1. Introduction
To prevent global warming and the exhaustion of fossil fuels, the introduction of renewable energies such as the photovoltaic generation (PV) and wind energy generation (WG), has been gaining attention in power systems. Moreover, an energy storage system (ESS) such as a battery is expected to play a significant role in stabilization of power flow fluctuation caused by the varying power output of the renewable generators due to unstable weather conditions [
1,
2]. Recently, in the residential side, the introduction of PV and battery (either electric vehicle or small fixed type) system is gaining attention, since the PV and batteries can contribute to reduction of peak load. In order to reduce peak load and electricity cost in demand side, an optimal scheduling of household appliances is required and have been investigated [
3,
4,
5,
6]. Where, it is very important to consider uncertainties of PV output in the dayahead scheduling, because the actual PV output often deviates from forecasted value.
In order to cope with the uncertainties imposed by the forecasting error of PV output, a stochastic optimization programming such as a scenariobased approach has been studied and used to solve the scheduling optimization. The scenariobased approach can deal with the forecast error deriving various scenarios based on a certain probability density function and can minimize the expected cost under certain conditions [
7,
8,
9,
10]. The literature [
9] provides a stochastic model for an optimal scheduling of REs and thermal units in micro grids to maximize the expected value of profit in electrical market. In the paper, the uncertainties of PV output are considered using scenariobased approach in which the scenarios of PV is generated on the basis of the probability distribution function of forecasted error. In Reference [
10], a stochastic programming model for an optimal scheduling of distributed energy resources such as ESS with PV system is presented. The paper is taking into account the uncertainties for PV output and presents the setting of the probability of solar irradiance in detail. The literature [
7,
8,
9] addresses the scenariobased method’s problem which the calculation time increases as the number of scenarios increase. This problem has been improved by a countermeasure such as reduction of scenario [
7,
8,
9].
Another approach that focuses on a forecasting method for uncertain REs output, is proposed in References [
11,
12,
13]. Reference [
12] describes the significance of the accuracy of the forecasted value using the reforecasting model in order to maximize the benefit in electricity market. Reference [
13] presents dispatch scheduling of generators and ESS where the probabilistic forecast method is used in order to deal with the uncertain REs output. Additionally, the data used to forecast is updated every hour and the forecast accuracy can be improved.
From above background, the uncertainties caused by the forecasted error of PV output should be considered in the scheduling of controllable appliances such as batteries in residential side. Moreover, the consideration of reforecast is necessary to improve forecast accuracy as well. In our previous work [
14], we proposed an optimal scheduling method by applying the scenariobased approach in a smart home, however reforecast and replan has not been considered in the optimization process in the paper. It is expected that operational cost can be even reduced using the latest data which are atmosphere, humidity, and so forth, and needed to forecast PV output. These factors are generally updated every several hours [
11].
The objectives of this paper are to minimize the expected operational cost for the resident using controllable loads such as batteries and electric vehicle (EV) installed in a smart home while also considering uncertain PV output. In this paper, it is assumed that the power company provides a suitable command value for power flow in the smart home. Then, the optimum schedule of controllable loads such as batteries is solved with using tabu search (TS) algorithm considering uncertainties for finding charge or discharge control of a small fixed battery and EV systems according to the condition of the PV generation power, where the PV power output is reforecasted every several hours by the neural network (NN) [
15] forecasting method. In addition, the scheduling in the optimization process is replanned every several hours according to reforecasted PV output. An additional purpose of this paper is to reveal how much the expected operational cost can be reduced by introducing reforecast and replan in addition to consideration of the uncertainties.
This paper provides a robust optimal scheduling decision method of controllable loads which are battery, EV and heat pump (HP) in a smart home, which can address the uncertainties by applying scenariobased approach in a stochastic optimization. Additionally, reforecast and replan are considered in the optimization process to further decrease the expected operational cost. In order to verify the effectiveness of the proposed method, the results obtained in four different cases are compared and discussed in the simulation results. Finally, the optimal schedules obtained in the four cases are tested for the PV output involving uncertainties in order to confirm the robustness. This is done using Monte Carlo simulation for the various scenarios. The comparison of simulation results obtained in four cases is stated and examined. The statistical analysis mentions the significance of consideration of the uncertainties, reforecast and replan in the scheduling of the controllable loads, and it implied that the capacity of batteries can be reduced by the proposed optimization method in the statistical analysis. The effectiveness of the proposed method is validated with using MATLAB${}^{\circledR}$.
