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Scheduling of price-sensitive residential storage devices and loads with thermal inertia in distribution grid
Applied Energy 183 (2016) 636–644
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Scheduling of price-sensitive residential storage devices and loads with thermal inertia in distribution grid Ehsan Reihani b, Saeed Sepasi a,⇑, Reza Ghorbani b a b
Hawaii Natural Energy Institute, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, 1680 East-West Road, POST 110, Honolulu, HI 96822, USA Renewable Energy Design Laboratory (REDLab), Department of Mechanical Engineering, University of Hawaii at Manoa, USA
h i g h l i g h t s Optimal control of common storage devices and loads is presented. Results of two solution methods for this device are presented and compared. Combined optimization of devices is compared with individual device optimization.
a r t i c l e
i n f o
Article history: Received 4 May 2016 Received in revised form 4 August 2016 Accepted 19 August 2016
Keywords: Demand response Residential storage devices Dynamic programming Loads with thermal inertia Distributed renewable energy resources
a b s t r a c t Demand response provides more flexibility in the operation of a power grid with high renewable energy penetration. In this paper, a mathematical model of storage devices and loads with thermal inertia, such as batteries, water heaters, and air conditioners, is presented. Dynamic and linear programming methods are used to solve the optimization problem subject to comfort constraints. The price signal for these devices can incorporate financial criteria as well as reliability criteria. Optimal operation of the storage devices and loads with thermal inertia are obtained considering the given price signal. Individual device performance optimization is compared with all of the devices as a single solution and optimal operation plots and results are presented and discussed. The presented optimization problem for house devices is solved using dynamic programming and mathematical programming and the results are compared. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Demand response (DR) management is considered to be an important resource in optimizing the operation of power systems. It is defined as a change in the power consumption of electrical utilities customers to better match the demand for power with the supply [1]. With increasing generation of distributed renewable energy, DR can help the operator of the power system to minimize the negative impacts of such volatile resources on the stability of the grid. Adding intelligence to the residential loads to control the power flow is a crucial step in the smart grid paradigm. Interaction of smart loads using widespread and bidirectional communication infrastructure turns DR into a useful resource for operators of power systems. This bidirectional power and information flow in the emerging utility environment paves the way for utilizing loads with intrinsic storage capacities, such as thermal loads, distributed battery storage systems, and air ⇑ Corresponding author. E-mail address: [email protected] (S. Sepasi). http://dx.doi.org/10.1016/j.apenergy.2016.08.115 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
conditioners (ACs), to provide services, such as peak load shaving, load smoothing, and reserve procurement. In general, energy storage can take over multiple roles as a necessary positioner to facilitate financial profitability [2]. Distributed chemical battery storage and thermal storage in the form of water heaters (WHs) and ACs comprise significant portions of responsive demand at the residential scale. Each of these storage devices provide different benefits incidental to the pattern or amount of power they consume. However, implementing DR correctly would help improve market efficiency and operational reliability [3]. DR could be utilized to procure reserves using a short-term, stochastic securityconstrained unit commitment (SCUC) model [4]. Developing such efficient DR models for these choices and efficient algorithms for performance optimization would enhance the stability of the power system [5,6]. In general, the management of DR requires the identification of an optimal operating point that reduces the cost of electricity without causing the users any significant discomfort [7]. Optimal scheduling of battery energy storage systems (BESSs) and residential appliances are investigated in [8] taking into account operational and planning constraints. They have
E. Reihani et al. / Applied Energy 183 (2016) 636–644
developed to the point where they can be considered mature enough to serve as generation resources [9]. Impacts of such bulk energy storage devices in terms of reduction in market prices, system production, and their economic viability are studied in [10]. In the following, battery storage, as well as loads with thermal inertia, is discussed. 1.1. Battery Electric vehicles (EVs) and typical batteries have been used over the past years as energy storage devices for residential buildings [11,12]. BESSs potentially offers two main applications in the residential sector to help the power grid (i.e. the increasing demand for EVs and the widespread storage of residential energy [8]). Residential BESSs can be used as an off-grid hybrid energy solution, specifically in rural and isolated areas that have no access to grid power [13]. Also, residential BESSs can be used as a backup system to the grid connection. In this case, residential BESSs could be used as a solution for optimal scheduling of residential appliances, taking into account operational and planning constraints [8]. This approach could be improved by including the utility’s desired goals in the objective functions. It could lead to more possibilities of using BESSs in ancillary markets and electrical grids [9]. 1.2. Water heater Heat pump water heaters (HPWHs) and electric resistor water heaters (ERWHs) account for 20–30% of the energy demand of the U.S. residential sector [14]. The power consumption and the daily load patterns of homes are correlated. They have the highest portion of usage in residential power demand during daily load peak times [14]. Moreover, controlling their switching action is very convenient and easy because their heating elements are resistors and there is no need for reactive power support [15]. By heating the water inside HPWHs to higher temperatures, loads with thermal inertia can act as energy storage devices. The experiments performed in [16] showed that HPWHs and ERWHs are capable of providing demand response services, such as the reduction of the peak load, load shifting, and regulation of the ramping rate. 1.3. Air conditioning Air conditioning (AC) units are power-intensive appliances in households. These units store thermal energy while their consumption is governed by variations in the ambient temperature. This requires that the thermal dynamics of AC units be modeled properly. Accurate modeling can help in simulating AC units’ consumption of electrical power as they respond to ancillary service signals based on changes in the ambient temperature [17]. Control and aggregated load modeling for ACs have been studied extensively and reported in the literatures [17,18]. One of the main concepts in modeling aggregated loads is characterizing the evolution of the temperature density of the population [19]. Potential solutions for this problem include stochastic differential equations [20,21] and the deterministic fluid dynamics approach [22], with both approaches leading to the same type of partial differential equation (PDE) (i.e. the Fokker-Planck type). In addition to this first principle-based approach, data-driven approaches, which are based on Markov chains, have been used by researchers [18,23]. Establishing a control method to regulate the aggregated power response is the next step after obtaining a good model. Examples of applied control methods are model predictive control [24] and Lyapunov-based control [25]. Considering the utility’s desired goals in the objective function enhances the approach for all of these appliances. The joint influence of price and CO2 signals on DR behavior is presented in
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[26], where the consumers’ inconvenience levels are also considered in the proposed model. The sixth Northwest power plan [27], based on the experience in the region and elsewhere, states that the achievable technical potential for demand response in the region is around 5% of the peak load over the 20-year-plan horizon. The DR resources in this program include direct load control for air conditioning, space heating, water heating, irrigation scheduling, aggregators, demand buyback, interruptible contracts, and dispatchable generators. The load change in DR can be divided into two categories (i.e., priced-based demand response and incentive-based demand response [1]). Real-time pricing, criticalpeak pricing, and time-of-use tariffs provide customers with time-varying rates that reflect the price of electricity during different time periods. Incentive-based demand response programs pay participating customers to reduce their loads at times requested by the program sponsor. The price signal can entail different factors, such as cost of reserve procurement, reliability studies, operational planning, and load forecasts [28]. The contributing factors on price signal can stem from operational as well as planning criteria. Reliability criteria can be embedded in unit commitment and economic dispatch problems such as security-constrained unit commitment [29] and security-constrained optimal power flow [30]. The locational marginal price (LMP) during 24 h is obtained by solving the above optimization problem and may be considered as a price signal. According to the definition in [31], time-sensitive pricing lies in the non-dispatchable class of demand response. A study on reliability standards and working group ancillary services performed by General Electric (GE) through the Hawaii Natural Energy Institute (HNEI) [16] pointed out that storage and DR should be allowed to provide primary frequency response and spinning reserve. The mentioned DR services are frequency response, regulation, spinning reserves, non-spinning reserves, and replacement reserves. Load commitment was addressed in [32] by proposing a framework to minimize the household payment. Since water heater storage and battery storage have significant impacts on the overall contribution of a house in DR programs, optimizing this portion of DR should receive special attention. A distributed system with a DR management scheme is presented in [33] to flatten the load curve while minimizing the customers’ electricity costs. A reliable communication infrastructure between house appliances and utility is necessary for implementing the proposed algorithm. DR scheduling in a deregulated environment is presented in [34], where a new market concept called DR exchange (DRX) is introduced. In this paper, battery, water heater, and air conditioning energy models are presented, and their optimal operations were obtained using a given price signal. Based on the results, heuristics for optimal operation are derived. Individual device performance optimization is compared with all of the devices as a single solution and obtained results are discussed. The presented optimization problem for house devices is solved using dynamic programming and mathematical programming and the results are compared.
