- Email: [email protected]
Optimal battery sizing of smart home via convex programming
Energy 140 (2017) 444e453
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Optimal battery sizing of smart home via convex programming Xiaohua Wu a, 1, Xiaosong Hu b, *, 1, Xiaofeng Yin a, Caiping Zhang c, Shide Qian d a
School of Automobile and Transportation, Xihua University, Chengdu, 610039, China The State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400044, China c National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing, 100044, China d Beijing Benz Automotive Co.Ltd, Beijing, 100176, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2017 Received in revised form 17 August 2017 Accepted 22 August 2017 Available online 26 August 2017
This paper develops a convex programming (CP) framework for optimal sizing and energy management of smart home with battery energy storage system (BESS) and photovoltaic (PV) power generation, for the goal of maximizing home economy, while satisfying home power demand. We analyse the historical electric energy data of three different homes located in California and Texas, and indicate the necessity and importance of a BESS. Based on the structures and system models of these smart homes, the CP problem is formulated to rapidly and efficiently solve the optimal design/control issue. Based on different time horizons, maximal powers to grid, prices of BESS, the optimal parameters of BESS and its potential to electric energy cost savings are systematically compared for the three homes. A deviation analysis between the results obtained by CP and DP (dynamic programming) is also presented. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Smart homes Energy management Optimal sizing PV arrays Battery energy storage
1. Introduction The present energy crisis (increasingly severe energy shortage and supply instability), and environmental crisis (global warming and air pollution) have promoted the rapid development of integrating renewable energy with the grid [1,2]. However, some renewable energy, such as solar and wind energy, is intermittent and unstable in nature due to metrological conditions [3,4]. Consequently, researchers have focused on developing effective control strategies to improve the performance and economy for buildings and homes integrating renewable energy [5e8]. This paper develops an optimal design/control method for battery sizing and energy management of smart home. The existing literature, e.g., the forgoing work, has presented several optimization methods, such as mixed-integer linear programming (MILP) [9,10], rolling horizon strategy [11], particle swarm optimization (PSO) approach [12], geometric program [13], model predictive control (MPC) [14,15], dynamic programming (DP) [16], adaptive dynamic programming (ADP) [17], and stochastic dynamic programming (SDP) [18], for creating efficient operational schedules or making good consumption and
* Corresponding author. E-mail address: [email protected] (X. Hu). 1 X. Wu and X. Hu equally contributed to this research work. http://dx.doi.org/10.1016/j.energy.2017.08.097 0360-5442/© 2017 Elsevier Ltd. All rights reserved.
production decisions to home energy management (HEM). An optimal HEM strategy under dynamic electrical and thermal constraints is developed through solving an MILP in Ref. [9], which is able to provide an optimal solution to power consumption and management of renewable resources. Similarly, considering photovoltaic (PV) arrays, battery energy storage, and electric vehicle (EV) in Ref. [10], the effects of the accurate SSUEP (Set of Sequential Uninterruptible Energy Phases)-based model on the day-ahead energy management of a residential microgrid are formulated as an MILP optimization framework. A novel energy management system based on a rolling horizon strategy for a microgrid is proposed with PV panels, two wind turbines, a diesel generator, and an energy storage system in Ref. [11]. An improved PSO approach is introduced to optimize distributed energy resources operation schedules for a smart home case study in Ref. [12]. The impacts of the response capability levels of consumers on the economic integration of distributed PV power in smart homes, and the impacts of PV and battery capacities on consumers power expenses are analyzed using non-cooperation game theoretical power market complementarity model in Ref. [13]. Based on short-term forecasts of residential renewable power generation, a dynamic HEM algorithm is put forward in Ref. [14] for decreasing the total grid energy cost while maximizing user comfort. Both simulation and experimental results show the ability of the devised algorithm to control both sources and loads. A nonlinear predictive
X. Wu et al. / Energy 140 (2017) 444e453
energy management method for buildings with PV system and battery storage is proposed in Ref. [15], which forecasted house load demand via artificial neural networks. Based on highly resolved energy consumption models, an automated dynamic energy management framework is established in Ref. [16] to find the optimal schedule of residential controllable appliances, where DP is utilized to find the global solution. A computationally feasible and self-learning optimal control scheme for a residential energy system with batteries is devised in Ref. [17], the idea of which is to approximate DP solution by using neural networks. A probability distribution model combining household power consumption, EV home-charging, and PV power production is built using a convolution approach to merge three separate existing probability distribution models in Ref. [18]. In Ref. [4], ZigBee technologies are exploited for comprehensive field tests, including monitoring PV and wind energy systems, and energy management of buildings/ homes. The literature provides a number of approaches for energy management of smart home with renewable energy, almost all of which share a common goal, namely, to meet overall home electric power demand while minimizing the total operational energy cost. Few studies explored the optimal battery size and control strategy simultaneously. Furthermore, a single home was often considered without a comparision of multiple homes with different electricity supply/demand patterns. This paper constructs an optimal design/control framework for exploring multiple smart homes with battery energy storage system (BESS) and PV arrays power generation. The key challenge addressed by this paper is to simultaneously optimize the battery size and energy management strategy with the consideration of the cost of BESS, computational efficiency, and home electricity difference. An emerging effective tool, convex programming (CP), which can rapidly and efficiently optimize both management strategy and parameters, as opposed to many other methods generally focusing on only controls, has been applied by researchers [19e21]. Demand response (DR) implementation considering a single home with different appliances is modeled using CP to minimize electricity payment in Ref. [22]. Similarly, A later study in Ref. [23] formulates a CP problem to minimize electricity payment and waiting time under real-time pricing for a multi-agent system, in order to evaluate optimal residential DR implementation in a distribution network. The previous two studies, however, do not involve optimal component design. Nevertheless, CP has been successfully applied to simultaneously optimize component size and energy controller for vehicles [24,25]. In this article, CP is leveraged to rapidly and efficiently optimize both HEM strategy and BESS size. In our previous study [26], a CP framework was built for the optimal integration of a hybrid solarbattery power source into smart home nanogrid with PEV load. The main differences between this endeavor and [26] (i.e., major contributions) include (i) the optimization results of three different homes with PV arrays located in California and Texas are systematically compared, instead of analyzing a single home in most of existing studies, and (ii) a deviation analysis between the outcomes procured by CP and DP is carried out to showcase the computational efficiency and effectiveness of CP. The remainder of the paper is arranged as follows. Section 2 analyses the historical electric energy data of the three homes. The structures and models are described in Section 3. The CP problem is formulated in Section 4. The optimization results are discussed and contrasted in Section 5, followed by conclusions summarized in Section 6.
445
emissions, residents are seriously concerned about installing PV panels on the roof of their houses, especially in the area with high electricity price and abundant sunshine, i.e., California, USA. Due to the high price of battery, some of the residents only stall PV arrays without BESS. They directly sell the redundant PV power to the utility grid, since PV cannot storage energy by itself. We analyse the PV power supply data and home load data from three different family homes with PV arrays. The collected data of Home 1 correspond to date range from 2014-01-01 to 2014-12-31 in California, US [27]. The collected data of Home 2 and Home 3 correspond to date range from 2016-01-01 to 2016-12-31 accessed from the Pecan Street Inc. Dataport [28]. The hourly grid power (the home load power minus the PV power) of the three homes on each day, as well as their averages, are shown in Fig. 1. The hourly grid power of the three homes vary from 2.26 kW to 4.58 kW, 4.26 kW to 7.42 kW, and 3.37 kW to 7.07 kW, respectively. The peak power to the grid of the three homes always happen from 10:00e14:00, 8:00e17:00, and 10:00e18:00, respectively, owing to the PV power generation. The peak power from the grid always occurs from 18:00e1:00 for Home 1, from 17:00e24:00 for Home 2, from 6:00e9:00 and 17:00e23:00 for Home 3, due to the heavy home load demand. At present, there are two types of electric rate plans for residential houses from Pacific Gas and Electric (PG&E) Company in California, tiered rated plan and time-of-use rate plan. Referring to the non-tiered, time-of-use plans, the hourly time-varying electric price in California, as well as that in Austin, Texas, is shown in Fig. 2(a) [29e32]. The PG&E electric price is lowest (10 cents/kWh) from 23:00 to 7:00, and more expensive during Peak (43 cents/kWh, 14:00e21:00) and Partial-Peak (22 cents/kWh, 7:00e14:00 and 21:00e23:00) periods. The Austin Energy electric price is lowest (2.18 cents/kWh) from 22:00 to 6:00, and more expensive on Peak (12.2 cents/kWh, 14:00e20:00) and Mild-Peak (7.13 cents/kWh, 6:00e14:00 and 20:00e22:00) hours. Obviously, the PG&E electric price is more expensive than the Austin Energy price. Fig. 2-(b to d) plot the hourly electric energy costs for the three homes in one year on each day and its average [29e33]. It is evident that all the three houses sell electric energy to the grid during Partial-Peak period and buy it during Peak period. If having a BESS, users can store the redundant PV power and buy electric energy with low price for the use of Peak period. The BESS can not only reduce household electric cost, but also supply electric power to the house during lacking of electric power, because of blackout [26]. Next, we apply CP approach to design the main parameters of BESS and synthesize an energy management controller.
3. Configuration and models 3.1. Configuration We consider a single smart home with BESS and PV arrays, as illustrated in Fig. 3 [26,34]. The smart home electric energy system comprises house appliances, utility grid, a home BESS, PV arrays, and associated power electronics. The power flow among them is managed by a smart home energy management system (SHEMS). Because the original house has had the PV arrays, but without BESS, an important mission is to determine desired parameters of the home BESS.