The rest of the paper is organized as follows.
Section 2 describes the power system model involved in the smart home.
Section 3 presents a formulation of the optimization problem with the TS algorithm, including the uncertainties.
Section 4 discusses simulation results which are derived from the optimization method. Furthermore, the results are statistically confirmed to verify the effectiveness of the optimal solution by using Monte Carlo techniques.
Section 5 presents the conclusion and future research avenue.
3. Formulation of Optimization Problem
In this section, an optimal scheduling decision of controllable loads in the smart home is described. The objective function involving the constraint conditions is explained in
Section 3.1, the optimization methodology is described in
Section 3.2. The TS is explained in
Section 3.3, the more detailed optimization procedure with the TS is described in
Section 3.4 and the insolation forecasting method is explained in
Section 3.5.
3.1. SetUp of Objective Function
The objective of this research is to minimize the expected operational cost in a day for the uncertainties of the forecasted error. The objective function and constraints are explained as follow.
The objective function (4) consists of two indexes. The first term
${E}_{ELE}$ represents conventional electricity payment which increases in proportion to energy usage amount, the electricity payment is obtained by multiplying the energy usage amount in a day (kWh) by the unit price
${C}_{t}$. In this paper, we set the unit price referencing one used in Tokyo Electric Power Company (TEPCO) [
21]. The second term
${E}_{DEV}$ represents the penalty cost which increases according to quantity of deviation power, which have been proposed in the literature [
1,
3] by our previous work and it has been verified that suppression of power flow can be reduced by the electricity price system.
The penalty cost is decreased if the power flow
${P}_{It}$ in the smart home lies within a bandwidth of a commanded power flow which power company sets. In this paper, it is assumed that the commanded power flow from the power company to residential side has been determined throughout electricity market between power generation side and retail one such as aggregator and the power company gives the determined power flow to the residential side. The penalty cost is obtained by multiplying the deviation power (kW) by the unit price
${D}_{t}$, where the
${D}_{t}$ is varying according to the difference between commanded power flow and actual power flow in the smart home as depicted in
Figure 2. For example, the penalty cost
${E}_{DEV}$ is calculated as follows. If the power flow exists within “region A”, the penalty cost
${E}_{DEV}$ is calculated by multiplying the deviation power flow (kW) by the unit price
${D}_{t}$ = 10 (Yen/kW). Besides, we set the constant unit price of sold power to 20 (Yen/kWh). The proposed electricity price is shown in
Table 2.
The forecast error of PV output is considered in the optimization process in this paper. We consider a number of possible scenarios for the uncertainties of PV output by applying the scenariobased approach where the scenario is produced by adding a forecast error based on normal distribution. Thus, the actual power flow changes in scenario s because of the forecast error and the actual power flow becomes ${P}_{It}+\Delta {P}_{It}^{s}$. Hence, the smallest expected cost obtained as a result of the optimization process indicates that the solution of scheduling of the controllable loads such as batteries is effectual for all scenarios.
Constraints conditions in solving the optimization problem to reveal the optimal scheduling of the controllable loads, are as following. Equations (5) and (6) mean inverter capacity constraints for the fixed battery and EV, respectively, these are
${P}_{Bmax}=1$ kW and
${P}_{EVmax}=3$ kW. Equations (7) and (8) indicate state of charge constraints for the battery and EV respectively where
${C}_{Bmin}=20$%,
${C}_{EVmin}=20$%,
${C}_{Bmax}=100$% and
${C}_{EVmax}=100$%, respectively. Equation (
9) indicates constraint for energy of battery in order to use at next day. Equation (
10) indicates constraint for remaining energy of EV in preparation to be used outside and it is set to be higher than 90% of capacity of EV.