2. Models of storage appliances and DR approaches 2.1. Battery model Batteries are used to store energy so that it can be transferred to the grid at different time intervals. Since there is little control over the generation of electrical energy from renewable resources, storage is used to add the flexibility of providing electrical power from intermittent resources. Batteries can be used at residences to store excess energy from solar panels during the day, and this stored energy can then be used to meet the load demand at night. They can also serve different ancillary services, such as the control of
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voltages and frequencies using active and reactive power. A utility or a demand response aggregator can send a signal to control the power flow of the battery. The price signal is a result of some planning and operation optimization given the available resources. In this section, the cost of operating a battery is optimized using the price signal from the utility or the dynamic response aggregator (DRA). The price signal is the echo of some of the reliability and operational optimization planning. For instance, the price signal can contribute to load leveling goal by which a utility shifts some of the peak load to period of times when the load is low. This price signal could be a reflection of a strategy to absorb fluctuations caused by PVs leading to a more reliable grid. The equations that govern the power flow and state of charge (SOC) of a battery are given in Eqs. (1) and (2):
Etþ1 ¼ Et þ P t Dt SOC tþ1 ¼ SOC t þ
ð1Þ P t Dt Etot
ð2Þ
The energy level of a battery at any given time is updated by the power flow at that time step. The state of charge of the battery is also calculated in a similar manner. The SOC of a battery can be taken from the experimental data of the battery, which would make the model more precise. The efficiency of charging/discharging can also be considered in calculating the SOC of a battery [35]. The battery has been modeled as Thevenin’s equivalent circuit model, where the voltage and impedance of the circuit models are obtained from a look-up table based on experimental data [36]. The model used in this paper ignores the nonlinearity of SOC change with respect to power, but other models can be easily used to find the SOC of a battery subject to power change. The power flow of a battery is constrained by the nominal power and the SOC of the battery, as shown in Eqs. (3) and (4):
Pt < min
ðSOC max Etot Et Þ ; Pnom Dt
ð3Þ
ðSOC min Etot Et Þ ; Pnom ; Dt
ð4Þ
Pt > max
where parameters Pnom, SOCt, Pt, and Et denote nominal power, state of charge, power flow, and energy of the battery at time t, respectively. The power flow of a battery should not exceed an amount that would violate the SOC limits of the battery, and it also should be in the range of its nominal power. The SOC of a battery should be in the range of [0.2–0.8] to increase the lifecycle of the BESS [37]. The initial SOC of a battery usually is considered to be 0.2, assuming the battery is completely discharged. However, the optimization problem can be solved from any point of time onwards with a given initial value of SOC. Since the shape of the load is repeated over a 24-h period, the optimization horizon is considered for the same period of time. The time step is given as 15 min in case the utility must redo the optimization based on an updated price signal. Contingencies or some unprecedented market prices of power may affect the planning of a portion of the capacity of battery storage of the utility or aggregator, thereby leading to an updated price signal. Since the aggregated storage of a residential battery can be a potential source of spinning reserve for the power system, the price signal of the battery can be influenced by the price of ancillary service in the power market. The objective function minimizes the cost of operating a battery during the 24-h-planning horizon subject to the constraints specified in Eqs. (3) and (4). The overall cost is formulated as a linear optimization problem with power flow levels as the decision variables:
X Min pðtÞcðtÞ t
ð5Þ
The price signal, c(t), is given as an array for the next 24 h and p(t) is the power flow of the battery. The dynamic programming and mathematical optimization methods that are used to determine the optimal solutions for Eq. (5) are presented in the next section. Other objectives for BESS can be also defined, Load peak shaving, power smoothing and frequency regulation are among those objectives [29,38]. If the objective of using BESS is to flatten the load curve, the following objective function can be used: N X 2 Min Lt þ Pt Pref
ð6Þ
t¼1
where Lt, Pt and Pref are the load, power flow of BESS and reference power curve at time t. The objective function uses the BESS power to follow the Pref power curve. In order to flatten the load curve, the reference power curve can be defined as a straight line. BESS constraints such as nominal power and SOC limits are considered with the objective function to find the optimal power flow of BESS at each time step. Load smoothing is another objective function of BESS which removes the fluctuation of the power curve usually caused by renewable energy intermittency. The capacity and nominal power of BESS should be chosen considering the maximum magnitude of load fluctuations. 2.1.1. Dynamic programming for BESS Dynamic programming is a planning tool that addresses decision making of an agent in a stochastic environment. Each agent determines the best course of action using the available environment information and by considering future stages. This tradeoff removes myopic attainments among a set of interactive actions and presents an efficient compromise between immediate costs and future costs. The mathematical model of dynamic programming is built around finite-discrete-time Markov decision processes. This process is characterized by several elements, i.e., states, environment, actions, and cost. An agent has a finite set of actions at each state. A certain cost is followed by taking specific actions at discrete times. The history of an agent’s interaction with the environment is summarized in the agent’s current state, which is comprised of one of the decision making elements and the cost associated with immediate action. The transition probability from state i to state j, denoted by pij, which depends solely on the current state i and the action that is taken, u, which is a Markov property, i.e.:
pij ¼ f ði; uÞ
ð7Þ
A discrete time dynamic system is defined as:
xkþ1 ¼ f k ðxk ; uk ; xk Þ;
k ¼ 0; 1; . . . ; N 1
ð8Þ
The state xk goes to state xk+1 following action uk e U(xk) with a probability of xk 2 Pðjxk ; uk Þ. The class of control laws makes up an admissible set of policies p ¼ fl0 ; . . . ; lN1 g where lk maps states xk into control laws uk = lk(xk). For all of the initial conditions, J 0 ð1Þ; . . . ; J 0 ðnÞ, the sequence Jk(i) proceeds backward in time from period N 1 to period 0 generated by the iteration:
( J kþ1 ðiÞ ¼ min
uk 2UðiÞ
gði; uÞ þ
) n X pij ðuÞJ k ðjÞ
ð9Þ
j¼1
In other words, Jk(i) gives the optimal cost starting from state i of a k -stage problem with cost per stage given by g and a special cost-free termination state. This problem is called the stochastic shortest-path problem, and its objective is to reach the terminal state with the minimum expected cost. The deterministic shortest path problem is a special case in which the transition probability is equal to 1 for each state-control pair (i, u). Therefore, if a control action, u, is applied at state i, it would move to state j with the
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probability of 1. The state transition probability will improve by observing the environmental behavior over time, which involves some sort of learning algorithm. This aim of learning mechanism is to drive the problem into the deterministic area, which, in turn, leads to additional optimum solutions. At each state of SOC in the battery, several charging and discharging actions with different magnitudes will occur, can be taken which and they will lead to their respective previous states, as shown in Fig. 1. The cost at each state is the sum of the immediate cost and the past experience of the agent, which is calculated by the Bellman DP equation, i.e., Eq. (8), in which the probabilities are all set to one due to the deterministic nature of storage. Using the given commands, the state propagation is calculated by Eq. (2), and the set of allowable actions at each state is limited by Eqs. (3) and (4). The immediate cost, g(i, u), depends on the cost of electricity at time t. After the values of all states are calculated for all time stages, the optimal trajectory is determined by following the optimal action from the state in the last stage with the minimum cost.
SOC opt ðNÞ ¼ arg min J N ðjÞ j ¼ 1; 2; . . . ; n SOC opt ðt 1Þ ¼ SOC opt ðtÞ þ
ð10Þ
Popt ðtÞDt Etot
ð11Þ
where SOCopt(t) and Popt(t) are the optimal SOC and power flow of BESS at time t. 2.1.2. Mathematical programming for BESS Mathematical optimization is used to determine the optimal solutions for Eq. (5) subject to the constraints in Eqs. (3) and (4), which can be specified as follows:
SOC min þ
P1 Dt P2 Dt Pk Dt þ þ ... þ 6 SOC max Etot Etot Etot
k ¼ 1; . . . ; n
ð12Þ
And the same relations can be written for the other part of the constraint in Eq. (12):
SOC min
P 1 Dt P 2 Dt P k Dt ... 6 SOC min Etot Etot Etot
k ¼ 1; . . . ; n ð13Þ
In all of these equations, the power flow of the battery is constrained by the nominal power:
Pnom 6 Pi 6 Pnom
minimization of the power flow of the battery weighted by the price coefficients. Other objective functions, such as peak shaving or power smoothing, also can be defined. 2.2. Water heater A water heater is modeled as a single heating element that heats up a nominal mass of water in a tank, as shown in Fig. 2 [22]. The goal of optimizing the operation of the water heater is to determine the optimal time steps for switching ON/OFF given the price signal. The objective function can be described as follows:
X Min xðtÞcðtÞ 0 6 xðtÞ 6 1;
ð15Þ
where x(t) is the decision variable for the water heater at time step t and c(t) is given price signal. The objective function is the same as battery storage with the constraint for keeping the water temperature in the comfort range, i.e., T min 6 TðtÞ 6 T max . The temperature of the water inside the tank is influenced by two factors, i.e., the volume of hot water removed, which is replaced by the same volume of inlet water, resulting in thermal loss. We assume that the temperature of the water is updated by the volume of water removed, and, then, the temperature of the water decreases based on the heat transfer equations. The water mixing temperature is given in Eq. (16) in which the volume of inlet water is determined from DHWcalc [39], which is a realistic domestic hot-water profile:
T n ðtÞ ¼
T curr Mcurr ðtÞ þ T inlet ðtÞ M inlet ðtÞ Mcurr ðtÞ þ Minlet ðtÞ
ð16Þ
The decrease in the temperature of the water is modeled with a resistor, and it changes based on Eq. (17) [40]:
M SHW
dT ¼ UAðT amb ðtÞ T n ðtÞÞ þ Q ; dt
TðtÞ ¼ T n ðtÞ þ dT
ð17Þ
where M = Mass of water in the tank (kg) SHW = Specific heat of water (J/kg/°C) T = Temperature of the water in the tank (°C) t = Time (h) UA = Standby heat loss coefficient times the cross sectional area of the storage tank (W/°C/s) Q = Rate of heat input to the tank from the heater (J/h). It is equal to zero when the heater is OFF.