3.2. System model 2. Historical data analysis In order to reduce the cost of electric energy and carbon
The power balance equation of the smart home with home BESS and PV power supply is
446
X. Wu et al. / Energy 140 (2017) 444e453
Fig. 1. Statistical hourly grid power data of three homes on each day (blue) and average (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Pgrid;k ¼ Pdem;k þ Pbatt;k Ppv;k ;
k ¼ 0; …; N 1;
(1)
where Pgrid,k, Pdem,k, Pbatt,k, and Ppv,k are the electric power from the grid, electric load demand of the house, electric power of battery, and power supply of PV arrays, respectively. Variable k is time index, and N is the final time step of one year. The controller also must maintain home battery energy and power within allowable bounds [35],
SOCmin Qeap Ek SOCmax Qeap ; Pmin Pbatt;k Pmax ;
k ¼ 0; …; N;
k ¼ 0; …; N 1;
(2) (3)
where Ek is the energy of home battery, and SOCmin and SOCmax are the home battery's minimal SOC (state of charge) and maximal SOC, respectively. The battery energy capacity is denoted by Qeap. Furthermore, Pmin and Pmax are the home battery minimal power and maximal power, respectively (Pmin ¼ Pmax ) [36]. The dynamics of the battery are governed by the following equation:
Ekþ1 ¼ Ek þ Dt Pbatt;k hPbatt;k ; E0 ¼ Einit ;
k ¼ 0; …; N 1;
(4) (5)
where Dt, h, and Einit are the time interval, lost efficiency of BESS, and initial home battery energy, respectively. The battery power is assumed to be positive, by convention. Considering battery's expensive price and limited lifetime, as well as the grid power
quality, Pgrid,k should be limited as below
Pgrid;k P max grid ;
(6)
where P max grid 0 is the maximal power that can be provided to the grid. The power from the smart home to the grid is assumed to be negative, by convention. However, selling power back to the grid can cause voltage increases in the distribution lines and reverse power flows. This can violate voltage constraints - a topic not addressed in this paper. Herein, P max grid is limited to be two cases: no power supply to the grid and constrained power supply to the grid. 4. Optimization problem formulation A standard CP problem is formulated as
minimize s: t:
FðxÞ fi ðxÞ 0; i ¼ 1; …; p; hj ðxÞ ¼ 0; j ¼ 1; …; q; x2Z
(7)
where Z2Rn is a convex set, F(x) and fi(x) are convex functions, and hj(x) are affine functions of optimization vector x [37]. Comparing CP problem with other optimization problems, the convex optimization problem has three additional requirements: (i) objective function F(x) must be convex; (ii) the inequality constraint functions fi(x) must be convex for all i ¼ 1; …; p; (iii) the equality constraint functions hj(x) must be affine for all j ¼ 1; …; q. The key work of CP is to guarantee the convexity of models. Recognizing a
X. Wu et al. / Energy 140 (2017) 444e453
447
Fig. 2. Electric price [29e33] and hourly electricity cost in one year on each day (blue) and average (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Structure of smart home with PV arrays and BESS [26,34].
448
X. Wu et al. / Energy 140 (2017) 444e453
convex optimization problem can be difficult, however. The challenge and art in using convex optimization is in recognizing and formulating the problem. Once this formulation is done, solving the problem is essentially an off-the-shelf technology. This section presents the CP approach used for solving the optimal parameters design and power management problem for the smart home. The optimization variables are Pbatt;k ; Pgrid;k ; Ek ; Pmax ; Qeap , while the constraints are the home power balance (1), the battery constraints (2), (3), (4), (5), and the grid limits (6). The convex objective function F(x), which is of great interest to the home owner, is formulated to minimize a summation of the total electric energy cost in time horizon and BESS cost, for which we mainly consider the battery cost and charger cost [26]:
initialization can be readily accomplished. The theoretical and algorithmic aspects of CP are detailed in Ref. [37].
F ¼ Cny þ cb Qeap þ cc Pmax ;
Based on historical home load demand and PV power generation data, as well as hourly time-varying electric price, the optimal parameters of BESS and energy management strategy can be attained via CP. According to the report of Avicenne Energy, the worldwide battery price will vary from 60 $/kWh to 203 $/kWh in 2020 [39]. Considering different values of P max grid , time horizons,
(8)
where Cny is n-year total electricity cost, cb is the battery price per kiloWatt-hour, and cc is charger price per kiloWatt. For simplicity, we assume that the total electric energy cost is the same in every year. As a result, we can deduce Cny as follows
Cny ¼ n
N1 X
ce;k Pgrid;k ∕100;
(9)
k¼0
where ce,k is the electricity price, and n is the time horizon. It's easy to see that the objective function F is linear, which is convex. The inequality constraint functions include Eqns (2), (3) and (6), which are linear and convex. The equality constraint functions include Eqns (1), (4) and (5). Eqns (1) and (5) are linear and affine. Eqn (4) is absolute function, which is not affine. Because in the convex optimization problem, only affine equality constraints are tolerated. The total original problem is not a convex problem due to the absolute equality constraint, which is also nonlinear. But relaxing (4) to inequalities gives a convex problem without qualitatively altering the original problem as follows [26].
Ekþ1 Ek þ Dt Pbatt;k hPbatt;k ;
k ¼ 0; …; N 1:
(10)
Now, Eqn (10) is absolute inequality, which is convex. We can fomulate the problem as a convex problem. The CP algorithm for simultaneously optimal energy management and component sizing of the PV/battery home is summarized in Table 1. A tool, CVX [37], is employed to parse the optimization problem, inducing a semi-definite program that can be efficiently solved by SeDuMi (Self-Dual-Minimization), which is an add-on for MATLAB and can solve optimization problems with linear, quadratic and semidefiniteness constraints [38]. It should be emphasized that thanks to the convexity, a globally optimal solution with arbitrary
Table 1 Component sizing and energy management via CP. Component sizing and energy management via CP For k ¼ 1,...,N, with N corresponding to the final time step of one year Optimization variables: Pbatt;k ; Pgrid;k ; Ek ; Pmax ; Qeap Expressions: F ¼ Cny þ cb Qeap þ cc Pmax Objective function: F Constraints: SOCmin Qeap Ek SOCmax Qeap ; k ¼ 0; …; N; Pmax Pbatt;k Pmax ; k ¼ 0; …; N 1; Ekþ1 Ek þ DtðPbatt;k hPbatt;k Þ; k ¼ 0; …; N 1; E0 ¼ Einit, Pgrid;k ¼ Pdem;k þ Pbatt;k Ppv;k ; P Cny ¼ n N1 k¼0 ce;k Pgrid;k ∕100, Pgrid;k P max grid :
k ¼ 0; …; N 1;
5. Results & discussion This section analyses the properties of the proposed CP approach for optimizing battery parameters and control strategy. The key parameters are listed in Table 2. All the simulations were run on a PC with a 2.50 GHz Intel Core i5-2450 M CPU and 4 GB of internal memory. 5.1. System parameters optimization
battery and charger prices, the parameters of BESS and the cost can be explored. First, we assume that the time horizon is 5 years. Independently of battery price (60 $/kWh to 203 $/kWh) and charger price (500 $/kW to 1500 $/kW) [40,41], the optimal values of maximum power Pmax for three different homes are shown in Table 3. The maximum power decreases as the P max grid increases from 0 kW to 3 kW (i.e., constrained power supply to the grid). Independently of charger price (500 $/kW to 1500 $/kW), the optimal values of battery energy capacity Qeap for the three different homes are shown in Table 4. Independently of the maximal power provided to the grid P max grid (0 kWe3 kW) and battery price (60 $/kWh to 203 $/kWh), the optimal value of battery energy capacity Qeap of Home 1 maintains constant, equals to 15 kWh. Those of Home 2 and Home 3 decrease until 3.33 kWh, as the battery price cc increases from 60 $/kWh to 203 $/kWh. It can be seen that the values of charger price cb do not affect the optimal battery energy capacity and the maximum power. If Pgrid is constrained as P max grid ¼ 0 kW, independently of the time horizon (1 yeare10 years), battery price (60 $/kWh to 203 $/kWh),
Table 2 Key parameters. Parameter Description
Symbol
Value
Unit
Step time Maximum battery SOC
Dt SOCmax
1 0.90
e
Minimum battery SOC
SOCmin
0.20
h
e
Lost efficiency
0.10
hour
and charger price (500 $/kW to 1500 $/kW), the maximum power Pmax maintains constant, equal to 2.26 kW, 4.26 kW, and 3.37 kW, for these three homes. The values of Pmax equal to the maximum power to the grid, in the case of no BESS. The optimal values of battery energy capacity Qeap are shown in Fig. 4. The battery energy capacity of Home 1 is augmented until 15 kWh, as the time horizon increases. Likewise, if the owner of Home 1 has a Nissan Leaf, the battery energy capacity is augmented, as the time horizon increases, and the maximum power maintains constant, equal to 2.26 kW [26]. Given the battery and charger prices of 100 $/kWh and 1000 $/kW, as well as different P max grid (0 kW or 3 kW) and time horizons, the optimal values of battery energy capacity Qeap, maximum
X. Wu et al. / Energy 140 (2017) 444e453 Table 3 Optimal results of maximum power [kW] based on different values of P max grid [kW]. P max grid
Home 1
Home 2
Home 3
0 1 2 3
2.26 1.46 1.46 1.46
4.26 3.26 2.26 1.26
3.37 2.37 1.37 0.37
Table 4 Optimal results of battery energy capacity [kWh] based on different values of P max grid [kW] and battery price [$/kWh]. Home
P max grid
cb ¼ 60
cb ¼ 100
cb ¼ 150
cb ¼ 203
1 2 2 2 2 3 3 3 3
0e3 0 1 2 3 0 1 2 3
15 15 15 15 11.89 15 15 12.96 3.53
15 10.25 13.69 13.48 9.91 8.13 10.78 8.66 3.53
15 3.33 5.19 3.33 3.33 3.33 3.33 3.33 3.33
15 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33
power Pmax, and cost are shown in Table 5, where FBESS, Fe, FnoB, and Fdiff are the cost of home BESS, the electric cost with BESS in one year, the electric cost without BESS in one year, and the electric cost difference with and without BESS in one year, respectively. If P max grid ¼ 0 kW, the battery energy capacities of the three homes increase until 15 kWh, as the time horizon increases until 4 years, 8 years, and 8 years, respectively. After that, the battery size remains
449
constant. The maximum power Pmax maintains constant, equal to 2.26 kW, 4.26 kW, and 3.37 kW, for the three households. If P max grid ¼ 3 kW, the battery energy capacities of the three homes increase until 15 kWh, as the time horizon increases until 3 years, 10 years, and 10 years, respectively. The maximum power Pmax increases until 1.46 kW, as the time horizon increases until 3 years for Home 1. After that, it remains constant. When time horizon increases from 1 year to 9 years, the maximum power Pmax for Home 2 and Home 3 is constant, equal to 1.26 kW and 0.37 kW, respectively. The electric costs with BESS in one year Fe for Home 1 and Home 3 are always less than that of the scenario without BESS, except for the time horizon of 1 year. The total cost savings become more significant with an increased time horizon. The cost savings in Home 1 are more significant than those in Home 2 and Home 3, due to higher PG&E electric price. The electric cost savings are larger, when P max grid ¼ 3 kW. However, prividing power to the grid will significantly increase the battery cost induced by accelerated degradation, as argued in Refs. [15,42]. In the following, we thus only consider the case of no power supply to the grid of Home 1. Without power supply to the grid, it is reasonable to assume a home battery life larger than 5 years [43]. The optimal battery energy capacity and the maximum power for Home 1 we consider are 15 kWh and 2.26 kW, and the cost of home BESS is 3760 $. With BESS, the electric energy cost in one year is 1043.1 $, whereas without the BESS, the electric energy cost in one year is 1881.5 $. The cost reduction reaches up to around 44.6%. If Home 1 has a Nissan Leaf that can not discharge power to the home, with the optimal values of battery energy capacity (17 kWh) and maximum power (2.26 kW), the electric cost with BESS over one year is 39.2% less than that without BESS [26]. With/without EV, home BESS can
Fig. 4. Results of battery energy capacity for different time horizons and BESS prices based on P max grid ¼ 0 kW.
450
X. Wu et al. / Energy 140 (2017) 444e453
Table 5 Optimal results based on cb ¼ 100 $/kWh and cc ¼ 1000 $/kW. Home
P max grid [kW]
n [year]
Qeap [kWh]
Pmax [kW]
FBESS [$]
Fe [$]
FnoB [$]
Fdiff [$]
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3
0 0 0 0 3 3 3 0 0 0 0 0 0 3 3 3 3 3 0 0 0 0 0 0 3 3 3
1 2 3 4e10 1 2 3e10 1e3 4 5 6 7 8e10 1e4 5 6 7e9 10 1e3 4 5 6 7 8e10 1e4 5e9 10
3.33 11.88 14.46 15 3.33 3.33 15 3.33 5.95 10.25 13.19 14.84 15 3.33 9.91 11.27 11.89 15 3.33 4.96 8.13 11.37 13.82 15 3.33 3.53 15
2.26 2.26 2.26 2.26 0.001 0.30 1.46 4.26 4.26 4.26 4.26 4.26 4.26 1.26 1.26 1.26 1.26 1.48 3.37 3.37 3.37 3.37 3.37 3.37 0.37 0.37 1.49
2593 3448 3706 3760 334 633 2960 4593 4855 5285 5579 5744 5760 1593 2251 2387 2449 2980 3703 3866 4183 4507 4752 4870 703 723 2990
1791.7 1165.8 1060.4 1043.1 1880.5 1650.3 809.3 483.2 413.7 317.7 264.2 238.7 236.4 143.7 1.3 26.4 36.4 89 694.9 651.4 580.9 521.9 483.6 467.3 581.5 577.3 343.2
1881.5 1881.5 1881.5 1881.5 1881.5 1881.5 1881.5 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 654.8 654.8 654.8 654.8 654.8 654.8 654.8 654.8 654.8
89.8 715.7 821.1 838.4 1 231.2 1072.2 268.6 199.1 103.1 49.6 24.1 21.8 70.9 215.9 241 251.0 303.6 40.1 3.4 73.9 132.9 171.2 187.5 73.3 77.5 311.6
obviously increase the home economy.
5.2. Example of energy management strategy Based on the Home 1's optimal results of battery energy capacity Qeap ¼ 15 kWh, the maximum power Pmax ¼ 2.26 kW, and the constraint of P max grid ¼ 0, this subsection presents the resulting CP control law simulated on two different operating samples, including summer day and winter day. The hourly power allocation and battery SOC over two summer days are described in Fig. 5-(a). It is evident that the majority of the home battery charging occurs during the low electricity price period: 24:00e7:00 and high PV power supply period: 10:00e15:00. Most of the battery discharging happens during the high electricity price period: 16:00e23:00. The electric power from the grid is zero during the period: 8:00e21:00. Similarly, the simulation results for two winter days are indicated in Fig. 5-(b). In winter, because of less PV power generation, the home battery charges during the low electricity price period and
discharges during the high electricity price period. The electric power from the grid is zero during the period: 16:00e21:00. In summary, the home does not buy electric energy from the grid during the peak periods of electric price. The outcomes of Home 2 and Home 3 are analogous. To demonstrate the potential economic benefits of equipping PV/battery with a home, we examine the electric energy cost curve of Home 1 in a comparative fashion. The hourly electric energy cost for two days in summer and winter is shown in Fig. 6-(a) and Fig. 6(b), respectively. With BESS, the two-day electric energy cost is 3.72 $ in summer and 7.16 $ in winter. Without BESS, the two-day electric energy cost is 7.87 $ in summer and 12.66 $ in winter. Therefore, the addition of BESS can save 52.7% and 43.4% electric cost in summer and winter, respectively. It is less cost-effective in winter, owing to larger household power demand and less PV power generation. It turns out that the BESS is needed both in summer and winter. With BESS, the electric power values from the grid of Home 1 in one year on each day (blue) and average (red) are shown in Fig. 7-
Fig. 5. Hourly power allocation and battery SOC of Home 1.
X. Wu et al. / Energy 140 (2017) 444e453
451
Fig. 6. Electric energy cost of Home 1.
(a). Compared with Fig. 1-(a), the peak power from the grid always happens from 24:00e7:00 when the electric price is low, and the PV/battery smart home does not supply electric energy to the grid. The electric energy cost values from the grid in one year on each day (blue) and average (red) are given in Fig. 7-(b). In contrast with Fig. 2-(b), at the presence of BESS, the electric energy cost from the grid is obviously less.
5.3. Rapid computation To demonstrate the advantages and efficacy of CP, we compare CP and DP approaches, in terms of precision and computational efficiency. Here, we first introduce the DP algorithms. DP is usually used as a benchmark, due to Bellman Principle of Optimality. Consider a multi-stage decision process, we define the instantaneous electricity cost functional gk(Ek,Pbatt,k) as follows:
gk Ek ; Pbatt;k ¼ ce;k ,Dt,Pgrid;k ¼ ce;k ,Dt, Pdem;k þ Pbatt;k Ppv;k
(11)
The controller must maintain battery Ek and power Pbatt,k within simple bounds as Eqns (2) and (3). The design problem is to search the control input Pbatt,k that minimizes the electricity cost. The controller will be synthesized as a time-varying state feedback control law. Namely, the control is the output of a mapping that depends on the current deterministic state Ek. We formalize this as a finite-time dynamic program,
minPbatt;k ;Ek
N1 X
ce;k ,Dt, Pdem;k þ Pbatt;k Ppv;k
(12)
k¼0
s: t:
Eqns ð2Þ ð6Þ:
(13)
Now we define the value function. Let Vk(Ek) be the minimum expected cost-to-go from time step k to N, given the current battery energy level-Ek. Then the principle of optimality is given by:
Vk ðEk Þ ¼ minPbatt;k 2D k gk Ek ; Pbatt;k þ Vkþ1 ðEkþ1 Þ ;
(14)
where gk ð,; ,Þ is the instantaneous cost in (11). The minimization operator is subject to a time-varying admissible control set D k characterized by (2)e(6). We also have the boundary condition
VN ðEN Þ ¼
EN Emax 0; for Emin N N ∞; otherwise:
(15)
Finally, the optimal control action is saved as
P *batt;k
¼ gk ðEk Þ ¼ argminPbatt;k 2D k gðEk ; Ik Þ þ Vkþ1 ðEkþ1 Þ:
(16)
In our work, the comparison between CP and DP aims not to show that CP works better than DP (we do not state such a claim
Fig. 7. Electric power and electric energy cost for Home 1 with BESS in one year on each day (blue) and average (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
452
X. Wu et al. / Energy 140 (2017) 444e453
throughout the paper). We only take DP as a benchmark (as usually done in optimal control problems) to manifest that the performance of our CP can be close to that of DP. In theory, DP can ensure a global optimal result according to Bellmans Principle of Optimality, and it has been used in previous residential energy management studies [16,17]. However, its numerical implementation (e.g., grid size) may lead to slightly inferior results, compared to CP. The difference between DP and CP is always marginal, demonstrating the effectiveness of CP. Moreover, we can clearly see the advantage of CP in computational efficiency. First, we assume the BESS of Home 1 has 15 kWh battery and 2.26 kW charger (optimal parameters) in the two methods. Over one-year data, the computational time of CP is 3.96 s, whereas that of DP is 2475 s. The CP is 625 times faster than DP. Battery charge power and the electric power from the grid in CP and DP are compared in Fig. 8. Both methods can make the battery charge with low electric price and supply energy to the home with high electric price. Now, we consider if the smart home with PV and battery energy storage can be optimally managed in the real world on a daily basis. Here we assume that the PV power generation and the home load demand in the future 24 h can be forecasted well. Both CP and DP are applied to optimize control policy. Given different time intervals Dt, the computational time and electric energy cost are summarized in Table 6, where tCP, tDP, FCP, FDP, and Lo denote the computational time to execute CP, computational time to execute DP, electric energy cost using CP, electric energy cost using DP, and loss of optimality comparing CP with DP, respectively. We define loss of optimality Lo as follows:
Lo ¼ ðFCP FDP Þ∕FDP :
(17)
The 24-h electric energy cost is virtually the same for CP and DP in cases of different time intervals (the maximal loss of optimality is about 6.9%). Despite that calculation steps increase from 48 to 1440, the computational time of CP is always less than 1 s, whereas that of DP drastically increases. The CP method is more than 30 times faster than DP. Hence, it is more suitable for fast system economic analysis.
