Control of Energy Storage in Home Energy Management Systems: NonSimultaneous Charging and Discharging Guarantees
Abstract
In this paper we provide nonsimultaneous charging and discharging guarantees for a linear energy storage system (ESS) model for a model predictive control (MPC) based home energy management system (HEMS) algorithm. The HEMS optimally controls the residential load and residentiallyowned power sources, such as photovoltaic (PV) power generation and energy storage, given residential customer preferences such as energy cost sensitivity and ESS lifetime. Under certain problem formulations with a linear ESS model, simultaneous charging and discharging can be observed as the optimal solution when there is high penetration of PV power. We present analysis for a proposed HEMS optimization formulation that ensures nonsimultaneous ESS charging and discharging operation for a linear ESS model that captures both charging and discharging efficiency of the ESS. The energy storage system model behavior guarantees are shown for various electricity pricing schemes such as time of use (TOU) pricing and net metering. Simulation results demonstrating desirable ESS behavior are provided for each electricity pricing scheme.
Nomenclature
Penalty coefficient for ESS charging  

Penalty coefficient for ESS discharging  
Cost of ($/kWh)  
ESS state of charge (kWh)  
Maximum energy storage in ESS (kWh)  
Minimum energy storage in ESS (kWh)  
ESS charging efficiency  
ESS discharging efficiency  
Curtailed solar power (kW)  
Power injected into ESS (kW)  
Power drawn from ESS (kW)  
Power consumed from the grid (kW)  
Total residential load (kW)  
Maximum charging power (kW)  
Maximum discharging power (kW)  
Available solar power (kW) 
I Introduction
Integrating renewable energy into the power grid has led to both generationside and demandside solutions to address the intermittent nature of these renewable energy resources. Electric energy storage systems (ESS) are commonly used to cope with the variability in renewable energy resources. In particular, demandside residential energy management solutions can be used to address stable renewable energy integration since residential buildings account for over 37.6% of total electricity consumption in the U.S. [buildingdata]. Home energy management systems (HEMS) provide residential demandside energy management by coordinating residentiallyowned power sources, appliances, user preferences, and renewable energy forecasts [jin2017foresee, wu2015stochastic, zhou2016smart, HUSSAIN2015, Garifi18pes]. Many HEMS systems are equipped with an ESS due to decreasing battery costs and gained flexibility in responding to demand response (DR), and energy cost savings for the customer in various electricity pricing schemes such as timeofuse (TOU) pricing [batteryecon, erdinc2015, Ghazvini2017, KHAN2016, HUSSAIN2015, Garifi18pes]. However, when incorporating ESS models into HEMS, ensuring proper ESS dynamics can be limiting due to the use of lossless or nonconvex ESS operation models, or the use of restrictive computation methods. In this work, we provide nonsimultaneous battery charging and discharging operation guarantees for a widelyused ESS model that captures both charging and discharging efficiency for use in a model predictive control (MPC) based HEMS algorithm that coordinates power drawn from the grid, available solar power, an ESS, and total residential load.
To address this, we use the KarushKuhnTucker (KKT) optimality conditions to show that solutions to the convex MPCbased HEMS algorithm where the ESS is simultaneously charging and discharging are suboptimal under certain conditions. Similar analysis on ESS charging and discharging dynamics derived from the KKT conditions has been applied to a multiperiod optimal power flow (OPF) problem with ESS assets [kkt_opf2013]. For the OPF problem, the authors in [kkt_opf2013] use the natural relation between locational marginal prices (LMPs) and the KKT conditions, to determine conditions for nonsimultaneous ESS charging and discharging dynamics. We apply similar analysis to give guarantees on proper ESS dynamics in an MPCbased HEMS optimization framework for certain electricity pricing schedules and in the presence or absence of net metering. The penalty approach for discouraging simultaneous charging and discharging in the same ESS model is used in [Zarrilli2018]; however, we provide a formal proof for the conditions under which this approach works. Additionally, the authors in [7792609] provide analysis on simultaneous charging and discharging using the same ESS model for a distributed power system with multiple gridconnected storage systems.
The outline of this paper is as follows: in Section II, we survey various ESS models used in HEMS literature. In Section III, the mathematical HEMS models and the overall MPCbased HEMS optimization problem are introduced. In Section LABEL:sec:proofs, we present the main theoretical results of this paper showing that simultaneous ESS charging and discharging is suboptimal for various scenarios. In Section LABEL:sec:sims, we provide simulation results highlighting proper ESS behavior for various electricity pricing schemes and net metering considered in this paper. We also provide simulation results showing when simultaneous ESS charging and discharging is an optimal solution. In Section LABEL:sec:conclusion, conclusions regarding the use of the proposed ESS model are discussed, as well as potential areas of future work.
Ii Commonly Used ESS Models
Energy storage systems are often included in renewable energy research due to their energy management flexibility [hemmati2017technical]. Additionally, an ESS can be used to account for uncertainties in renewable energy such as solar and wind energy [baker2017optEss, hemmati2017technical]. In HEMS research, many homes are equipped with an ESS due to its flexibility to participate in grid services such as demand response [jin2017foresee, Garifi18pes, HUSSAIN2015] or help with consumer energy cost savings in realtime or demand pricing energy markets [wu2015stochastic, batteryecon, zhou2016smart]. However, ensuring proper ESS dynamics (i.e. ensuring the model does not allow a physically unrealizable optimal control policy where the ESS simultaneously charges and discharges in the same time step) when incorporating ESS models into the HEMS framework leads to the use of lossless, binary, or nonconvex models.
To ensure nonsimultaneous ESS charging and discharging, a lossless ESS model is often used. A common lossless model used in HEMS literature is:
(1a)  
(1b)  
(1c) 
for all time where is the power injected or drawn into the ESS [yu2017online, CDC2016]. The model in (1a)(1c) ensures nonsimultaneous ESS charging and discharging since the power injected into the ESS and drawn from the ESS are captured in one variable which represents discharging when negative and charging when positive. While this model is linear, it assumes perfect ESS charging and discharging efficiency, implying the roundtrip efficiency of the ESS is 100% [yu2017online, CASTILLO2014885], which does not accurately model the nonnegligible losses of the system.
Additionally, nonconvex ESS models are used in HEMS research to capture ESS charging and discharging efficiency. A common nonconvex ESS model that includes both charging and discharging terms and ensures the ESS does not simultaneously charge and discharge uses binary variables and is given by [hemmati2017technical, chen2013mpc, 2015robustOpt_wBatt_wRERs, wu2015stochastic, erdinc2015, Ghazvini2017, ParisioEssMILP]:
(2a)  
(2b)  
(2c)  
(2d)  
(2e) 
for all time . The use of separate terms for ESS charging and discharging, and , respectively, allow for a roundtrip efficiency of less than 100% which accounts for losses in ESStogrid interactions [CASTILLO2014885, 2015robustOpt_wBatt_wRERs]. However, the use of binary variables in ESS models results in a nonconvex model requiring the use of computationally restrictive methods such as mixed integer (non)linear programming (MILP/MINLP) [hemmati2017technical, chen2013mpc, 2015robustOpt_wBatt_wRERs, wu2015stochastic]. Alternatively, the following nonlinear, nonconvex ESS model without binary variables can be used:
(3a)  
(3b)  
(3c)  
(3d)  
(3e) 
for all time where the nonconvex constraint in (3e) is included in the model to ensure nonsimultaneous ESS charging and discharging. Similar to the ESS model with binary variables, the model in (3a)(3e) also requires the use of computationally restrictive nonconvex optimization solvers.
Thus, the common models used to ensure nonsimultaneous ESS charging and discharging either assume perfect 100% roundtrip efficiency or are nonconvex requiring the use of computationally limiting numerical methods. In this work, we provide nonsimultaneous ESS charging and discharging guarantees for a frequently used linear ESS model that captures both charging and discharging efficiency. The main contributions of this work are the following:

We present a formalized and thorough analysis that shows simultaneous charging and discharging is suboptimal for a frequently used linear ESS model that captures both charging and discharging efficiency under the proposed HEMS optimization formulation, including under various pricing schemes and net metering. We believe this to be the first manuscript that formally proves these claims.

The use of nonconvex or mixed integer ESS models to ensure nonsimultaneous charging and discharging are unnecessary under the proposed HEMS optimization formulation.

We present situations under which simultaneous charging and discharging is optimal in the HEMS optimization, and provide a means to ensure an equivalent solution where simultaneous charging and discharging does not occur without adopting a nonconvex ESS model.
Iii Problem Formulation
In this section, we provide the mathematical models for the HEMS and the overall MPC optimization problem. In this work, we assume that the HEMS must coordinate the residential PV generation, the ESS, the total residential load, and power drawn from the grid needed to satisfy the load, as shown in Fig. 1.
The residential energy storage system state of charge (SOC) and power charged/discharged are modeled by:
(4a)  
(4b)  
(4c)  
(4d)  
(4e) 
for all where , define the SOC limits. The charging efficiency is denoted and the discharging efficiency is denoted . The condition in (4e) can be relaxed and the results we present in this paper hold for any and satisfying ; however, we constrain and as in (4e) for a more practical roundtrip ESStogrid efficiency. While there exist more complex models that capture nonlinear ESS dynamics, we specifically consider this linear ESS model since it is widely used for gridconnected ESS applications [jin2017foresee, Zarrilli2018, kkt_opf2013, 7792609, baker2017optEss].
The available solar power generated from PV, denoted , is a function of the solar irradiance, area of the PV array, and array and inverter efficiency. The curtailed solar power is denoted . The overall power consumption from building loads is denoted . The power balance is given by:
(5) 
Next, we provide the overall MPCbased optimization problem for the HEMS. The objective function , which captures both customer electricity price sensitivity and ESS lifetime considerations, is given by: