FINANCE , STATISTICS, FORECASTING. ACCURACY TESTING AND ERROR MATRICES

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Energy and Buildings 158 (2018) 1632–1639

Contents lists available at ScienceDirect

Energy and Buildings

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n b u i l d

tochastic receding horizon control minimizing mean-variance with emand forecasting for home EMSs

kira Yoshida a,∗, Jun Yoshikawa b, Yu Fujimoto c, Yoshiharu Amano a,d, Yasuhiro Hayashi e

Department of Applied Mechanics and Aerospace Engineering, Waseda University, 17 Kikui-cho, Shinju-ku, 162-0044, Tokyo, Japan Department of Applied Mechanics, Waseda University, 17 Kikui-cho, Shinju-ku, 162-0044, Tokyo, Japan Advanced Collaborative Research Organization for Smart Society, Waseda University, 1-4-3 Okubo, Shinjuku-ku, 169-8555, Tokyo, Japan Research Institute for Science and Engineering, Waseda University, 17 Kikui-cho, Shinjuku-ku, 162-0044, Tokyo, Japan Department of Electrical Engineering and Bioscience, Waseda University, 1-4-3 Okubo, Shinjuku-ku, 169-8555, Tokyo, Japan

r t i c l e i n f o

rticle history: eceived 31 January 2017 eceived in revised form 5 October 2017 ccepted 25 November 2017 vailable online 29 November 2017

a b s t r a c t

As demand-side energy management system (EMS) has not only a function to maximize owner’s utility by automatic control but also a potential capability to respond a demand activation, demand-side EMS is expected to fulfill the key role for intelligent control of community energy management. To achieve the stable control result of EMS against the uncertain environment, it needs approach from both forecast and operation sides. The objective of this study is to construct a control scheme for a home EMS from a

eywords: nergy management system tochastic receding horizon control ean-variance model

IT modeling uel cell

practicable implementation perspective and to evaluate the control performance of the scheme proposed. The scheme is composed of operational planning with receding horizon and energy demand forecasting. The goal of the scheme is to maximize householder’s utility only by retrofitting the home EMS to a residential household. Throughout numerical results, it is revealed that the better control performance comes from the conservative operational strategy derived from the multiple scenarios.

© 2017 Elsevier B.V. All rights reserved.

. Introduction

Demand-side energy management is an attractive concept to chieve intelligent control for communicty energy system. Home nergy Management System (HEMS) and Building Energy Man- gement System (BEMS) have not only a function to maximize wner’s utility by automatic control but also a potential capabil- ty to respond a demand activation, for example, shifting energy emand. These EMSs centralize supervised control of each control-

able device. Supervised control of a device requires operational trategy, e.g., a set-point sequence for the device, due to the degree f freedom of it. For example, energy storage devices, e.g., hot ater tank and electrical battery (BT), need the plan concern-

ng charge/discharge timing in advance, and some devices should

ot switch on/off status frequently from machine protection per- pective. An operational strategy is derived from an operational lanning problem as dispatching problem. The operational plan-

∗ Corresponding author. E-mail addresses: [email protected] (A. Yoshida),

[email protected] (J. Yoshikawa), [email protected] Y. Fujimoto), [email protected] (Y. Amano), [email protected] (Y. Hayashi).

ttps://doi.org/10.1016/j.enbuild.2017.11.064 378-7788/© 2017 Elsevier B.V. All rights reserved.

ning problem decides not only optimal scheduling but also optimal energy supply path, from the massive number of alternative oper- ational strategies. Alternatives are caused from that the energy system for the problem could have some kind of devices to meet energy demand. Potential device candidates to install into resi- dential household are, for example, photovoltaic device (PV), fuel cell cogeneration unit (FC-CGU), gas-fired water heater (GH), heat pump water heater, thermal energy storage (TES), electrical bat- tery (BT), electric vehicle (EV), the heating, ventilation, and cooling (HVAC) unit such as room-air conditioner (AC), and so on.

The contributions in EMS research is divided into follows: modeling device, analyzing/forecasting time-series such as energy demand, constructing an operational strategy, implementing con- trol logic, analyzing thermal environment of dwelling, designing system and so on. Barbato et al. [1], Vardakas et al. [2], and Lee et al. [3] surveyed overall EMS research.