Table 6 Comparison of CP and DP for Home 1 on a 24-h basis. N
Dt [min]
tCP [s]
tDP [s]
FCP [$]
FDP [$]
Lo [%]
48 96 288 1440
30 15 5 1
0.50 0.56 0.60 0.81
15.68 30.49 93.16 452.64
1.86 1.95 1.94 1.76
1.74 1.94 1.99 1.83
6.9 0.5 2.5 3.8
Table 7 Comparison of CP and DP for Home 1 based on sub-optimal parameters over one year. Qeap [kWh]
Pmax [kW]
FCP [$]
FDP [$]
Lo [%]
15 15 15 14 13 12
3 2.5 2.3 2.3 2.3 2.3
1040 1041 1043 1076 1143 1160
1097 1098 1098 1130 1166 1210
5.2 5.2 5.0 4.8 2.0 4.1
parameters of BESS, the control results of CP and DP over one year are shown in Table 7. When Qeap is 15 kWh, and Pmax changes from 3 kW to 2.3 kW, the results of CP are almost the same, similar to DP. When Pmax is 2.3 kW, and Qeap alters from 15 kWh to 12 kWh, the total costs of CP increase, similar to DP. Based on different suboptimal parameters of BESS for Home 1, the total cost of CP over one year is close to that of DP (the maximal loss of optimality is about 5.2%). 6. Conclusion The CP problem is mathematically formulated and demonstrated to simultaneously optimize the electric power allocation and BESS parameters for three different homes, located in California and Texas. The optimization results are systematically compared, taking various time horizons, maximal values of power to grid, BESS prices, and home electricy use/demand patterns into accout. Key findings are summarized below:
5.4. Sensitivity to parameters of BESS In this subsection, we account for a simple but nontrivially useful sensitivity analysis to the parameters of BESS, taking Home 1 as an example, where sub-optimal parameters of BESS are contrasted to the optimal parameters of CP. Given sub-optimal
(1) If there is no power supply to the grid, independently of the BESS price, the optimal values of maximum battery power for these three homes are 2.26 kW, 4.26 kW, and 3.37 kW, respectively. The values of charger price cb do not affect the optimal battery energy capacity and the maximum power for the three homes. Compared to the electric cost of a home without BESS, the usefulness of BESS to improve the home economy is pronouced. The total cost savings become more significant with increased time horizon and P max grid . The cost savings in Home 1 are more significant than those in Home 2 and Home 3, due to higher PG&E electric price. (2) The CP framework is substantiated to be much faster than DP, with close accuracy and optimality, which is favorable to real-time control and efficient design of smart home in practice. The maximal loss of optimality in the 24-h electric energy cost, relative to DP, is approximately 6.9%. The CP method is distinctly more than 30 times faster than DP. The future work could incorporate explicit thermal and ageing dynamics of the home battery into the optimization framework, in order to quantify the battery health implication. Acknowledgements
Fig. 8. Battery Charge power and grid power with CP and DP for Home 1.
This work was supported in part by the Sichuan Provincial
X. Wu et al. / Energy 140 (2017) 444e453
Scientific Research Innovation Team Program(Grant No. 17TD0035), Science and Technology Department of Sichuan Province (Grant Nos. 2017TD0026, 2015TD0021, 2016HH0010), NSF of China (Grant No. 51375402), Xihua University Fund (Grant No. z1220315), and the Fundamental Research Funds for the Central Universities (Project No. 0903005203398). References [1] Lund PD, Lindgren J, Mikkola J, Salpakari J. Review of energy system flexibility measures to enable high levels of variable renewable electricity. Renew Sustain Energy Rev 2015;45(0):785e807. [2] Entchev E, Yang L, Ghorab M, Lee E. Simulation of hybrid renewable microgeneration systems in load sharing applications. Energy 2013;50:252e61. [3] Aghajani G, Shayanfar H, Shayeghi H. Demand side management in a smart micro-grid in the presence of renewable generation and demand response. Energy 2017;126:622e37. [4] Batista N, Melcio R, Matias J, Catalao J. Photovoltaic and wind energy systems monitoring and building/home energy management using zigbee devices within a smart grid. Energy 2013;49:306e15. [5] Sechilariu M, Wang B, Locment F. Building integrated photovoltaic system with energy storage and smart grid communication. IEEE Trans Ind Electron 2013;60(4):1607e18. [6] Shirazi E, Jadid S. Cost reduction and peak shaving through domestic load shifting and ders. Energy 2017;124:146e59. [7] Amirioun MH, Kazemi A. A new model based on optimal scheduling of combined energy exchange modes for aggregation of electric vehicles in a residential complex. Energy 2014;69:186e98. [8] Prinsloo G, Mammoli A, Dobson R. Customer domain supply and load coordination: a case for smart villages and transactive control in rural off-grid microgrids. Energy 2017;135:430e41. [9] De Angelis F, Boaro M, Fuselli D, Squartini S, Piazza F, Wei Q. Optimal home energy management under dynamic electrical and thermal constraints. IEEE Trans Ind Inf 2013;9(3):1518e27. [10] Mohseni A, Mortazavi SS, Ghasemi A, Nahavandi A, Abdi MT. The application of household appliances' flexibility by set of sequential uninterruptible energy phases model in the day-ahead planning of a residential microgrid. Energy 2017;139:315e28. [11] Palma-Behnke R, Benavides C, Lanas F, Severino B, Reyes L, Llanos J, et al. A microgrid energy management system based on the rolling horizon strategy. IEEE Trans Smart Grid 2013;4(2):996e1006. [12] Pedrasa M, Spooner T, MacGill I. Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. IEEE Trans Smart Grid 2010;1(2):134e43. [13] Wang G, Zhang Q, Li H, McLellan BC, Chen S, Li Y, et al. Study on the promotion impact of demand response on distributed pv penetration by using noncooperative game theoretical analysis. Appl Energy 2017;185(Part 2): 1869e78. [14] Elma O, Tasc1karaoglu A, Ince AT, Selamogullar US. Implementation of a dynamic energy management system using real time pricing and local renewable energy generation forecasts. Energy 2017;134:206e20. [15] Sun C, Sun F, Moura SJ. Nonlinear predictive energy management of residential buildings with photovoltaics & batteries. J Power Sources 2016;325: 723e31. [16] Muratori M, Rizzoni G. Residential demand response: dynamic energy management and time-varying electricity pricing. IEEE Trans Power Syst 2016;31(2):1108e17. [17] Huang T, Liu D. Residential energy system control and management using adaptive dynamic programming. In: The 2011 international joint conference on neural networks; 2011. p. 119e24. [18] Munkhammar J, Widn J, Rydn J. On a probability distribution model
[19] [20] [21]
[22] [23]
[24] [25]
[26]
[27] [28] [29]
[30] [31] [32] [33]
[34]
[35]
[36]
[37] [38] [39] [40]
[41] [42]
[43]
453
combining household power consumption, electric vehicle home-charging and photovoltaic power production. Appl Energy 2015;142:135e43. Han D, Chesi G, Hung YS. Robust consensus for a class of uncertain multiagent dynamical systems. IEEE Trans Ind Inf 2013;9(1):306e12. Fattahi S, Ashraphijuo M, Lavaei J, Atamtrk A. Conic relaxations of the unit commitment problem. Energy 2017;134:1079e95. Shen J, Cheng C, Wu X, Cheng X, Li W, Lu J. Optimization of peak loads among multiple provincial power grids under a central dispatching authority. Energy 2014;74:494e505. Tsui K, Chan S. Demand response optimization for smart home scheduling under real-time pricing. IEEE Trans Smart Grid 2012;3(4):1812e21. Wang Z, Paranjape R. Optimal residential demand response for multiple heterogeneous homes with real-time price prediction in a multiagent framework. IEEE Trans Smart Grid 2017;8(3):1173e84. Hu X, Zou Y, Yang Y. Greener plug-in hybrid electric vehicles incorporating renewable energy and rapid system optimization. Energy 2016;111:971e80. Hu X, Murgovski N, Johannesson L, Egardt B. Optimal dimensioning and power management of a fuel cell/battery hybrid bus via convex programming. IEEE/ASME Trans Mechatronics 2015;20(1):457e68. Wu X, Hu X, Teng Y, Qian S, Cheng R. Optimal integration of a hybrid solarbattery power source into smart home nanogrid with plug-in electric vehicle. J Power Sources 2017;363:277e83. SolarCity. Solarcity. 2014. Tech. rep., SolarCity, http://www.solarcity.com/. P. S. Inc. Pecan street dataport. 2017. Tech. rep. https://dataport.pecanstreet. org. Gas P, Company E. Electric vehicles making sense of the rates. Pacific Gas and Electric Company; 2015. Tech. rep, http://www.pge.com/en/myhome/ saveenergymoney/pev/rates/index.page. Wu X, Hu X, Yin X, Moura S. Stochastic optimal energy management of smart home with pev energy storage. IEEE Trans Smart Grid PP 2017;99:1. 1. Zachar M, Daoutidis P. Microgrid/macrogrid energy exchange: a novel market structure and stochastic scheduling. IEEE Trans Smart Grid 2017;8(1):178e89. Fares RL, Webber ME. The impacts of storing solar energy in the home to reduce reliance on the utility. Nat Energy 2017;2(17001):1e10. Austin energy city of austin electric tariff. 2016. Tech. rep. http://austinenergy. com/wps/wcm/connect/ab6d045c-643e-4c16-921f-c76fa0fee2bf/ FY2016aeElectricRateSchedule.pdf?MOD¼AJPERES. Wu X, Hu X, Moura S, Yin X, Pickert V. Stochastic control of smart home energy management with plug-in electric vehicle battery energy storage and photovoltaic array. J Power Sources 2016;333:203e12. Li J, Wu Z, Zhou S, Fu H, Zhang XP. Aggregator service for pv and battery energy storage systems of residential building. CSEE J Power Energy Syst 2015;1(4):3e11. Wu D, Zeng H, Lu C, Bouletu B. Two-stage energy management for office buildings with workplace ev charging and renewable energy. IEEE Trans Transp Electrification 2017;3(1):225e37. Boyd S, Vandenberghe L. Convex optimization. Cambrige, U.K: Cabridge Univ. Press; 2004. Sturm JF. Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optim Methods Softw 1999;11(1e4):625e53. Pillot C. The worldwide battery market 2012-2025. 2013. Tech rep., Avicenne Energy. Yilmaz M, Krein PT. Review of battery charger topologies, charging power levels, and infrastructure for plug-in electric and hybrid vehicles. IEEE Trans Power Electron 2013;28(5):2151e69. Morrowa DKK, Francfort J. Plug-in hybrid electric vehicle charging infrastructure review. U.S. Department of Energy; Nov. 2008. Tech. rep. Yilmaz M, Krein PT. Review of the impact of vehicle-to-grid technologies on distribution systems and utility interfaces. IEEE Trans Power Electron 2013;28(12):5673e89. Calculate the cost of photovoltaic systems home solar electricity. New Mexico Solar Energy Association; 2015. Tech. rep.
