.

9 Tions from the aggregated Battery set point Nonlinear convex relaxations of the AC-OPF problem also z Battery degradation variable for one Battery exist; however, the resulting semi definite [15], [16] and system second order cone [17], [18] programs are computationally zk Z decision vector for Battery degradation large and not suitable for computationally-limited controllers. variables Linear programming approximations of the second order cone programming problem have been developed, but the number of the linear constraints is large [19]. Ref. [20] derives I. I NTRODUCTION. an AC power flow approximation for distribution networks I N the next few years, it can be expected that many Battery systems will be installed in the low voltage (LV). distribution grid to cope with high in-feed from photovaltaics that is linear between the complex voltage and the complex power injections, but does not demonstrate how to use the approximation within an Optimal power Flow (OPF) problem.

10 (PV) [1] and other fluctuating energy sources also connected Our contribution is the development of a control strategy to the distribution grid. In particular, Battery systems can miti- that leverages the linearized AC-OPF approximation to opti- gate voltage violations and thermal line overloading, allowing mize Distributed Battery Operation within a LV grid. We incor- Distribution System Operators (DSOs) to defer line and trans- porate the linear AC-OPF into a two-stage Model Predictive former upgrades. Some recent papers [2], [3] developed decen- Control (MPC) control scheme that consists of a scheduler tralized Battery control strategies to provide voltage support. and a real-time (RT) controller. The scheduler is a robust MPC. Decentralized control strategies have the advantage that they that solves a multi-period OPF minimizing Battery degradation rely only on local measurements.