ð14Þ
In other words, the battery can accumulate or discharge power at most at the nominal power. The objective function is the
The comfortable temperature range can be represented as follows:
T min 6 T n ðtÞ þ
Fig. 1. Actions and costs at each state.
xðtÞ 2 Z
t
UAðT amb ðtÞ T n ðtÞÞ þ xðtÞQ 6 T max M SHw
Fig. 2. Water heater model.
ð18Þ
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Model used in [41] uses ordinary differential equations (ODE) to find the temperature change which is caused by several factors such as electrical energy input, losses and energy transfer due to hot water draw. One node and two node water heater models are discussed in [42]. In one node model, it is assumed that water heater tank has a uniform temperature which is determined by first order differential equation similar to [20]. In two node model, it is assumed that cold and hot water inside the tank have different temperatures. This model is used to determine the height of the hot water inside the tank. The temperature in our model is considered to be uniform in the tank and changes according to (16). The current model can be enhanced by experimental data from the water heater which can be collected from the grid interactive water heaters. The relationship between the input power and temperature change, hot water draw pattern and cooling rate data can be used to tune the model or just make up a look up table. 2.2.1. Dynamic programming for WH The temperature of the water in the water heater evolves probabilistically based on the volume of hot water withdrawn from the water heater. The allowable set of actions consists of switching ON and switching OFF, while keeping the temperature in the comfortable range for the customer. A cost is incurred for the use of electricity once the agent takes an action. All of the states, actions, and costs are observed at discrete times. The temperature at each time has sufficient information for the agent to take an action in the stochastic environment. Therefore, the water heater’s past experience with the environment is summarized in the current temperature state realizing a Markov property. After laying the mathematical foundation for water heater planning problem acting as an agent, dynamic programming is utilized to provide the optimal sequence of actions in an elegant and principled manner. The value at each state is a probabilistic minimization obtained by (19), shown graphically in Fig. 3. The probabilistic nature of state changes arises from the stochastic volume of hot water withdrawn. After all the state values for all time stages are obtained, the optimal trajectory is found by following the optimal action at each time step moving backwards. The following equations present the optimal state at the final time and at other times.
T opt;N ¼ arg min J N ðjÞ j ¼ 1; 2; . . . ; n T opt ðt 1Þ ¼
X
pðj; uopt ðtÞÞ T opt ðtÞ t – N
j2T opt ðtÞ
ð19Þ ð20Þ
where T, J, u, p are temperature, cost, decision, and probability, respectively. 2.2.2. B2. Second solution for the optimal control of the water heater The water heater optimization problem is very similar to that of battery programming, with the variables being the only difference in decision making. In the case of programming a battery, the decision variable, which was the power flow of the battery, was a continuous variable. But, for the case of the water heater, the decision variables are integer variables of 0 and 1, which represent switching the water heater ON and OFF. Combining Eqs. (16) and (21), the constraints can be expressed as follows:
Tð2Þ ¼ Tð1Þð1 þ kð1Þ cÞ þ að1Þ þ bð1Þ cað1Þ þ xð1Þxð1Þ 6 T max ð21Þ Tð3Þ ¼ ½Tð1Þð1 þ kð1Þ cÞ þ að1Þ þ bð1Þ cað1Þ þ xxð1Þ½1 þ kð2Þ c það2Þ þ bð2Þ cað2Þ þ xxð2Þ 6 T max ð22Þ In terms of decision variables, these equations change to:
1
xð1Þ 6
x
ðT max Tð1Þð1 þ kð1Þ cÞ að1Þ bð1Þ þ cað1ÞÞ
ð23Þ
xð1 þ kð2Þ cÞxð1Þ þ xxð2Þ 6 T max ð1 þ kð2Þ cÞðTð1Þð1 þ kð1Þ cÞ ð24Þ þ að1Þ þ bð1Þ cað1ÞÞ að2Þ bð2Þ þ cað2Þ and
aðiÞ ¼
M in ðiÞT in ðiÞ M c ðiÞ þ M in ðiÞ
ð25Þ
bðiÞ ¼
UA T amb ðiÞ M SHw
ð26Þ
c¼
UA M SHw
kðiÞ ¼
x¼
M c ðiÞ Mc ðiÞ þ M in ðiÞ
Q M SHw
ð27Þ
ð28Þ
ð29Þ
where Mc is mass of water left in tank (kg) Min is mass of water from cold water inlet (kg) Tin is the inlet temperature of the cold water (kg) Tamb is ambient temperature (°C). The same constraints can be written for the minimum comfort temperature. The constraint can be reformulated as a linear matrix with the objective function or as a linear weighted sum of decision variables. The problem can be solved via integer linear programming or evolutionary algorithms. 2.3. AC
Fig. 3. Value of each state in water heater planning problem.
Air conditioning systems consume a significant portion of the total power used in the residential and commercial sectors. AC units keep the room temperature within a comfortable temperature range, and they can be adjusted manually or remotely. In this work, comfortable temperature interval for AC is [18.8 °C, 25 °C], and initial indoor temperature is set to be 29.5 °C. This comfortable temperature interval is similar to other researcher’s defined comfort period [43,44]. It is desired to minimize the cost of electricity
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given a price signal. The mathematical model of the thermal behavior of a room, as taken from [45], is given as follows:
T i ðtÞ ¼
C1 C1 T i1 ðti1 Þ xi1 eC 2 :ðtti1 Þ C2 C2
T 0 ðt 0 Þ ¼ T 0
ð30Þ ð31Þ
Tðt; ðxi ; t i Þ; T 0 Þ ¼ Tg i ðtÞ
t i1 6 t 6 ti
ð32Þ
The external thermal source is modeled as a thermal pulse train with different magnitudes that is defined below:
WðtÞ ¼
m X
xi dðt ti Þ
ð33Þ
i¼1
In Eq. (30), C1 and C2 are constants that were derived in [46]. Also, d(t) is the Dirac delta function, and xi is the level of thermal intensity entering the room at time t. The objective function is similar to that of the water heater, minimizing a linear cost throughout a time horizon. The constraints also are similar, keeping the temperature of the room in the comfortable range, as defined by the user. Validity of the model can be improved by real time measurement of room temperature and finding the relationship between the parameters of AC model and temperature change. The uncertainty of temperature noise can be reduced using a probabilistic function which is tuned by collecting temperature measurement. 2.3.1. AC optimal control solutions: Dynamic programming In the optimal control of AC, the temperature evolves deterministically with the decision variables. The uncertainty in the change in the temperature is modeled with the thermal pulse train. Therefore, the optimal trajectory of the temperature and the optimal decision variables are obtained using deterministic dynamic programming. The thermal noise associated with the outdoor temperature also can be modeled probabilistically. In this state, the resultant temperature state can be determined using a relationship similar to that in Eq. (20).
power flow of the battery given the price function shown in Fig. 4 [47]. The optimal SOC trajectory of linear and dynamic programming is shown in Fig. 5, which shows that the two methods generated almost the same trajectory, with DP giving a slightly better result. Since the optimal SOC had some fluctuations, it was smoothed to increase the battery’s storage lifetime. The power flow is sensitive to the local minima in the price signal. When there is a valley in the price signal, the battery charges until it reaches the local maximum, and, later, it discharges the stored energy to minimize the overall cost. Fig. 6 shows the power flow that resulted from the optimal SOC. Since the fluctuations in the trajectory of the SOC may affect the battery’s lifetime, a smoothing spline was applied to the SOC’s trajectory shown in Fig. 7. The smoothed SOC trajectory s(x) is obtained from minimizing (19) given smoothing parameter p, the weights wi and the input data x and y:
Z X p wi ðyi sðxi ÞÞ2 þ ð1 pÞ i
2
d s 2
dx
!2 dx
The smoothing parameter and the weights are assumed to be 0.099 and 1, respectively. The smoothing is performed to reduce the stress on the battery and increase its lifetime. The price signal, as well as the decision variables, has been concatenated 10 times to increase the search space. The obtained results are the same as before, showing a slight improvement of DP over mathematical programming. 3.2. Water heater An electric water heater with a 50-gallon capacity and a power requirement of 4.5 kW was used for the HPWH test case. This
2.4. Combined DR device optimization In this section, we discuss house level simultaneous optimization of water heater, AC and battery storage assuming that the house price signal is given. The objective function is to minimize the overall cost of electricity usage during 24 h ahead. The objective function is the sum of the individual devices costs shown below:
XX Min xði; tÞcðtÞ i
ð34Þ
t
The constraints are the same for each individual device. This optimization problem is mixed integer nonlinear programming. Since the states of the problem are of different types, dynamic programming cannot be used to find the shortest cost path. Hence, mathematical optimization method is sued to find the optimal decision variables.
Fig. 4. Price signal.
3. Simulation and results 3.1. Battery A Powerwall battery specification was used as the test case for the optimal control. The battery was a Li-ion, wallmounted battery that had a capacity of 6.4 kW h and a nominal power of 3.3 kW [46]. Dynamic programming was used to determine the optimal SOC state trajectory using Eqs. (9) and (10). Linear programming also was used to produce the optimal
ð35Þ
Fig. 5. Optimal SOC trajectory obtained by DP and LP.
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tion, while LP fell to local minima. The results in Fig. 7(b) and (c) show that the water heater was switched ON during the lowest points of the price signal while keeping the temperature in the comfort range. If the volume of hot water removed from the water heater is high, the water heater is switched ON during other time intervals to satisfy the constraints. 3.3. AC
Fig. 6. Optimal power obtained by DP.
The solutions for AC with DP and with LP overlap and give the optimal result while satisfying the constraints. In Fig. 8, the outdoor temperate is assumed to be around 29 °C with the thermal noise shown in the following subplot. AC keeps the temperature in the allowed range by switching ON in the local minima of the price signal. The outdoor temperature has been increased 5° in Fig. 9, which has doubled the frequency of AC on states. In the next figure, Fig. 10, the outdoor temperature is 29 °C while the input thermal noise has been doubled. The amount of input thermal noise during noon time increases which results in AC to switch ON more often during that time. Seasonal temperature change could affect this comfort zone and its range could vary. It could be applied as constrain changes to the optimization problem, as well. 3.4. Combined device simultaneous optimization
Fig. 7. (a) Hot water draw profile, (b) optimal switching of water heater and (c) temperature change of water heater following the optimal switching dictions points.
capacity of the water heater is good enough for a household with a size of 3–5 people, assuming their appliances typical flow rate is within the normal range [48]. The problem with optimizing the operation of the water heater was it had a smaller search scope because the decision variables were integers. A sample hot water draw profile is given in Fig. 8(a), and the corresponding optimal decision variables are shown in Fig. 7(b). Fig. 7(c) shows the temperature of the water heater as this test was being conducted. The water heater was turned on at the minimum value points in the cost curve while keeping the temperature of the water in the comfort zone, the temperature of which was assumed to be between 37.77 °C and 60 °C. The DP and LP gave the same results, but, when the search space was larger, DP gave the optimum solu-
The result of simultaneous optimization of DR devices is shown in Fig. 11. The AC has been switched ON one fewer times while the water heater has been switched ON more. The overall cost of optimization considering home as a single unit is more than the sum of the costs of optimizing individual unit operations. However, as the optimization of each DR unit is independent, considering each device separately yields better results compared to when all of the devices are considered as a single solution. Therefore, it is desirable to optimize the DR devices individually. DRA needs to consider the impact of DR devices operations on the power grid. In other words, if all the houses connected to a same distribution grid transformer receive a same price signal, they will draw power at the valley of the price signal. On the other hand, the aggregate power flowing through the transformer should not exceed the rating. For example, a 25 kV A transformer current connected to a single phase 120 V feeder should not exceed 70 A. If the transformer is providing power to four houses, each house should not draw more than 17.5 A. Assuming the breaker cutoff current setting to be 15 A, then the transformer is already protected by the breaker fuses in the houses. Therefore, installing
Fig. 8. (a) Optimal switching of AC assuming the outdoor temperate is 29 °C, (b) thermal noise and (c) room temperature.
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Fig. 9. (a) Optimal switching of AC assuming the outdoor temperate is 34 °C, (b) thermal noise and (c) room temperature.
Fig. 10. (a) Optimal switching of AC assuming the outdoor temperate is 29 °C and doubled temperature noise compare to Fig. 8, (b) thermal noise and (c) room temperature.
Fig. 11. Combined device simultaneous optimization: (a) BESS SOC, (b) water heater and (c) air conditioner.
more loads such as BESS should be done considering maximum load of the house. 4. Conclusions In this paper, we presented models of residential storage devices, including a battery, a water heater, and an AC. The optimal operation of these devices and loads were determined using dynamic and linear programming given a signal price, and the corresponding results are presented. It is shown that dynamic pro-
gramming gives the optimal solution while the formulation and implementation takes more effort compared with using available mixed-integer linear/nonlinear solution packages. Moreover, it is shown that individual optimization of devices yields better results than considering all of them as a single solution. Since the DR devices might draw power at the valley of price signal simultaneously, installing more batteries for DR purpose should consider the rating of the house breaker as well as distribution transformer rating. Heuristics are found for optimal performance of DR devices in case devices act autonomously without communication. The
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heuristics show that the devices should draw power at the global minimum, and if the constraints are violated, more power should be drawn at the local minima. Feeding back power is done at the peak of signal price to maximize the savings. The experimental results can be used to tune in the model or as a look-up table to enhance the accuracy of the results. References [1] QDR Q. Benefits of demand response in electricity markets and recommendations for achieving them. Tech Rep. Washington, DC (USA): US Dept. Energy; 2006. [2] Zafirakis Dimitrios et al. The value of arbitrage for energy storage: evidence from European electricity markets. Appl Energy 2016. [3] Rahimi Farrokh, Ipakchi Ali. Demand response as a market resource under the smart grid paradigm. IEEE Trans Smart Grid 2010;1(1):82–8. [4] Parvania Masood, Fotuhi-Firuzabad Mahmud. Demand response scheduling by stochastic SCUC. IEEE Trans Smart Grid 2010;1(1):89–98. [5] Mohsenian-Rad Amir-Hamed et al. Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid. IEEE Trans Smart Grid 2010;1(3):320–31. [6] Yoon Ji Hoon, Bladick Ross, Novoselac Atila. Demand response for residential buildings based on dynamic price of electricity. Energy Build 2014;80:531–41. [7] Tsui Kai Man, Chan Shing-Chow. Demand response optimization for smart home scheduling under real-time pricing. IEEE Trans Smart Grid 2012;3 (4):1812–21. [8] Setlhaolo Ditiro, Xia Xiaohua. Optimal scheduling of household appliances with a battery storage system and coordination. Energy Build 2015;94:61–70. [9] Cho Joohyun, Kleit Andrew N. Energy storage systems in energy and ancillary markets: a backwards induction approach. Appl Energy 2015;147:176–83. [10] Das Trishna, Krishnan Venkat, McCalley James D. Assessing the benefits and economics of bulk energy storage technologies in the power grid. Appl Energy 2015;139:104–18. [11] Leadbetter Jason, Swan Lukas. Battery storage system for residential electricity peak demand shaving. Energy Build 2012;55:685–92. [12] Sepasi Saeed, Roose Leon R, Matsuura Marc M. Extended kalman filter with a fuzzy method for accurate battery pack state of charge estimation. Energies 2015;8(6):5217–33. [13] Tutkun Nedim. Minimization of operational cost for an off-grid renewable hybrid system to generate electricity in residential buildings through the SVM and the BCGA methods. Energy Build 2014;76:470–5. [14] Pourmousavi S Ali, Patrick Stasha N, Nehrir M Hashem. Real-time demand response through aggregate electric water heaters for load shifting and balancing wind generation. IEEE Trans Smart Grid 2014;5(2):769–78. [15] Vrettos Evangelos, Koch Stephan, Andersson Goran. Load frequency control by aggregations of thermally stratified electric water heaters. In: 3rd IEEE PES international conference and exhibition on innovative smart grid technologies (ISGT Europe). IEEE; 2012. [16] Mayhorn ET, Parker SA, Chassin FS. Evaluation of the demand response performance of large capacity electric water heaters; 2015. [17] Lu Ning. An evaluation of the HVAC load potential for providing load balancing service. IEEE Trans Smart Grid 2012;3(3):1263–70. [18] Lu Ning, Chassin David P, Widergren Steve E. Modeling uncertainties in aggregated thermostatically controlled loads using a state queueing model. IEEE Trans Power Syst 2005;20(2):725–33. [19] Zhang Wei et al. Aggregated modeling and control of air conditioning loads for demand response. IEEE Trans Power Syst 2013;28(4):4655–64. [20] Callaway Duncan S. Tapping the energy storage potential in electric loads to deliver load following and regulation, with application to wind energy. Energy Convers Manage 2009;50(5):1389–400. [21] Malhame Roland, Chong Chee-Yee. Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system. IEEE Trans Automatic Control 1985;30(9):854–60. [22] Faruqui Ahmad, Sergici Sanem, Sharif Ahmed. The impact of informational feedback on energy consumption—a survey of the experimental evidence. Energy 2010;35(4):1598–608. [23] Mathieu Johanna L, Callaway Duncan S. State estimation and control of heterogeneous thermostatically controlled loads for load following. In: 45th Hawaii international conference on system science (HICSS). IEEE; 2012.
[24] Koch Stephan, Mathieu Johanna L, Callaway Duncan S. Modeling and control of aggregated heterogeneous thermostatically controlled loads for ancillary services. Proc. PSCC 2011. [25] Bashash Saeid, Fathy Hosam K. Modeling and control insights into demandside energy management through setpoint control of thermostatic loads. In: American control conference (ACC). IEEE; 2011. [26] Song Meng et al. Estimating the impacts of demand response by simulating household behaviours under price and CO2 signals. Electr Power Syst Res 2014;111:103–14. [27] Power, Northwest, and Conservation Council. Sixth northwest conservation and electric power plan. Northwest Power and Conservation Council; 2010. [28] Reihani Ehsan, Ghorbani Reza. Load commitment of distribution grid with high penetration of photovoltaics (PV) using hybrid series-parallel prediction algorithm and storage. Electr Power Syst Res 2016;131:224–30. [29] Mehrtash Mahdi, Raoofat Mahdi, Mohammadi Mohammad, Zareipour Hamidreza. Fast stochastic security-constrained unit commitment using point estimation method. Int Trans Electr Energy Syst 2016(26):671–88. [30] Wanga Qin, McCalleyb James D, Zhengc Tongxin, Litvinovc Eugene. Solving corrective risk-based security-constrained optimal power flow with Lagrangian relaxation and Benders decomposition. Int J Electr Power Energy Syst 2016;75:255–64. [31] NERC. 2011 demand response availability report. Atlanta (GA 30326): North American Electric Reliability Corporation; 2013. [32] Rastegar Mohammad, Fotuhi-Firuzabad Mahmud, Aminifar Farrokh. Load commitment in a smart home. Appl Energy 2012;96:45–54. [33] Safdarian Amir, Fotuhi-Firuzabad Mahmud, Lehtonen Matti. A distributed algorithm for managing residential demand response in smart grids. IEEE Trans Ind Inform 2014;10(4):2385–93. [34] Nguyen Duy Thanh, Negnevitsky Michael, de Groot Martin. Market-based demand response scheduling in a deregulated environment. IEEE Trans Smart Grid 2013;4(4):1948–56. [35] Lu Chao et al. Optimal sizing and control of battery energy storage system for peak load shaving. Energies 2014;7(12):8396–410. [36] Li Xiangjun, Hui Dong, Lai Xiaokang. Battery energy storage station (BESS)based smoothing control of photovoltaic (PV) and wind power generation fluctuations. IEEE Trans Sustain Energy 2013;4(2):464–73. [37] Sepasi Saeed, Ghorbani Reza, Liaw Bor Yann. Inline state of health estimation of lithium-ion batteries using state of charge calculation. J Power Sources 2015;299:246–54. [38] Reihani Ehsan, Sepasi Saeed, Roose Leon R, Matsuura Marc. Energy management at the distribution grid using a battery energy storage system (BESS). Int J Electr Power Energy Syst 2016;77:337–44. [39] Jordan Ulrike, Vajen Klaus. DHWcalc: program to generate domestic hot water profiles with statistical means for user defined conditions. In: Proc ISES solar world congress. [40] Goh Chong Hock K, Apt Jay. Consumer strategies for controlling electric water heaters under dynamic pricing. Carnegie Mellon Electricity Industry Center working paper, No. CEIC-04-02; 2004. [41] Kepplinger Peter, Huber Gerhard, Petrasch Jörg. Autonomous optimal control for demand side management with resistive domestic hot water heaters using linear optimization. Energy Build 2015;100:50–5. [42] Diao Ruisheng et al. Electric water heater modeling and control strategies for demand response. In: IEEE power and energy society general meeting. IEEE; 2012. [43] Chen Zhi, Wu Lei, Fu Yong. Real-time price-based demand response management for residential appliances via stochastic optimization and robust optimization. IEEE Trans Smart Grid 2012;3(4):1822–31. [44] Shao Shengnan, Pipattanasomporn Manisa, Rahman Saifur. Grid integration of electric vehicles and demand response with customer choice. IEEE Trans Smart Grid 2012;3(1):543–50. [45] Aftab Muhammad, Chau Sid Chi-Kin, Armstrong Peter. Smart air-conditioning control by wireless sensors: an online optimization approach. In: Proceedings of the fourth international conference on future energy systems. ACM; 2013. [46] Motors Tesla. Powerwall; tesla home battery. URL: http://www. teslamotors.com/powerwall [accessed: May 2, 2015]. [47] ISO NE fifteen-minute preliminary real-time locational marginal price. Online: http://www.iso-ne.com/isoexpress/web/reports/pricing/-/tree/lmp-by-node. [48] http://energy.gov/energysaver/sizing-new-water-heater [accessed: Aug 2, 2016].
Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Scheduling of price-sensitive residential storage devices and loads with thermal inertia in distribution grid Ehsan Reihani b, Saeed Sepasi a,⇑, Reza Ghorbani b a b
Hawaii Natural Energy Institute, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, 1680 East-West Road, POST 110, Honolulu, HI 96822, USA Renewable Energy Design Laboratory (REDLab), Department of Mechanical Engineering, University of Hawaii at Manoa, USA
h i g h l i g h t s Optimal control of common storage devices and loads is presented. Results of two solution methods for this device are presented and compared. Combined optimization of devices is compared with individual device optimization.
a r t i c l e
i n f o
Article history: Received 4 May 2016 Received in revised form 4 August 2016 Accepted 19 August 2016
Keywords: Demand response Residential storage devices Dynamic programming Loads with thermal inertia Distributed renewable energy resources
a b s t r a c t Demand response provides more flexibility in the operation of a power grid with high renewable energy penetration. In this paper, a mathematical model of storage devices and loads with thermal inertia, such as batteries, water heaters, and air conditioners, is presented. Dynamic and linear programming methods are used to solve the optimization problem subject to comfort constraints. The price signal for these devices can incorporate financial criteria as well as reliability criteria. Optimal operation of the storage devices and loads with thermal inertia are obtained considering the given price signal. Individual device performance optimization is compared with all of the devices as a single solution and optimal operation plots and results are presented and discussed. The presented optimization problem for house devices is solved using dynamic programming and mathematical programming and the results are compared. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Demand response (DR) management is considered to be an important resource in optimizing the operation of power systems. It is defined as a change in the power consumption of electrical utilities customers to better match the demand for power with the supply [1]. With increasing generation of distributed renewable energy, DR can help the operator of the power system to minimize the negative impacts of such volatile resources on the stability of the grid. Adding intelligence to the residential loads to control the power flow is a crucial step in the smart grid paradigm. Interaction of smart loads using widespread and bidirectional communication infrastructure turns DR into a useful resource for operators of power systems. This bidirectional power and information flow in the emerging utility environment paves the way for utilizing loads with intrinsic storage capacities, such as thermal loads, distributed battery storage systems, and air ⇑ Corresponding author. E-mail address: [email protected] (S. Sepasi). http://dx.doi.org/10.1016/j.apenergy.2016.08.115 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
conditioners (ACs), to provide services, such as peak load shaving, load smoothing, and reserve procurement. In general, energy storage can take over multiple roles as a necessary positioner to facilitate financial profitability [2]. Distributed chemical battery storage and thermal storage in the form of water heaters (WHs) and ACs comprise significant portions of responsive demand at the residential scale. Each of these storage devices provide different benefits incidental to the pattern or amount of power they consume. However, implementing DR correctly would help improve market efficiency and operational reliability [3]. DR could be utilized to procure reserves using a short-term, stochastic securityconstrained unit commitment (SCUC) model [4]. Developing such efficient DR models for these choices and efficient algorithms for performance optimization would enhance the stability of the power system [5,6]. In general, the management of DR requires the identification of an optimal operating point that reduces the cost of electricity without causing the users any significant discomfort [7]. Optimal scheduling of battery energy storage systems (BESSs) and residential appliances are investigated in [8] taking into account operational and planning constraints. They have
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developed to the point where they can be considered mature enough to serve as generation resources [9]. Impacts of such bulk energy storage devices in terms of reduction in market prices, system production, and their economic viability are studied in [10]. In the following, battery storage, as well as loads with thermal inertia, is discussed. 1.1. Battery Electric vehicles (EVs) and typical batteries have been used over the past years as energy storage devices for residential buildings [11,12]. BESSs potentially offers two main applications in the residential sector to help the power grid (i.e. the increasing demand for EVs and the widespread storage of residential energy [8]). Residential BESSs can be used as an off-grid hybrid energy solution, specifically in rural and isolated areas that have no access to grid power [13]. Also, residential BESSs can be used as a backup system to the grid connection. In this case, residential BESSs could be used as a solution for optimal scheduling of residential appliances, taking into account operational and planning constraints [8]. This approach could be improved by including the utility’s desired goals in the objective functions. It could lead to more possibilities of using BESSs in ancillary markets and electrical grids [9]. 1.2. Water heater Heat pump water heaters (HPWHs) and electric resistor water heaters (ERWHs) account for 20–30% of the energy demand of the U.S. residential sector [14]. The power consumption and the daily load patterns of homes are correlated. They have the highest portion of usage in residential power demand during daily load peak times [14]. Moreover, controlling their switching action is very convenient and easy because their heating elements are resistors and there is no need for reactive power support [15]. By heating the water inside HPWHs to higher temperatures, loads with thermal inertia can act as energy storage devices. The experiments performed in [16] showed that HPWHs and ERWHs are capable of providing demand response services, such as the reduction of the peak load, load shifting, and regulation of the ramping rate. 1.3. Air conditioning Air conditioning (AC) units are power-intensive appliances in households. These units store thermal energy while their consumption is governed by variations in the ambient temperature. This requires that the thermal dynamics of AC units be modeled properly. Accurate modeling can help in simulating AC units’ consumption of electrical power as they respond to ancillary service signals based on changes in the ambient temperature [17]. Control and aggregated load modeling for ACs have been studied extensively and reported in the literatures [17,18]. One of the main concepts in modeling aggregated loads is characterizing the evolution of the temperature density of the population [19]. Potential solutions for this problem include stochastic differential equations [20,21] and the deterministic fluid dynamics approach [22], with both approaches leading to the same type of partial differential equation (PDE) (i.e. the Fokker-Planck type). In addition to this first principle-based approach, data-driven approaches, which are based on Markov chains, have been used by researchers [18,23]. Establishing a control method to regulate the aggregated power response is the next step after obtaining a good model. Examples of applied control methods are model predictive control [24] and Lyapunov-based control [25]. Considering the utility’s desired goals in the objective function enhances the approach for all of these appliances. The joint influence of price and CO2 signals on DR behavior is presented in
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[26], where the consumers’ inconvenience levels are also considered in the proposed model. The sixth Northwest power plan [27], based on the experience in the region and elsewhere, states that the achievable technical potential for demand response in the region is around 5% of the peak load over the 20-year-plan horizon. The DR resources in this program include direct load control for air conditioning, space heating, water heating, irrigation scheduling, aggregators, demand buyback, interruptible contracts, and dispatchable generators. The load change in DR can be divided into two categories (i.e., priced-based demand response and incentive-based demand response [1]). Real-time pricing, criticalpeak pricing, and time-of-use tariffs provide customers with time-varying rates that reflect the price of electricity during different time periods. Incentive-based demand response programs pay participating customers to reduce their loads at times requested by the program sponsor. The price signal can entail different factors, such as cost of reserve procurement, reliability studies, operational planning, and load forecasts [28]. The contributing factors on price signal can stem from operational as well as planning criteria. Reliability criteria can be embedded in unit commitment and economic dispatch problems such as security-constrained unit commitment [29] and security-constrained optimal power flow [30]. The locational marginal price (LMP) during 24 h is obtained by solving the above optimization problem and may be considered as a price signal. According to the definition in [31], time-sensitive pricing lies in the non-dispatchable class of demand response. A study on reliability standards and working group ancillary services performed by General Electric (GE) through the Hawaii Natural Energy Institute (HNEI) [16] pointed out that storage and DR should be allowed to provide primary frequency response and spinning reserve. The mentioned DR services are frequency response, regulation, spinning reserves, non-spinning reserves, and replacement reserves. Load commitment was addressed in [32] by proposing a framework to minimize the household payment. Since water heater storage and battery storage have significant impacts on the overall contribution of a house in DR programs, optimizing this portion of DR should receive special attention. A distributed system with a DR management scheme is presented in [33] to flatten the load curve while minimizing the customers’ electricity costs. A reliable communication infrastructure between house appliances and utility is necessary for implementing the proposed algorithm. DR scheduling in a deregulated environment is presented in [34], where a new market concept called DR exchange (DRX) is introduced. In this paper, battery, water heater, and air conditioning energy models are presented, and their optimal operations were obtained using a given price signal. Based on the results, heuristics for optimal operation are derived. Individual device performance optimization is compared with all of the devices as a single solution and obtained results are discussed. The presented optimization problem for house devices is solved using dynamic programming and mathematical programming and the results are compared.
2. Models of storage appliances and DR approaches 2.1. Battery model Batteries are used to store energy so that it can be transferred to the grid at different time intervals. Since there is little control over the generation of electrical energy from renewable resources, storage is used to add the flexibility of providing electrical power from intermittent resources. Batteries can be used at residences to store excess energy from solar panels during the day, and this stored energy can then be used to meet the load demand at night. They can also serve different ancillary services, such as the control of
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voltages and frequencies using active and reactive power. A utility or a demand response aggregator can send a signal to control the power flow of the battery. The price signal is a result of some planning and operation optimization given the available resources. In this section, the cost of operating a battery is optimized using the price signal from the utility or the dynamic response aggregator (DRA). The price signal is the echo of some of the reliability and operational optimization planning. For instance, the price signal can contribute to load leveling goal by which a utility shifts some of the peak load to period of times when the load is low. This price signal could be a reflection of a strategy to absorb fluctuations caused by PVs leading to a more reliable grid. The equations that govern the power flow and state of charge (SOC) of a battery are given in Eqs. (1) and (2):
Etþ1 ¼ Et þ P t Dt SOC tþ1 ¼ SOC t þ
ð1Þ P t Dt Etot
ð2Þ
The energy level of a battery at any given time is updated by the power flow at that time step. The state of charge of the battery is also calculated in a similar manner. The SOC of a battery can be taken from the experimental data of the battery, which would make the model more precise. The efficiency of charging/discharging can also be considered in calculating the SOC of a battery [35]. The battery has been modeled as Thevenin’s equivalent circuit model, where the voltage and impedance of the circuit models are obtained from a look-up table based on experimental data [36]. The model used in this paper ignores the nonlinearity of SOC change with respect to power, but other models can be easily used to find the SOC of a battery subject to power change. The power flow of a battery is constrained by the nominal power and the SOC of the battery, as shown in Eqs. (3) and (4):
Pt < min
ðSOC max Etot Et Þ ; Pnom Dt
ð3Þ
ðSOC min Etot Et Þ ; Pnom ; Dt
ð4Þ
Pt > max
where parameters Pnom, SOCt, Pt, and Et denote nominal power, state of charge, power flow, and energy of the battery at time t, respectively. The power flow of a battery should not exceed an amount that would violate the SOC limits of the battery, and it also should be in the range of its nominal power. The SOC of a battery should be in the range of [0.2–0.8] to increase the lifecycle of the BESS [37]. The initial SOC of a battery usually is considered to be 0.2, assuming the battery is completely discharged. However, the optimization problem can be solved from any point of time onwards with a given initial value of SOC. Since the shape of the load is repeated over a 24-h period, the optimization horizon is considered for the same period of time. The time step is given as 15 min in case the utility must redo the optimization based on an updated price signal. Contingencies or some unprecedented market prices of power may affect the planning of a portion of the capacity of battery storage of the utility or aggregator, thereby leading to an updated price signal. Since the aggregated storage of a residential battery can be a potential source of spinning reserve for the power system, the price signal of the battery can be influenced by the price of ancillary service in the power market. The objective function minimizes the cost of operating a battery during the 24-h-planning horizon subject to the constraints specified in Eqs. (3) and (4). The overall cost is formulated as a linear optimization problem with power flow levels as the decision variables:
X Min pðtÞcðtÞ t
ð5Þ
The price signal, c(t), is given as an array for the next 24 h and p(t) is the power flow of the battery. The dynamic programming and mathematical optimization methods that are used to determine the optimal solutions for Eq. (5) are presented in the next section. Other objectives for BESS can be also defined, Load peak shaving, power smoothing and frequency regulation are among those objectives [29,38]. If the objective of using BESS is to flatten the load curve, the following objective function can be used: N X 2 Min Lt þ Pt Pref
ð6Þ
t¼1
where Lt, Pt and Pref are the load, power flow of BESS and reference power curve at time t. The objective function uses the BESS power to follow the Pref power curve. In order to flatten the load curve, the reference power curve can be defined as a straight line. BESS constraints such as nominal power and SOC limits are considered with the objective function to find the optimal power flow of BESS at each time step. Load smoothing is another objective function of BESS which removes the fluctuation of the power curve usually caused by renewable energy intermittency. The capacity and nominal power of BESS should be chosen considering the maximum magnitude of load fluctuations. 2.1.1. Dynamic programming for BESS Dynamic programming is a planning tool that addresses decision making of an agent in a stochastic environment. Each agent determines the best course of action using the available environment information and by considering future stages. This tradeoff removes myopic attainments among a set of interactive actions and presents an efficient compromise between immediate costs and future costs. The mathematical model of dynamic programming is built around finite-discrete-time Markov decision processes. This process is characterized by several elements, i.e., states, environment, actions, and cost. An agent has a finite set of actions at each state. A certain cost is followed by taking specific actions at discrete times. The history of an agent’s interaction with the environment is summarized in the agent’s current state, which is comprised of one of the decision making elements and the cost associated with immediate action. The transition probability from state i to state j, denoted by pij, which depends solely on the current state i and the action that is taken, u, which is a Markov property, i.e.:
pij ¼ f ði; uÞ
ð7Þ
A discrete time dynamic system is defined as:
xkþ1 ¼ f k ðxk ; uk ; xk Þ;
k ¼ 0; 1; . . . ; N 1
ð8Þ
The state xk goes to state xk+1 following action uk e U(xk) with a probability of xk 2 Pðjxk ; uk Þ. The class of control laws makes up an admissible set of policies p ¼ fl0 ; . . . ; lN1 g where lk maps states xk into control laws uk = lk(xk). For all of the initial conditions, J 0 ð1Þ; . . . ; J 0 ðnÞ, the sequence Jk(i) proceeds backward in time from period N 1 to period 0 generated by the iteration:
( J kþ1 ðiÞ ¼ min
uk 2UðiÞ
gði; uÞ þ
) n X pij ðuÞJ k ðjÞ
ð9Þ
j¼1
In other words, Jk(i) gives the optimal cost starting from state i of a k -stage problem with cost per stage given by g and a special cost-free termination state. This problem is called the stochastic shortest-path problem, and its objective is to reach the terminal state with the minimum expected cost. The deterministic shortest path problem is a special case in which the transition probability is equal to 1 for each state-control pair (i, u). Therefore, if a control action, u, is applied at state i, it would move to state j with the
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probability of 1. The state transition probability will improve by observing the environmental behavior over time, which involves some sort of learning algorithm. This aim of learning mechanism is to drive the problem into the deterministic area, which, in turn, leads to additional optimum solutions. At each state of SOC in the battery, several charging and discharging actions with different magnitudes will occur, can be taken which and they will lead to their respective previous states, as shown in Fig. 1. The cost at each state is the sum of the immediate cost and the past experience of the agent, which is calculated by the Bellman DP equation, i.e., Eq. (8), in which the probabilities are all set to one due to the deterministic nature of storage. Using the given commands, the state propagation is calculated by Eq. (2), and the set of allowable actions at each state is limited by Eqs. (3) and (4). The immediate cost, g(i, u), depends on the cost of electricity at time t. After the values of all states are calculated for all time stages, the optimal trajectory is determined by following the optimal action from the state in the last stage with the minimum cost.