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Optimal battery sizing of smart home via convex programming Xiaohua Wu a, 1, Xiaosong Hu b, *, 1, Xiaofeng Yin a, Caiping Zhang c, Shide Qian d a
School of Automobile and Transportation, Xihua University, Chengdu, 610039, China The State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400044, China c National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing, 100044, China d Beijing Benz Automotive Co.Ltd, Beijing, 100176, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2017 Received in revised form 17 August 2017 Accepted 22 August 2017 Available online 26 August 2017
This paper develops a convex programming (CP) framework for optimal sizing and energy management of smart home with battery energy storage system (BESS) and photovoltaic (PV) power generation, for the goal of maximizing home economy, while satisfying home power demand. We analyse the historical electric energy data of three different homes located in California and Texas, and indicate the necessity and importance of a BESS. Based on the structures and system models of these smart homes, the CP problem is formulated to rapidly and efficiently solve the optimal design/control issue. Based on different time horizons, maximal powers to grid, prices of BESS, the optimal parameters of BESS and its potential to electric energy cost savings are systematically compared for the three homes. A deviation analysis between the results obtained by CP and DP (dynamic programming) is also presented. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Smart homes Energy management Optimal sizing PV arrays Battery energy storage
1. Introduction The present energy crisis (increasingly severe energy shortage and supply instability), and environmental crisis (global warming and air pollution) have promoted the rapid development of integrating renewable energy with the grid [1,2]. However, some renewable energy, such as solar and wind energy, is intermittent and unstable in nature due to metrological conditions [3,4]. Consequently, researchers have focused on developing effective control strategies to improve the performance and economy for buildings and homes integrating renewable energy [5e8]. This paper develops an optimal design/control method for battery sizing and energy management of smart home. The existing literature, e.g., the forgoing work, has presented several optimization methods, such as mixed-integer linear programming (MILP) [9,10], rolling horizon strategy [11], particle swarm optimization (PSO) approach [12], geometric program [13], model predictive control (MPC) [14,15], dynamic programming (DP) [16], adaptive dynamic programming (ADP) [17], and stochastic dynamic programming (SDP) [18], for creating efficient operational schedules or making good consumption and
* Corresponding author. E-mail address: [email protected] (X. Hu). 1 X. Wu and X. Hu equally contributed to this research work. http://dx.doi.org/10.1016/j.energy.2017.08.097 0360-5442/© 2017 Elsevier Ltd. All rights reserved.
production decisions to home energy management (HEM). An optimal HEM strategy under dynamic electrical and thermal constraints is developed through solving an MILP in Ref. [9], which is able to provide an optimal solution to power consumption and management of renewable resources. Similarly, considering photovoltaic (PV) arrays, battery energy storage, and electric vehicle (EV) in Ref. [10], the effects of the accurate SSUEP (Set of Sequential Uninterruptible Energy Phases)-based model on the day-ahead energy management of a residential microgrid are formulated as an MILP optimization framework. A novel energy management system based on a rolling horizon strategy for a microgrid is proposed with PV panels, two wind turbines, a diesel generator, and an energy storage system in Ref. [11]. An improved PSO approach is introduced to optimize distributed energy resources operation schedules for a smart home case study in Ref. [12]. The impacts of the response capability levels of consumers on the economic integration of distributed PV power in smart homes, and the impacts of PV and battery capacities on consumers power expenses are analyzed using non-cooperation game theoretical power market complementarity model in Ref. [13]. Based on short-term forecasts of residential renewable power generation, a dynamic HEM algorithm is put forward in Ref. [14] for decreasing the total grid energy cost while maximizing user comfort. Both simulation and experimental results show the ability of the devised algorithm to control both sources and loads. A nonlinear predictive
X. Wu et al. / Energy 140 (2017) 444e453
energy management method for buildings with PV system and battery storage is proposed in Ref. [15], which forecasted house load demand via artificial neural networks. Based on highly resolved energy consumption models, an automated dynamic energy management framework is established in Ref. [16] to find the optimal schedule of residential controllable appliances, where DP is utilized to find the global solution. A computationally feasible and self-learning optimal control scheme for a residential energy system with batteries is devised in Ref. [17], the idea of which is to approximate DP solution by using neural networks. A probability distribution model combining household power consumption, EV home-charging, and PV power production is built using a convolution approach to merge three separate existing probability distribution models in Ref. [18]. In Ref. [4], ZigBee technologies are exploited for comprehensive field tests, including monitoring PV and wind energy systems, and energy management of buildings/ homes. The literature provides a number of approaches for energy management of smart home with renewable energy, almost all of which share a common goal, namely, to meet overall home electric power demand while minimizing the total operational energy cost. Few studies explored the optimal battery size and control strategy simultaneously. Furthermore, a single home was often considered without a comparision of multiple homes with different electricity supply/demand patterns. This paper constructs an optimal design/control framework for exploring multiple smart homes with battery energy storage system (BESS) and PV arrays power generation. The key challenge addressed by this paper is to simultaneously optimize the battery size and energy management strategy with the consideration of the cost of BESS, computational efficiency, and home electricity difference. An emerging effective tool, convex programming (CP), which can rapidly and efficiently optimize both management strategy and parameters, as opposed to many other methods generally focusing on only controls, has been applied by researchers [19e21]. Demand response (DR) implementation considering a single home with different appliances is modeled using CP to minimize electricity payment in Ref. [22]. Similarly, A later study in Ref. [23] formulates a CP problem to minimize electricity payment and waiting time under real-time pricing for a multi-agent system, in order to evaluate optimal residential DR implementation in a distribution network. The previous two studies, however, do not involve optimal component design. Nevertheless, CP has been successfully applied to simultaneously optimize component size and energy controller for vehicles [24,25]. In this article, CP is leveraged to rapidly and efficiently optimize both HEM strategy and BESS size. In our previous study [26], a CP framework was built for the optimal integration of a hybrid solarbattery power source into smart home nanogrid with PEV load. The main differences between this endeavor and [26] (i.e., major contributions) include (i) the optimization results of three different homes with PV arrays located in California and Texas are systematically compared, instead of analyzing a single home in most of existing studies, and (ii) a deviation analysis between the outcomes procured by CP and DP is carried out to showcase the computational efficiency and effectiveness of CP. The remainder of the paper is arranged as follows. Section 2 analyses the historical electric energy data of the three homes. The structures and models are described in Section 3. The CP problem is formulated in Section 4. The optimization results are discussed and contrasted in Section 5, followed by conclusions summarized in Section 6.
445
emissions, residents are seriously concerned about installing PV panels on the roof of their houses, especially in the area with high electricity price and abundant sunshine, i.e., California, USA. Due to the high price of battery, some of the residents only stall PV arrays without BESS. They directly sell the redundant PV power to the utility grid, since PV cannot storage energy by itself. We analyse the PV power supply data and home load data from three different family homes with PV arrays. The collected data of Home 1 correspond to date range from 2014-01-01 to 2014-12-31 in California, US [27]. The collected data of Home 2 and Home 3 correspond to date range from 2016-01-01 to 2016-12-31 accessed from the Pecan Street Inc. Dataport [28]. The hourly grid power (the home load power minus the PV power) of the three homes on each day, as well as their averages, are shown in Fig. 1. The hourly grid power of the three homes vary from 2.26 kW to 4.58 kW, 4.26 kW to 7.42 kW, and 3.37 kW to 7.07 kW, respectively. The peak power to the grid of the three homes always happen from 10:00e14:00, 8:00e17:00, and 10:00e18:00, respectively, owing to the PV power generation. The peak power from the grid always occurs from 18:00e1:00 for Home 1, from 17:00e24:00 for Home 2, from 6:00e9:00 and 17:00e23:00 for Home 3, due to the heavy home load demand. At present, there are two types of electric rate plans for residential houses from Pacific Gas and Electric (PG&E) Company in California, tiered rated plan and time-of-use rate plan. Referring to the non-tiered, time-of-use plans, the hourly time-varying electric price in California, as well as that in Austin, Texas, is shown in Fig. 2(a) [29e32]. The PG&E electric price is lowest (10 cents/kWh) from 23:00 to 7:00, and more expensive during Peak (43 cents/kWh, 14:00e21:00) and Partial-Peak (22 cents/kWh, 7:00e14:00 and 21:00e23:00) periods. The Austin Energy electric price is lowest (2.18 cents/kWh) from 22:00 to 6:00, and more expensive on Peak (12.2 cents/kWh, 14:00e20:00) and Mild-Peak (7.13 cents/kWh, 6:00e14:00 and 20:00e22:00) hours. Obviously, the PG&E electric price is more expensive than the Austin Energy price. Fig. 2-(b to d) plot the hourly electric energy costs for the three homes in one year on each day and its average [29e33]. It is evident that all the three houses sell electric energy to the grid during Partial-Peak period and buy it during Peak period. If having a BESS, users can store the redundant PV power and buy electric energy with low price for the use of Peak period. The BESS can not only reduce household electric cost, but also supply electric power to the house during lacking of electric power, because of blackout [26]. Next, we apply CP approach to design the main parameters of BESS and synthesize an energy management controller.
3. Configuration and models 3.1. Configuration We consider a single smart home with BESS and PV arrays, as illustrated in Fig. 3 [26,34]. The smart home electric energy system comprises house appliances, utility grid, a home BESS, PV arrays, and associated power electronics. The power flow among them is managed by a smart home energy management system (SHEMS). Because the original house has had the PV arrays, but without BESS, an important mission is to determine desired parameters of the home BESS.