Constraints:
where,
t:  Index for time (20 min time step) 
T:  Total schedule hours (T = 24 in this paper) 
s:  Individual scenario (S = 100) 
${P}^{s}$:  Probability in scenario s 
${E}_{day}$:  Expected operational cost in a day 
${E}_{ELE}$:  Expected electric charge amount in a day 
${E}_{DEV}$:  Expected penalty charge amount in a day 
${C}_{t}$:  Unit price on electric charge 
${D}_{t}$:  Unit price on penalty charge 
${B}_{It}$:  Command value of power flow to smart home 
${P}_{It}$:  Power flow from power system to smart home 
$\Delta {P}_{It}$:  Variation of power flow caused by forecasted error 
${P}_{{B}_{t}}$:  Charge/discharge power of fixed battery 
${P}_{E{V}_{t}}$:  Charge/discharge power of EV 
${P}_{Bmax}$:  Maximum of charge/discharge power in fixed battery 
${P}_{EVmax}$:  Maximum of charge/discharge power in EV 
${C}_{{B}_{t}}$:  State of charge of fixed battery in hour t 
${C}_{E{V}_{t}}$:  State of charge of EV in hour t 
${C}_{Bmax}$:  Maximum value of fixed battery capacity 
${C}_{EVmax}$:  Maximum value of EV capacity 
${C}_{Bmin}$:  Minimum value of fixed battery capacity 
${C}_{EVmin}$:  Minimum value of EV capacity 
3.2. Optimization Methodology
In this paper, a meta heuristic optimization technique is employed utilizing the TS [
22,
23] in order to achieve the optimal solution based on objective function (4) satisfying the constraints conditions (5)∼(10). The TS is widely employed for solving the optimization problem such as scheduling problems, with relatively shorter calculation time than the genetic algorithm (GA) [
24,
25,
26]. In this work, the simultaneous reforecast for PV output and plan controllable loads are executed every 3 h; the optimization problem can be effectively solved by the TS within a short period. Also, uncertainties of PV output are considered using a scenariobased approach which is widely employed for solving the scheduling problem and expansion planning problem of transmission network including uncertainties in power system [
7,
8,
9,
27,
28]. The PV output can be expressed in several scenarios where each scenario is derived by adding the forecast error based on normal distribution into forecasted value in the optimization process. The solutions are operated for the possible scenarios and evaluate with expected value in the iterative step of TS optimization. Furthermore, the TS algorithm and optimization procedure are described in
Section 3.3 and
Section 3.4, respectively.
After the optimization using TS, the optimal scheduling obtained from the optimization is operated for the real PV output, including uncertainties. This obtained simulation result is described in
Section 4. In order to verify the usefulness of the proposed scheme, the scheduling of controllable loads obtained by the proposed method is tested by Monte Carlo simulation where the uncertainty of PV output expressed as possible many scenarios which are derived based on normal probability distribution.
3.3. Tabu Search
The TS is one of a meta heuristic global optimization method and it is discovered by Glover [
29]. The TS has been effectively utilized for a combined optimization problem such a scheduling one. The problem to minimize
$f\left(x\right)$ can be formulated as bellow where
x indicates the optimal solution to minimize the function
$f\left(x\right)$ [
22,
30].
The Equation (
11) indicates that minimization of the function of
$f\left(x\right)$ provided that constraint (12) is satisfied. As first step in the TS, neighborhood solutions
${x}_{i}^{*}$ which is slightly moved from a present solution
${x}_{pre}$, are produced.
${x}_{i}^{*}$ which is the best solution in the neighborhood solutions, is picked out and next neighborhood solution
${x}_{i+1}^{*}$ is derived from the best solution in previous time. TS can arrive at the optimal solution by executing the iteration step until a specific criterion is satisfied. Tabu list which has function of memory system is used in order to prevent same loop which is often arising from simple iteration. The use of the tabu list which the latest moves are recorded, can decrease the possibility of remaining unsolved in a local loop. The highest evaluated solution checked in the tabu list in the neighborhood ones, is chosen as the next solution
${x}_{nex}$.
The implementation parameters which are employed for the TS, are as follows. The maximum value of global iteration is 2000 and the number of the tabu list is 500 and the considered number of scenarios is 100. The TS algorithm is embedded into the program to solve the optimization problem. The simulation is implemented on a desktop computer with a 2.20GHz Intel(R) Xeon(R) E52660 processor with 128 GB RAM using MATLAB (R2018b).
The detail procedure of the optimization for determining the optimal scheduling of batteries and HP using the TS algorithm is described in
Section 3.4.
3.4. Optimization Procedure
In this subsection, the procedure of the proposed optimization method to determine the optimal schedule of controllable loads, is explained. In this paper, it is assumed that power consumption except for controllable loads and heat load which the residential people use can be forecasted. Thus, the variable number to be solved, are charge/discharge power of the fixed battery, charge/discharge of EV and operation starting time of the HP. The optimization step with using TS is as follows.
 Step 1
The initial values for charge/discharge power of the battery and EV are set in addition to the operation starting time of the HP.
 Step 2
The neighborhood solutions which is slightly moved from initial solution or the chosen solution in a previous iteration, are produced and evaluated.