SOC opt ðNÞ ¼ arg min J N ðjÞ j ¼ 1; 2; . . . ; n SOC opt ðt 1Þ ¼ SOC opt ðtÞ þ
ð10Þ
Popt ðtÞDt Etot
ð11Þ
where SOCopt(t) and Popt(t) are the optimal SOC and power flow of BESS at time t. 2.1.2. Mathematical programming for BESS Mathematical optimization is used to determine the optimal solutions for Eq. (5) subject to the constraints in Eqs. (3) and (4), which can be specified as follows:
SOC min þ
P1 Dt P2 Dt Pk Dt þ þ ... þ 6 SOC max Etot Etot Etot
k ¼ 1; . . . ; n
ð12Þ
And the same relations can be written for the other part of the constraint in Eq. (12):
SOC min
P 1 Dt P 2 Dt P k Dt ... 6 SOC min Etot Etot Etot
k ¼ 1; . . . ; n ð13Þ
In all of these equations, the power flow of the battery is constrained by the nominal power:
Pnom 6 Pi 6 Pnom
minimization of the power flow of the battery weighted by the price coefficients. Other objective functions, such as peak shaving or power smoothing, also can be defined. 2.2. Water heater A water heater is modeled as a single heating element that heats up a nominal mass of water in a tank, as shown in Fig. 2 [22]. The goal of optimizing the operation of the water heater is to determine the optimal time steps for switching ON/OFF given the price signal. The objective function can be described as follows:
X Min xðtÞcðtÞ 0 6 xðtÞ 6 1;
ð15Þ
where x(t) is the decision variable for the water heater at time step t and c(t) is given price signal. The objective function is the same as battery storage with the constraint for keeping the water temperature in the comfort range, i.e., T min 6 TðtÞ 6 T max . The temperature of the water inside the tank is influenced by two factors, i.e., the volume of hot water removed, which is replaced by the same volume of inlet water, resulting in thermal loss. We assume that the temperature of the water is updated by the volume of water removed, and, then, the temperature of the water decreases based on the heat transfer equations. The water mixing temperature is given in Eq. (16) in which the volume of inlet water is determined from DHWcalc [39], which is a realistic domestic hot-water profile:
T n ðtÞ ¼
T curr Mcurr ðtÞ þ T inlet ðtÞ M inlet ðtÞ Mcurr ðtÞ þ Minlet ðtÞ
ð16Þ
The decrease in the temperature of the water is modeled with a resistor, and it changes based on Eq. (17) [40]:
M SHW
dT ¼ UAðT amb ðtÞ T n ðtÞÞ þ Q ; dt
TðtÞ ¼ T n ðtÞ þ dT
ð17Þ
where M = Mass of water in the tank (kg) SHW = Specific heat of water (J/kg/°C) T = Temperature of the water in the tank (°C) t = Time (h) UA = Standby heat loss coefficient times the cross sectional area of the storage tank (W/°C/s) Q = Rate of heat input to the tank from the heater (J/h). It is equal to zero when the heater is OFF.
ð14Þ
In other words, the battery can accumulate or discharge power at most at the nominal power. The objective function is the
The comfortable temperature range can be represented as follows:
T min 6 T n ðtÞ þ
Fig. 1. Actions and costs at each state.
xðtÞ 2 Z
t
UAðT amb ðtÞ T n ðtÞÞ þ xðtÞQ 6 T max M SHw
Fig. 2. Water heater model.
ð18Þ
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Model used in [41] uses ordinary differential equations (ODE) to find the temperature change which is caused by several factors such as electrical energy input, losses and energy transfer due to hot water draw. One node and two node water heater models are discussed in [42]. In one node model, it is assumed that water heater tank has a uniform temperature which is determined by first order differential equation similar to [20]. In two node model, it is assumed that cold and hot water inside the tank have different temperatures. This model is used to determine the height of the hot water inside the tank. The temperature in our model is considered to be uniform in the tank and changes according to (16). The current model can be enhanced by experimental data from the water heater which can be collected from the grid interactive water heaters. The relationship between the input power and temperature change, hot water draw pattern and cooling rate data can be used to tune the model or just make up a look up table. 2.2.1. Dynamic programming for WH The temperature of the water in the water heater evolves probabilistically based on the volume of hot water withdrawn from the water heater. The allowable set of actions consists of switching ON and switching OFF, while keeping the temperature in the comfortable range for the customer. A cost is incurred for the use of electricity once the agent takes an action. All of the states, actions, and costs are observed at discrete times. The temperature at each time has sufficient information for the agent to take an action in the stochastic environment. Therefore, the water heater’s past experience with the environment is summarized in the current temperature state realizing a Markov property. After laying the mathematical foundation for water heater planning problem acting as an agent, dynamic programming is utilized to provide the optimal sequence of actions in an elegant and principled manner. The value at each state is a probabilistic minimization obtained by (19), shown graphically in Fig. 3. The probabilistic nature of state changes arises from the stochastic volume of hot water withdrawn. After all the state values for all time stages are obtained, the optimal trajectory is found by following the optimal action at each time step moving backwards. The following equations present the optimal state at the final time and at other times.
T opt;N ¼ arg min J N ðjÞ j ¼ 1; 2; . . . ; n T opt ðt 1Þ ¼
X
pðj; uopt ðtÞÞ T opt ðtÞ t – N
j2T opt ðtÞ
ð19Þ ð20Þ
where T, J, u, p are temperature, cost, decision, and probability, respectively. 2.2.2. B2. Second solution for the optimal control of the water heater The water heater optimization problem is very similar to that of battery programming, with the variables being the only difference in decision making. In the case of programming a battery, the decision variable, which was the power flow of the battery, was a continuous variable. But, for the case of the water heater, the decision variables are integer variables of 0 and 1, which represent switching the water heater ON and OFF. Combining Eqs. (16) and (21), the constraints can be expressed as follows:
Tð2Þ ¼ Tð1Þð1 þ kð1Þ cÞ þ að1Þ þ bð1Þ cað1Þ þ xð1Þxð1Þ 6 T max ð21Þ Tð3Þ ¼ ½Tð1Þð1 þ kð1Þ cÞ þ að1Þ þ bð1Þ cað1Þ þ xxð1Þ½1 þ kð2Þ c það2Þ þ bð2Þ cað2Þ þ xxð2Þ 6 T max ð22Þ In terms of decision variables, these equations change to:
1
xð1Þ 6
x
ðT max Tð1Þð1 þ kð1Þ cÞ að1Þ bð1Þ þ cað1ÞÞ
ð23Þ
xð1 þ kð2Þ cÞxð1Þ þ xxð2Þ 6 T max ð1 þ kð2Þ cÞðTð1Þð1 þ kð1Þ cÞ ð24Þ þ að1Þ þ bð1Þ cað1ÞÞ að2Þ bð2Þ þ cað2Þ and
aðiÞ ¼
M in ðiÞT in ðiÞ M c ðiÞ þ M in ðiÞ
ð25Þ
bðiÞ ¼
UA T amb ðiÞ M SHw
ð26Þ
c¼
UA M SHw
kðiÞ ¼
x¼
M c ðiÞ Mc ðiÞ þ M in ðiÞ
Q M SHw
ð27Þ
ð28Þ
ð29Þ
where Mc is mass of water left in tank (kg) Min is mass of water from cold water inlet (kg) Tin is the inlet temperature of the cold water (kg) Tamb is ambient temperature (°C). The same constraints can be written for the minimum comfort temperature. The constraint can be reformulated as a linear matrix with the objective function or as a linear weighted sum of decision variables. The problem can be solved via integer linear programming or evolutionary algorithms. 2.3. AC
Fig. 3. Value of each state in water heater planning problem.