3.2. System model 2. Historical data analysis In order to reduce the cost of electric energy and carbon
The power balance equation of the smart home with home BESS and PV power supply is
446
X. Wu et al. / Energy 140 (2017) 444e453
Fig. 1. Statistical hourly grid power data of three homes on each day (blue) and average (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Pgrid;k ¼ Pdem;k þ Pbatt;k Ppv;k ;
k ¼ 0; …; N 1;
(1)
where Pgrid,k, Pdem,k, Pbatt,k, and Ppv,k are the electric power from the grid, electric load demand of the house, electric power of battery, and power supply of PV arrays, respectively. Variable k is time index, and N is the final time step of one year. The controller also must maintain home battery energy and power within allowable bounds [35],
SOCmin Qeap Ek SOCmax Qeap ; Pmin Pbatt;k Pmax ;
k ¼ 0; …; N;
k ¼ 0; …; N 1;
(2) (3)
where Ek is the energy of home battery, and SOCmin and SOCmax are the home battery's minimal SOC (state of charge) and maximal SOC, respectively. The battery energy capacity is denoted by Qeap. Furthermore, Pmin and Pmax are the home battery minimal power and maximal power, respectively (Pmin ¼ Pmax ) [36]. The dynamics of the battery are governed by the following equation:
Ekþ1 ¼ Ek þ Dt Pbatt;k hPbatt;k ; E0 ¼ Einit ;
k ¼ 0; …; N 1;
(4) (5)
where Dt, h, and Einit are the time interval, lost efficiency of BESS, and initial home battery energy, respectively. The battery power is assumed to be positive, by convention. Considering battery's expensive price and limited lifetime, as well as the grid power
quality, Pgrid,k should be limited as below
Pgrid;k P max grid ;
(6)
where P max grid 0 is the maximal power that can be provided to the grid. The power from the smart home to the grid is assumed to be negative, by convention. However, selling power back to the grid can cause voltage increases in the distribution lines and reverse power flows. This can violate voltage constraints - a topic not addressed in this paper. Herein, P max grid is limited to be two cases: no power supply to the grid and constrained power supply to the grid. 4. Optimization problem formulation A standard CP problem is formulated as
minimize s: t:
FðxÞ fi ðxÞ 0; i ¼ 1; …; p; hj ðxÞ ¼ 0; j ¼ 1; …; q; x2Z
(7)
where Z2Rn is a convex set, F(x) and fi(x) are convex functions, and hj(x) are affine functions of optimization vector x [37]. Comparing CP problem with other optimization problems, the convex optimization problem has three additional requirements: (i) objective function F(x) must be convex; (ii) the inequality constraint functions fi(x) must be convex for all i ¼ 1; …; p; (iii) the equality constraint functions hj(x) must be affine for all j ¼ 1; …; q. The key work of CP is to guarantee the convexity of models. Recognizing a
X. Wu et al. / Energy 140 (2017) 444e453
447
Fig. 2. Electric price [29e33] and hourly electricity cost in one year on each day (blue) and average (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Structure of smart home with PV arrays and BESS [26,34].
448
X. Wu et al. / Energy 140 (2017) 444e453
convex optimization problem can be difficult, however. The challenge and art in using convex optimization is in recognizing and formulating the problem. Once this formulation is done, solving the problem is essentially an off-the-shelf technology. This section presents the CP approach used for solving the optimal parameters design and power management problem for the smart home. The optimization variables are Pbatt;k ; Pgrid;k ; Ek ; Pmax ; Qeap , while the constraints are the home power balance (1), the battery constraints (2), (3), (4), (5), and the grid limits (6). The convex objective function F(x), which is of great interest to the home owner, is formulated to minimize a summation of the total electric energy cost in time horizon and BESS cost, for which we mainly consider the battery cost and charger cost [26]:
initialization can be readily accomplished. The theoretical and algorithmic aspects of CP are detailed in Ref. [37].
F ¼ Cny þ cb Qeap þ cc Pmax ;
Based on historical home load demand and PV power generation data, as well as hourly time-varying electric price, the optimal parameters of BESS and energy management strategy can be attained via CP. According to the report of Avicenne Energy, the worldwide battery price will vary from 60 $/kWh to 203 $/kWh in 2020 [39]. Considering different values of P max grid , time horizons,
(8)
where Cny is n-year total electricity cost, cb is the battery price per kiloWatt-hour, and cc is charger price per kiloWatt. For simplicity, we assume that the total electric energy cost is the same in every year. As a result, we can deduce Cny as follows
Cny ¼ n
N1 X
ce;k Pgrid;k ∕100;
(9)
k¼0
where ce,k is the electricity price, and n is the time horizon. It's easy to see that the objective function F is linear, which is convex. The inequality constraint functions include Eqns (2), (3) and (6), which are linear and convex. The equality constraint functions include Eqns (1), (4) and (5). Eqns (1) and (5) are linear and affine. Eqn (4) is absolute function, which is not affine. Because in the convex optimization problem, only affine equality constraints are tolerated. The total original problem is not a convex problem due to the absolute equality constraint, which is also nonlinear. But relaxing (4) to inequalities gives a convex problem without qualitatively altering the original problem as follows [26].
Ekþ1 Ek þ Dt Pbatt;k hPbatt;k ;
k ¼ 0; …; N 1:
(10)
Now, Eqn (10) is absolute inequality, which is convex. We can fomulate the problem as a convex problem. The CP algorithm for simultaneously optimal energy management and component sizing of the PV/battery home is summarized in Table 1. A tool, CVX [37], is employed to parse the optimization problem, inducing a semi-definite program that can be efficiently solved by SeDuMi (Self-Dual-Minimization), which is an add-on for MATLAB and can solve optimization problems with linear, quadratic and semidefiniteness constraints [38]. It should be emphasized that thanks to the convexity, a globally optimal solution with arbitrary
Table 1 Component sizing and energy management via CP. Component sizing and energy management via CP For k ¼ 1,...,N, with N corresponding to the final time step of one year Optimization variables: Pbatt;k ; Pgrid;k ; Ek ; Pmax ; Qeap Expressions: F ¼ Cny þ cb Qeap þ cc Pmax Objective function: F Constraints: SOCmin Qeap Ek SOCmax Qeap ; k ¼ 0; …; N; Pmax Pbatt;k Pmax ; k ¼ 0; …; N 1; Ekþ1 Ek þ DtðPbatt;k hPbatt;k Þ; k ¼ 0; …; N 1; E0 ¼ Einit, Pgrid;k ¼ Pdem;k þ Pbatt;k Ppv;k ; P Cny ¼ n N1 k¼0 ce;k Pgrid;k ∕100, Pgrid;k P max grid :
k ¼ 0; …; N 1;
5. Results & discussion This section analyses the properties of the proposed CP approach for optimizing battery parameters and control strategy. The key parameters are listed in Table 2. All the simulations were run on a PC with a 2.50 GHz Intel Core i5-2450 M CPU and 4 GB of internal memory. 5.1. System parameters optimization
battery and charger prices, the parameters of BESS and the cost can be explored. First, we assume that the time horizon is 5 years. Independently of battery price (60 $/kWh to 203 $/kWh) and charger price (500 $/kW to 1500 $/kW) [40,41], the optimal values of maximum power Pmax for three different homes are shown in Table 3. The maximum power decreases as the P max grid increases from 0 kW to 3 kW (i.e., constrained power supply to the grid). Independently of charger price (500 $/kW to 1500 $/kW), the optimal values of battery energy capacity Qeap for the three different homes are shown in Table 4. Independently of the maximal power provided to the grid P max grid (0 kWe3 kW) and battery price (60 $/kWh to 203 $/kWh), the optimal value of battery energy capacity Qeap of Home 1 maintains constant, equals to 15 kWh. Those of Home 2 and Home 3 decrease until 3.33 kWh, as the battery price cc increases from 60 $/kWh to 203 $/kWh. It can be seen that the values of charger price cb do not affect the optimal battery energy capacity and the maximum power. If Pgrid is constrained as P max grid ¼ 0 kW, independently of the time horizon (1 yeare10 years), battery price (60 $/kWh to 203 $/kWh),
Table 2 Key parameters. Parameter Description
Symbol
Value
Unit
Step time Maximum battery SOC
Dt SOCmax
1 0.90
e
Minimum battery SOC
SOCmin
0.20
h
e
Lost efficiency
0.10
hour
and charger price (500 $/kW to 1500 $/kW), the maximum power Pmax maintains constant, equal to 2.26 kW, 4.26 kW, and 3.37 kW, for these three homes. The values of Pmax equal to the maximum power to the grid, in the case of no BESS. The optimal values of battery energy capacity Qeap are shown in Fig. 4. The battery energy capacity of Home 1 is augmented until 15 kWh, as the time horizon increases. Likewise, if the owner of Home 1 has a Nissan Leaf, the battery energy capacity is augmented, as the time horizon increases, and the maximum power maintains constant, equal to 2.26 kW [26]. Given the battery and charger prices of 100 $/kWh and 1000 $/kW, as well as different P max grid (0 kW or 3 kW) and time horizons, the optimal values of battery energy capacity Qeap, maximum
X. Wu et al. / Energy 140 (2017) 444e453 Table 3 Optimal results of maximum power [kW] based on different values of P max grid [kW]. P max grid
Home 1
Home 2
Home 3
0 1 2 3
2.26 1.46 1.46 1.46
4.26 3.26 2.26 1.26
3.37 2.37 1.37 0.37
Table 4 Optimal results of battery energy capacity [kWh] based on different values of P max grid [kW] and battery price [$/kWh]. Home
P max grid
cb ¼ 60
cb ¼ 100
cb ¼ 150
cb ¼ 203
1 2 2 2 2 3 3 3 3
0e3 0 1 2 3 0 1 2 3
15 15 15 15 11.89 15 15 12.96 3.53
15 10.25 13.69 13.48 9.91 8.13 10.78 8.66 3.53
15 3.33 5.19 3.33 3.33 3.33 3.33 3.33 3.33
15 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33
power Pmax, and cost are shown in Table 5, where FBESS, Fe, FnoB, and Fdiff are the cost of home BESS, the electric cost with BESS in one year, the electric cost without BESS in one year, and the electric cost difference with and without BESS in one year, respectively. If P max grid ¼ 0 kW, the battery energy capacities of the three homes increase until 15 kWh, as the time horizon increases until 4 years, 8 years, and 8 years, respectively. After that, the battery size remains
449
constant. The maximum power Pmax maintains constant, equal to 2.26 kW, 4.26 kW, and 3.37 kW, for the three households. If P max grid ¼ 3 kW, the battery energy capacities of the three homes increase until 15 kWh, as the time horizon increases until 3 years, 10 years, and 10 years, respectively. The maximum power Pmax increases until 1.46 kW, as the time horizon increases until 3 years for Home 1. After that, it remains constant. When time horizon increases from 1 year to 9 years, the maximum power Pmax for Home 2 and Home 3 is constant, equal to 1.26 kW and 0.37 kW, respectively. The electric costs with BESS in one year Fe for Home 1 and Home 3 are always less than that of the scenario without BESS, except for the time horizon of 1 year. The total cost savings become more significant with an increased time horizon. The cost savings in Home 1 are more significant than those in Home 2 and Home 3, due to higher PG&E electric price. The electric cost savings are larger, when P max grid ¼ 3 kW. However, prividing power to the grid will significantly increase the battery cost induced by accelerated degradation, as argued in Refs. [15,42]. In the following, we thus only consider the case of no power supply to the grid of Home 1. Without power supply to the grid, it is reasonable to assume a home battery life larger than 5 years [43]. The optimal battery energy capacity and the maximum power for Home 1 we consider are 15 kWh and 2.26 kW, and the cost of home BESS is 3760 $. With BESS, the electric energy cost in one year is 1043.1 $, whereas without the BESS, the electric energy cost in one year is 1881.5 $. The cost reduction reaches up to around 44.6%. If Home 1 has a Nissan Leaf that can not discharge power to the home, with the optimal values of battery energy capacity (17 kWh) and maximum power (2.26 kW), the electric cost with BESS over one year is 39.2% less than that without BESS [26]. With/without EV, home BESS can
Fig. 4. Results of battery energy capacity for different time horizons and BESS prices based on P max grid ¼ 0 kW.