 Step 3
The neighborhood solutions are evaluated in the Equation (
4), the best neighborhood solution which is not recorded is chosen and recorded in the tabu list. If the best solution already lies in the tabu list, the next best solution in the neighborhood ones is selected. Even if the selected solution is worse than a solution selected in the previous iteration, its registration to the tabu list is executed. Old recorded solutions are overwritten with the new chosen one in turns and new solutions are recorded.
 Step 4
If the chosen solution is better than the solution which is previously reserved as the optimal solution, the chosen solution is reserved as the optimal one.
 Step 5
If number of the global iteration achieves criteria one which was set in advance, the search process is finished. Otherwise, the algorithm proceeds to Step 6.
 Step 6
For the best solution obtained in Step 3, the process goes to Step 2 where the neighborhood solutions are derived from the best one again.
where, the pseudocode of the optimization algorithm using TS is as follows Algorithm 1 [
22].
Algorithm 1 Optimization algorithm using TS 
$x={x}_{0}$; % Set of initial solution forj = 1 to ${N}_{S}$ do Generate a set A which is a neighborhood solutions from x or previous one ${x}_{ne{i}_{N}}$; Find best neighborhood solution ${x}^{{}^{\prime}}$ of A; % check of constraints conditions, the solutions is not recorded in the tabu list; $x={x}^{{}^{\prime}}$; Update tabu list; if $f\left(x\right)<f\left({x}_{BEST}\right)$ then ${x}_{BEST}=x$ ${x}_{ne{i}_{N+1}}=x$ end if end for ${x}_{BEST}$ is the bestfound solution.

3.5. Insolation Forecasting Method
A NN is widely employed as one of the forecasted methods for insolation [
31,
32,
33]. In this paper, an NN is used for the insolation forecasting method. Parameters entered into the NN toolbox in MATLAB
^{®} are depicted in
Table 3. Input factors such as a data of atmospheric, temperature, humidity and hours of sunlight for the previous 24 h are entered into the NN. It can then forecast insolation for the proceeding 24 h. Furthermore, reforecast and replan are conducted in the optimization for the scheduling of controllable loads. The data fed into the NN is updated every 3 h. The overview of reforecast and updating data is shown in
Figure 3. Learning (in which optimize synaptic weight is optimized) is carried out for 365 days. After learning of the NN, the weights are applied for another year in order to confirm the availability of the optimized synaptic weight. A probability density of the forecast error of insolation for the year is depicted in
Figure 4. In addition to this, a probability density for the case of updating data is depicted in same figure. It is observed that the forecast error can be reduced by updating the data such as atmospheric and temperature which are used to forecast.
The whole flow chart of optimization including the forecasting with the NN and consideration of the uncertainties is depicted in
Figure 5. At first, PV output is obtained in forecasting the insolation with NN. Next, the forecasted error based on
Figure 6 is added into the forecasted PV output. In this paper, scenariobased approach is applied for the uncertainties. A number of scenarios are derived by adding forecast error which is expressed as shown in
Figure 6. In process step 3∼step 6, the neighborhood solutions are tried to operate for all scenarios and evaluated by the objective function Equation (
4) until the global iterations are satisfied. In this paper, reforecast and replan are considered. Thus, the schedule once planned is renewed every 3 h by using the latest data in the optimization process with the TS algorithm as shown in step 7. Hence, the reforecast and replan are executed for 8 times in a day.
5. Conclusions
This paper proposes an optimal scheduling method of controllable loads considering uncertainties, reforecast and replan. A fixed battery, EV and HP are utilized as controllable loads in order to control the power flow of interconnected point in the smart home. The optimization of determining the optimal scheduling of controllable loads is conducted by using TS. Moreover, to solve the optimization problem involving uncertainties caused by the forecasted error, a scenariobased approach is applied to the proposed scheme. The optimal scheduling is obtained from the optimization and it is observed that the power flow in the smart house can be controlled within the given bandwidth from the power company. Furthermore, to confirm the robustness of the proposed scheme, 1000 scenarios are tested, including uncertainties of PV output using the Monte Carlo simulation. The statistical analysis shows that the expected operational cost can be reduced by applying the proposed optimization method considering the uncertainties, reforecast and replan. Furthermore, the statistical analysis shows that the frequency of the state of charge on that battery considering the uncertainties, reforecast and replan exists at almost 40∼60% from 10:00 to 24:00 although the frequency of that without considering uncertainties, reforecast and replan exists at almost 80% in the same period. The statistical analysis indicates that there is the possibility of battery capacity reduction by introducing the proposed method.