Air conditioning systems consume a significant portion of the total power used in the residential and commercial sectors. AC units keep the room temperature within a comfortable temperature range, and they can be adjusted manually or remotely. In this work, comfortable temperature interval for AC is [18.8 °C, 25 °C], and initial indoor temperature is set to be 29.5 °C. This comfortable temperature interval is similar to other researcher’s defined comfort period [43,44]. It is desired to minimize the cost of electricity
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given a price signal. The mathematical model of the thermal behavior of a room, as taken from [45], is given as follows:
T i ðtÞ ¼
C1 C1 T i1 ðti1 Þ xi1 eC 2 :ðtti1 Þ C2 C2
T 0 ðt 0 Þ ¼ T 0
ð30Þ ð31Þ
Tðt; ðxi ; t i Þ; T 0 Þ ¼ Tg i ðtÞ
t i1 6 t 6 ti
ð32Þ
The external thermal source is modeled as a thermal pulse train with different magnitudes that is defined below:
WðtÞ ¼
m X
xi dðt ti Þ
ð33Þ
i¼1
In Eq. (30), C1 and C2 are constants that were derived in [46]. Also, d(t) is the Dirac delta function, and xi is the level of thermal intensity entering the room at time t. The objective function is similar to that of the water heater, minimizing a linear cost throughout a time horizon. The constraints also are similar, keeping the temperature of the room in the comfortable range, as defined by the user. Validity of the model can be improved by real time measurement of room temperature and finding the relationship between the parameters of AC model and temperature change. The uncertainty of temperature noise can be reduced using a probabilistic function which is tuned by collecting temperature measurement. 2.3.1. AC optimal control solutions: Dynamic programming In the optimal control of AC, the temperature evolves deterministically with the decision variables. The uncertainty in the change in the temperature is modeled with the thermal pulse train. Therefore, the optimal trajectory of the temperature and the optimal decision variables are obtained using deterministic dynamic programming. The thermal noise associated with the outdoor temperature also can be modeled probabilistically. In this state, the resultant temperature state can be determined using a relationship similar to that in Eq. (20).
power flow of the battery given the price function shown in Fig. 4 [47]. The optimal SOC trajectory of linear and dynamic programming is shown in Fig. 5, which shows that the two methods generated almost the same trajectory, with DP giving a slightly better result. Since the optimal SOC had some fluctuations, it was smoothed to increase the battery’s storage lifetime. The power flow is sensitive to the local minima in the price signal. When there is a valley in the price signal, the battery charges until it reaches the local maximum, and, later, it discharges the stored energy to minimize the overall cost. Fig. 6 shows the power flow that resulted from the optimal SOC. Since the fluctuations in the trajectory of the SOC may affect the battery’s lifetime, a smoothing spline was applied to the SOC’s trajectory shown in Fig. 7. The smoothed SOC trajectory s(x) is obtained from minimizing (19) given smoothing parameter p, the weights wi and the input data x and y:
Z X p wi ðyi sðxi ÞÞ2 þ ð1 pÞ i
2
d s 2
dx
!2 dx
The smoothing parameter and the weights are assumed to be 0.099 and 1, respectively. The smoothing is performed to reduce the stress on the battery and increase its lifetime. The price signal, as well as the decision variables, has been concatenated 10 times to increase the search space. The obtained results are the same as before, showing a slight improvement of DP over mathematical programming. 3.2. Water heater An electric water heater with a 50-gallon capacity and a power requirement of 4.5 kW was used for the HPWH test case. This
2.4. Combined DR device optimization In this section, we discuss house level simultaneous optimization of water heater, AC and battery storage assuming that the house price signal is given. The objective function is to minimize the overall cost of electricity usage during 24 h ahead. The objective function is the sum of the individual devices costs shown below:
XX Min xði; tÞcðtÞ i
ð34Þ
t
The constraints are the same for each individual device. This optimization problem is mixed integer nonlinear programming. Since the states of the problem are of different types, dynamic programming cannot be used to find the shortest cost path. Hence, mathematical optimization method is sued to find the optimal decision variables.
Fig. 4. Price signal.
3. Simulation and results 3.1. Battery A Powerwall battery specification was used as the test case for the optimal control. The battery was a Li-ion, wallmounted battery that had a capacity of 6.4 kW h and a nominal power of 3.3 kW [46]. Dynamic programming was used to determine the optimal SOC state trajectory using Eqs. (9) and (10). Linear programming also was used to produce the optimal
ð35Þ
Fig. 5. Optimal SOC trajectory obtained by DP and LP.
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tion, while LP fell to local minima. The results in Fig. 7(b) and (c) show that the water heater was switched ON during the lowest points of the price signal while keeping the temperature in the comfort range. If the volume of hot water removed from the water heater is high, the water heater is switched ON during other time intervals to satisfy the constraints. 3.3. AC
Fig. 6. Optimal power obtained by DP.
The solutions for AC with DP and with LP overlap and give the optimal result while satisfying the constraints. In Fig. 8, the outdoor temperate is assumed to be around 29 °C with the thermal noise shown in the following subplot. AC keeps the temperature in the allowed range by switching ON in the local minima of the price signal. The outdoor temperature has been increased 5° in Fig. 9, which has doubled the frequency of AC on states. In the next figure, Fig. 10, the outdoor temperature is 29 °C while the input thermal noise has been doubled. The amount of input thermal noise during noon time increases which results in AC to switch ON more often during that time. Seasonal temperature change could affect this comfort zone and its range could vary. It could be applied as constrain changes to the optimization problem, as well. 3.4. Combined device simultaneous optimization
Fig. 7. (a) Hot water draw profile, (b) optimal switching of water heater and (c) temperature change of water heater following the optimal switching dictions points.
capacity of the water heater is good enough for a household with a size of 3–5 people, assuming their appliances typical flow rate is within the normal range [48]. The problem with optimizing the operation of the water heater was it had a smaller search scope because the decision variables were integers. A sample hot water draw profile is given in Fig. 8(a), and the corresponding optimal decision variables are shown in Fig. 7(b). Fig. 7(c) shows the temperature of the water heater as this test was being conducted. The water heater was turned on at the minimum value points in the cost curve while keeping the temperature of the water in the comfort zone, the temperature of which was assumed to be between 37.77 °C and 60 °C. The DP and LP gave the same results, but, when the search space was larger, DP gave the optimum solu-
The result of simultaneous optimization of DR devices is shown in Fig. 11. The AC has been switched ON one fewer times while the water heater has been switched ON more. The overall cost of optimization considering home as a single unit is more than the sum of the costs of optimizing individual unit operations. However, as the optimization of each DR unit is independent, considering each device separately yields better results compared to when all of the devices are considered as a single solution. Therefore, it is desirable to optimize the DR devices individually. DRA needs to consider the impact of DR devices operations on the power grid. In other words, if all the houses connected to a same distribution grid transformer receive a same price signal, they will draw power at the valley of the price signal. On the other hand, the aggregate power flowing through the transformer should not exceed the rating. For example, a 25 kV A transformer current connected to a single phase 120 V feeder should not exceed 70 A. If the transformer is providing power to four houses, each house should not draw more than 17.5 A. Assuming the breaker cutoff current setting to be 15 A, then the transformer is already protected by the breaker fuses in the houses. Therefore, installing
Fig. 8. (a) Optimal switching of AC assuming the outdoor temperate is 29 °C, (b) thermal noise and (c) room temperature.
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Fig. 9. (a) Optimal switching of AC assuming the outdoor temperate is 34 °C, (b) thermal noise and (c) room temperature.
Fig. 10. (a) Optimal switching of AC assuming the outdoor temperate is 29 °C and doubled temperature noise compare to Fig. 8, (b) thermal noise and (c) room temperature.
Fig. 11. Combined device simultaneous optimization: (a) BESS SOC, (b) water heater and (c) air conditioner.
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gramming gives the optimal solution while the formulation and implementation takes more effort compared with using available mixed-integer linear/nonlinear solution packages. Moreover, it is shown that individual optimization of devices yields better results than considering all of them as a single solution. Since the DR devices might draw power at the valley of price signal simultaneously, installing more batteries for DR purpose should consider the rating of the house breaker as well as distribution transformer rating. Heuristics are found for optimal performance of DR devices in case devices act autonomously without communication. The
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