450
X. Wu et al. / Energy 140 (2017) 444e453
Table 5 Optimal results based on cb ¼ 100 $/kWh and cc ¼ 1000 $/kW. Home
P max grid [kW]
n [year]
Qeap [kWh]
Pmax [kW]
FBESS [$]
Fe [$]
FnoB [$]
Fdiff [$]
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3
0 0 0 0 3 3 3 0 0 0 0 0 0 3 3 3 3 3 0 0 0 0 0 0 3 3 3
1 2 3 4e10 1 2 3e10 1e3 4 5 6 7 8e10 1e4 5 6 7e9 10 1e3 4 5 6 7 8e10 1e4 5e9 10
3.33 11.88 14.46 15 3.33 3.33 15 3.33 5.95 10.25 13.19 14.84 15 3.33 9.91 11.27 11.89 15 3.33 4.96 8.13 11.37 13.82 15 3.33 3.53 15
2.26 2.26 2.26 2.26 0.001 0.30 1.46 4.26 4.26 4.26 4.26 4.26 4.26 1.26 1.26 1.26 1.26 1.48 3.37 3.37 3.37 3.37 3.37 3.37 0.37 0.37 1.49
2593 3448 3706 3760 334 633 2960 4593 4855 5285 5579 5744 5760 1593 2251 2387 2449 2980 3703 3866 4183 4507 4752 4870 703 723 2990
1791.7 1165.8 1060.4 1043.1 1880.5 1650.3 809.3 483.2 413.7 317.7 264.2 238.7 236.4 143.7 1.3 26.4 36.4 89 694.9 651.4 580.9 521.9 483.6 467.3 581.5 577.3 343.2
1881.5 1881.5 1881.5 1881.5 1881.5 1881.5 1881.5 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 214.6 654.8 654.8 654.8 654.8 654.8 654.8 654.8 654.8 654.8
89.8 715.7 821.1 838.4 1 231.2 1072.2 268.6 199.1 103.1 49.6 24.1 21.8 70.9 215.9 241 251.0 303.6 40.1 3.4 73.9 132.9 171.2 187.5 73.3 77.5 311.6
obviously increase the home economy.
5.2. Example of energy management strategy Based on the Home 1's optimal results of battery energy capacity Qeap ¼ 15 kWh, the maximum power Pmax ¼ 2.26 kW, and the constraint of P max grid ¼ 0, this subsection presents the resulting CP control law simulated on two different operating samples, including summer day and winter day. The hourly power allocation and battery SOC over two summer days are described in Fig. 5-(a). It is evident that the majority of the home battery charging occurs during the low electricity price period: 24:00e7:00 and high PV power supply period: 10:00e15:00. Most of the battery discharging happens during the high electricity price period: 16:00e23:00. The electric power from the grid is zero during the period: 8:00e21:00. Similarly, the simulation results for two winter days are indicated in Fig. 5-(b). In winter, because of less PV power generation, the home battery charges during the low electricity price period and
discharges during the high electricity price period. The electric power from the grid is zero during the period: 16:00e21:00. In summary, the home does not buy electric energy from the grid during the peak periods of electric price. The outcomes of Home 2 and Home 3 are analogous. To demonstrate the potential economic benefits of equipping PV/battery with a home, we examine the electric energy cost curve of Home 1 in a comparative fashion. The hourly electric energy cost for two days in summer and winter is shown in Fig. 6-(a) and Fig. 6(b), respectively. With BESS, the two-day electric energy cost is 3.72 $ in summer and 7.16 $ in winter. Without BESS, the two-day electric energy cost is 7.87 $ in summer and 12.66 $ in winter. Therefore, the addition of BESS can save 52.7% and 43.4% electric cost in summer and winter, respectively. It is less cost-effective in winter, owing to larger household power demand and less PV power generation. It turns out that the BESS is needed both in summer and winter. With BESS, the electric power values from the grid of Home 1 in one year on each day (blue) and average (red) are shown in Fig. 7-
Fig. 5. Hourly power allocation and battery SOC of Home 1.
X. Wu et al. / Energy 140 (2017) 444e453
451
Fig. 6. Electric energy cost of Home 1.
(a). Compared with Fig. 1-(a), the peak power from the grid always happens from 24:00e7:00 when the electric price is low, and the PV/battery smart home does not supply electric energy to the grid. The electric energy cost values from the grid in one year on each day (blue) and average (red) are given in Fig. 7-(b). In contrast with Fig. 2-(b), at the presence of BESS, the electric energy cost from the grid is obviously less.
5.3. Rapid computation To demonstrate the advantages and efficacy of CP, we compare CP and DP approaches, in terms of precision and computational efficiency. Here, we first introduce the DP algorithms. DP is usually used as a benchmark, due to Bellman Principle of Optimality. Consider a multi-stage decision process, we define the instantaneous electricity cost functional gk(Ek,Pbatt,k) as follows:
gk Ek ; Pbatt;k ¼ ce;k ,Dt,Pgrid;k ¼ ce;k ,Dt, Pdem;k þ Pbatt;k Ppv;k
(11)
The controller must maintain battery Ek and power Pbatt,k within simple bounds as Eqns (2) and (3). The design problem is to search the control input Pbatt,k that minimizes the electricity cost. The controller will be synthesized as a time-varying state feedback control law. Namely, the control is the output of a mapping that depends on the current deterministic state Ek. We formalize this as a finite-time dynamic program,
minPbatt;k ;Ek
N1 X
ce;k ,Dt, Pdem;k þ Pbatt;k Ppv;k
(12)
k¼0
s: t:
Eqns ð2Þ ð6Þ:
(13)
Now we define the value function. Let Vk(Ek) be the minimum expected cost-to-go from time step k to N, given the current battery energy level-Ek. Then the principle of optimality is given by:
Vk ðEk Þ ¼ minPbatt;k 2D k gk Ek ; Pbatt;k þ Vkþ1 ðEkþ1 Þ ;
(14)
where gk ð,; ,Þ is the instantaneous cost in (11). The minimization operator is subject to a time-varying admissible control set D k characterized by (2)e(6). We also have the boundary condition
VN ðEN Þ ¼
EN Emax 0; for Emin N N ∞; otherwise:
(15)
Finally, the optimal control action is saved as
P *batt;k
¼ gk ðEk Þ ¼ argminPbatt;k 2D k gðEk ; Ik Þ þ Vkþ1 ðEkþ1 Þ:
(16)
In our work, the comparison between CP and DP aims not to show that CP works better than DP (we do not state such a claim
Fig. 7. Electric power and electric energy cost for Home 1 with BESS in one year on each day (blue) and average (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
452
X. Wu et al. / Energy 140 (2017) 444e453
throughout the paper). We only take DP as a benchmark (as usually done in optimal control problems) to manifest that the performance of our CP can be close to that of DP. In theory, DP can ensure a global optimal result according to Bellmans Principle of Optimality, and it has been used in previous residential energy management studies [16,17]. However, its numerical implementation (e.g., grid size) may lead to slightly inferior results, compared to CP. The difference between DP and CP is always marginal, demonstrating the effectiveness of CP. Moreover, we can clearly see the advantage of CP in computational efficiency. First, we assume the BESS of Home 1 has 15 kWh battery and 2.26 kW charger (optimal parameters) in the two methods. Over one-year data, the computational time of CP is 3.96 s, whereas that of DP is 2475 s. The CP is 625 times faster than DP. Battery charge power and the electric power from the grid in CP and DP are compared in Fig. 8. Both methods can make the battery charge with low electric price and supply energy to the home with high electric price. Now, we consider if the smart home with PV and battery energy storage can be optimally managed in the real world on a daily basis. Here we assume that the PV power generation and the home load demand in the future 24 h can be forecasted well. Both CP and DP are applied to optimize control policy. Given different time intervals Dt, the computational time and electric energy cost are summarized in Table 6, where tCP, tDP, FCP, FDP, and Lo denote the computational time to execute CP, computational time to execute DP, electric energy cost using CP, electric energy cost using DP, and loss of optimality comparing CP with DP, respectively. We define loss of optimality Lo as follows:
Lo ¼ ðFCP FDP Þ∕FDP :
(17)
The 24-h electric energy cost is virtually the same for CP and DP in cases of different time intervals (the maximal loss of optimality is about 6.9%). Despite that calculation steps increase from 48 to 1440, the computational time of CP is always less than 1 s, whereas that of DP drastically increases. The CP method is more than 30 times faster than DP. Hence, it is more suitable for fast system economic analysis.
Table 6 Comparison of CP and DP for Home 1 on a 24-h basis. N
Dt [min]
tCP [s]
tDP [s]
FCP [$]
FDP [$]
Lo [%]
48 96 288 1440
30 15 5 1
0.50 0.56 0.60 0.81
15.68 30.49 93.16 452.64
1.86 1.95 1.94 1.76
1.74 1.94 1.99 1.83
6.9 0.5 2.5 3.8
Table 7 Comparison of CP and DP for Home 1 based on sub-optimal parameters over one year. Qeap [kWh]
Pmax [kW]
FCP [$]
FDP [$]
Lo [%]
15 15 15 14 13 12
3 2.5 2.3 2.3 2.3 2.3
1040 1041 1043 1076 1143 1160
1097 1098 1098 1130 1166 1210
5.2 5.2 5.0 4.8 2.0 4.1
parameters of BESS, the control results of CP and DP over one year are shown in Table 7. When Qeap is 15 kWh, and Pmax changes from 3 kW to 2.3 kW, the results of CP are almost the same, similar to DP. When Pmax is 2.3 kW, and Qeap alters from 15 kWh to 12 kWh, the total costs of CP increase, similar to DP. Based on different suboptimal parameters of BESS for Home 1, the total cost of CP over one year is close to that of DP (the maximal loss of optimality is about 5.2%). 6. Conclusion The CP problem is mathematically formulated and demonstrated to simultaneously optimize the electric power allocation and BESS parameters for three different homes, located in California and Texas. The optimization results are systematically compared, taking various time horizons, maximal values of power to grid, BESS prices, and home electricy use/demand patterns into accout. Key findings are summarized below:
5.4. Sensitivity to parameters of BESS In this subsection, we account for a simple but nontrivially useful sensitivity analysis to the parameters of BESS, taking Home 1 as an example, where sub-optimal parameters of BESS are contrasted to the optimal parameters of CP. Given sub-optimal
(1) If there is no power supply to the grid, independently of the BESS price, the optimal values of maximum battery power for these three homes are 2.26 kW, 4.26 kW, and 3.37 kW, respectively. The values of charger price cb do not affect the optimal battery energy capacity and the maximum power for the three homes. Compared to the electric cost of a home without BESS, the usefulness of BESS to improve the home economy is pronouced. The total cost savings become more significant with increased time horizon and P max grid . The cost savings in Home 1 are more significant than those in Home 2 and Home 3, due to higher PG&E electric price. (2) The CP framework is substantiated to be much faster than DP, with close accuracy and optimality, which is favorable to real-time control and efficient design of smart home in practice. The maximal loss of optimality in the 24-h electric energy cost, relative to DP, is approximately 6.9%. The CP method is distinctly more than 30 times faster than DP. The future work could incorporate explicit thermal and ageing dynamics of the home battery into the optimization framework, in order to quantify the battery health implication. Acknowledgements
Fig. 8. Battery Charge power and grid power with CP and DP for Home 1.
This work was supported in part by the Sichuan Provincial
X. Wu et al. / Energy 140 (2017) 444e453
Scientific Research Innovation Team Program(Grant No. 17TD0035), Science and Technology Department of Sichuan Province (Grant Nos. 2017TD0026, 2015TD0021, 2016HH0010), NSF of China (Grant No. 51375402), Xihua University Fund (Grant No. z1220315), and the Fundamental Research Funds for the Central Universities (Project No. 0903005203398). References [1] Lund PD, Lindgren J, Mikkola J, Salpakari J. Review of energy system flexibility measures to enable high levels of variable renewable electricity. Renew Sustain Energy Rev 2015;45(0):785e807. [2] Entchev E, Yang L, Ghorab M, Lee E. Simulation of hybrid renewable microgeneration systems in load sharing applications. Energy 2013;50:252e61. [3] Aghajani G, Shayanfar H, Shayeghi H. Demand side management in a smart micro-grid in the presence of renewable generation and demand response. Energy 2017;126:622e37. [4] Batista N, Melcio R, Matias J, Catalao J. Photovoltaic and wind energy systems monitoring and building/home energy management using zigbee devices within a smart grid. Energy 2013;49:306e15. [5] Sechilariu M, Wang B, Locment F. Building integrated photovoltaic system with energy storage and smart grid communication. IEEE Trans Ind Electron 2013;60(4):1607e18. [6] Shirazi E, Jadid S. Cost reduction and peak shaving through domestic load shifting and ders. Energy 2017;124:146e59. [7] Amirioun MH, Kazemi A. A new model based on optimal scheduling of combined energy exchange modes for aggregation of electric vehicles in a residential complex. Energy 2014;69:186e98. [8] Prinsloo G, Mammoli A, Dobson R. Customer domain supply and load coordination: a case for smart villages and transactive control in rural off-grid microgrids. Energy 2017;135:430e41. [9] De Angelis F, Boaro M, Fuselli D, Squartini S, Piazza F, Wei Q. Optimal home energy management under dynamic electrical and thermal constraints. IEEE Trans Ind Inf 2013;9(3):1518e27. [10] Mohseni A, Mortazavi SS, Ghasemi A, Nahavandi A, Abdi MT. The application of household appliances' flexibility by set of sequential uninterruptible energy phases model in the day-ahead planning of a residential microgrid. Energy 2017;139:315e28. [11] Palma-Behnke R, Benavides C, Lanas F, Severino B, Reyes L, Llanos J, et al. A microgrid energy management system based on the rolling horizon strategy. IEEE Trans Smart Grid 2013;4(2):996e1006. [12] Pedrasa M, Spooner T, MacGill I. Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. IEEE Trans Smart Grid 2010;1(2):134e43. [13] Wang G, Zhang Q, Li H, McLellan BC, Chen S, Li Y, et al. Study on the promotion impact of demand response on distributed pv penetration by using noncooperative game theoretical analysis. Appl Energy 2017;185(Part 2): 1869e78. [14] Elma O, Tasc1karaoglu A, Ince AT, Selamogullar US. Implementation of a dynamic energy management system using real time pricing and local renewable energy generation forecasts. Energy 2017;134:206e20. [15] Sun C, Sun F, Moura SJ. Nonlinear predictive energy management of residential buildings with photovoltaics & batteries. J Power Sources 2016;325: 723e31. [16] Muratori M, Rizzoni G. Residential demand response: dynamic energy management and time-varying electricity pricing. IEEE Trans Power Syst 2016;31(2):1108e17. [17] Huang T, Liu D. Residential energy system control and management using adaptive dynamic programming. In: The 2011 international joint conference on neural networks; 2011. p. 119e24. [18] Munkhammar J, Widn J, Rydn J. On a probability distribution model
[19] [20] [21]
[22] [23]
[24] [25]
[26]
[27] [28] [29]
[30] [31] [32] [33]
[34]
[35]
[36]
[37] [38] [39] [40]
[41] [42]
[43]
453
combining household power consumption, electric vehicle home-charging and photovoltaic power production. Appl Energy 2015;142:135e43. Han D, Chesi G, Hung YS. Robust consensus for a class of uncertain multiagent dynamical systems. IEEE Trans Ind Inf 2013;9(1):306e12. Fattahi S, Ashraphijuo M, Lavaei J, Atamtrk A. Conic relaxations of the unit commitment problem. Energy 2017;134:1079e95. Shen J, Cheng C, Wu X, Cheng X, Li W, Lu J. Optimization of peak loads among multiple provincial power grids under a central dispatching authority. Energy 2014;74:494e505. Tsui K, Chan S. Demand response optimization for smart home scheduling under real-time pricing. IEEE Trans Smart Grid 2012;3(4):1812e21. Wang Z, Paranjape R. Optimal residential demand response for multiple heterogeneous homes with real-time price prediction in a multiagent framework. IEEE Trans Smart Grid 2017;8(3):1173e84. Hu X, Zou Y, Yang Y. Greener plug-in hybrid electric vehicles incorporating renewable energy and rapid system optimization. Energy 2016;111:971e80. Hu X, Murgovski N, Johannesson L, Egardt B. Optimal dimensioning and power management of a fuel cell/battery hybrid bus via convex programming. IEEE/ASME Trans Mechatronics 2015;20(1):457e68. Wu X, Hu X, Teng Y, Qian S, Cheng R. Optimal integration of a hybrid solarbattery power source into smart home nanogrid with plug-in electric vehicle. J Power Sources 2017;363:277e83. SolarCity. Solarcity. 2014. Tech. rep., SolarCity, http://www.solarcity.com/. P. S. Inc. Pecan street dataport. 2017. Tech. rep. https://dataport.pecanstreet. org. Gas P, Company E. Electric vehicles making sense of the rates. Pacific Gas and Electric Company; 2015. Tech. rep, http://www.pge.com/en/myhome/ saveenergymoney/pev/rates/index.page. Wu X, Hu X, Yin X, Moura S. Stochastic optimal energy management of smart home with pev energy storage. IEEE Trans Smart Grid PP 2017;99:1. 1. Zachar M, Daoutidis P. Microgrid/macrogrid energy exchange: a novel market structure and stochastic scheduling. IEEE Trans Smart Grid 2017;8(1):178e89. Fares RL, Webber ME. The impacts of storing solar energy in the home to reduce reliance on the utility. Nat Energy 2017;2(17001):1e10. Austin energy city of austin electric tariff. 2016. Tech. rep. http://austinenergy. com/wps/wcm/connect/ab6d045c-643e-4c16-921f-c76fa0fee2bf/ FY2016aeElectricRateSchedule.pdf?MOD¼AJPERES. Wu X, Hu X, Moura S, Yin X, Pickert V. Stochastic control of smart home energy management with plug-in electric vehicle battery energy storage and photovoltaic array. J Power Sources 2016;333:203e12. Li J, Wu Z, Zhou S, Fu H, Zhang XP. Aggregator service for pv and battery energy storage systems of residential building. CSEE J Power Energy Syst 2015;1(4):3e11. Wu D, Zeng H, Lu C, Bouletu B. Two-stage energy management for office buildings with workplace ev charging and renewable energy. IEEE Trans Transp Electrification 2017;3(1):225e37. Boyd S, Vandenberghe L. Convex optimization. Cambrige, U.K: Cabridge Univ. Press; 2004. Sturm JF. Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optim Methods Softw 1999;11(1e4):625e53. Pillot C. The worldwide battery market 2012-2025. 2013. Tech rep., Avicenne Energy. Yilmaz M, Krein PT. Review of battery charger topologies, charging power levels, and infrastructure for plug-in electric and hybrid vehicles. IEEE Trans Power Electron 2013;28(5):2151e69. Morrowa DKK, Francfort J. Plug-in hybrid electric vehicle charging infrastructure review. U.S. Department of Energy; Nov. 2008. Tech. rep. Yilmaz M, Krein PT. Review of the impact of vehicle-to-grid technologies on distribution systems and utility interfaces. IEEE Trans Power Electron 2013;28(12):5673e89. Calculate the cost of photovoltaic systems home solar electricity. New Mexico Solar Energy Association; 2015. Tech. rep.