In [4–6], modeling devices are reported. Tischer et al. [4] mod- eled new generation residential devices such as PV, BT, EV, FC-CGU, as well as TES. They demonstrated an EMS scheme formulated as

dynamic programming (DP). Di Somma et al. [5] showed exergy modeling and optimal operational strategy for a combined cool- ing/heating and power system, biomass boiler, and solar plant using linear programming (LP). Yoshida et al. [6] modeled a PEM type

A. Yoshida et al. / Energy and Build

Nomenclature

a Slope b Intercept c Coefficient · E Electricity f Coefficient i Device index I Number of device j Primary energy consumption k Number of scenarios l Coefficient M Number of samples in dataset P Event probability ·

Q Heat flow rate s,S Amount of stored energy t Time index t Time index vector T Horizon u,U Input and output energy flow x Input vector y Output vector

Greek Symbols � Risk aversion parameter ı Binary variable � Standard deviation D Database

Subscripts and Superscripts D Energy demand k Scenario index K Number of scenarios i Device index I Number of device m Dataset index p Planning horizon q Query t Time index u Control horizon + Input − Output ·̂ Forecasted value

f s

r l s s d c g c a a b r [ l m

·̃ Operational strategy * Realized value

uel cell cogeneration unit(PEFC-CGU) composed of four statuses; tarting up, load following, steady state and shutting down.

In [7–13], analyzing/forecasting time-series methods are eported. Graditi et al. [7] modeled residential shiftable electrical oad using a Gaussian function. They also proposed a method for olving distributed dispatch problems by extending a glowworm warm optimization algorithm. For the classification of energy emand times-series, Yoshida et al. [8] proposed a hierarchical lustering technique using the dissimilarity measure derived by eneralized Kullback-Leibler divergence. Kanchev et al. [9] dis- ussed day-ahead power scheduling based on PV power predictions nd load forecasting against a microgrid including PV, gas turbine, nd BT. Tascikaraoglu et al. [10] discussed energy management ased on PV output forecasting derived from an artificial neu-

al network approach. For estimation of PV output, Graditi et al. 11] proposed methods using physical medialization and multi- ayer perceptron. Monteiro et al. [12] proposed a power forecasting

ethod based on historical datasets for spot values. Fujimoto

ings 158 (2018) 1632–1639 1633

et al. [13] proposed an energy demand forecasting method, which uses combined metric learning and K nearest neighbors (K-NNs) method.

In [4–10,14–18], constructing operational strategy methods are reported. Siano et al. [14] proposed a decision support method for EMS using a finite state machine. Keerthisinghe et al. [15] analyzed a HEMS composed of stochastic mixed integer linear programming (MILP) problem, and DP approaches. The HEMS managed PV, FC- CGU, BT, EV, and TES. To tackle the non-convexity issue, Kocuk et al. [16] proposed a mixed integer second-order cone program- ming (MISOCP) relaxation for non-convex power flow equation. Bai et al. [17] proposed a conic approximation of alternating current power flow equation in non-convex mixed integer nonlinear pro- gramming (MINLP). Macedo et al. [18] proposed a MISOCP model converted from mixed integer quadratic programming (MIQP) for a dispatching problem.

In [19–23], implementing control logic are reported. Graditi et al. [19] evaluated a control logic for shifting electricity and thermal demand. As control scheme should be switched from open- loop control to closed-loop control when forecasting is not enough high accuracy, Hokayem et al. [20] presented stochastic receding horizon control (SRHC) with state estimation of the Kalman filter. Oldewurtel et al. [21] discussed stochastic model predictive control (SMPC) involving comfort bounds as chance constraints to control building climate. Cariano et al. [22] proposed SMPC with Markov chain learning, that represents driver behavior, for energy man- agement of hybrid EVs. Ma et al. [23] presented SMPC minimizing expected energy costs for HVAC systems.

Angrisani et al. [24] and Anvari-Moghaddam et al. [25] reported the analyzing thermal environment of the dwelling. The latter mentioned an operational strategy for HEMS that considered the thermal comfort level for people by Predicted Mean Vote (PMV) [26].

In [27,28], designing systems are reported. Yokoyama et al. [27] proposed a robust optimal design method using minimax regret for energy supply system. Fuss et al. [28] proposed a Markowitz mean-variance model for portfolio problem to minimize risk.

As communication standard, e.g., SEP2.0 [29] and ECHONET Lite (EL) [30], has been established, HEMS and devices could communi- cate on/off signal and device status and so on, via them. Therefore, what is needed is operational strategy planned in reasonable calcu- lation time and control implementation for executing the strategy.

Although there is a broad range of theoretical studies concern- ing EMS element technologies, there is room for a development that how to integrate element technologies. Especially, to achieve the stable control result against the uncertain environment, it needs approach from both forecast and operation sides. This paper focuses on optimal energy management over a whole residential house- hold. Issues that we tackle in this article is to take into account the uncertainty of future events, to compensate forecast error through- out robust operational strategy, and to obtain operational strategy in reasonable calculation time.

The objective of this study is to construct a control scheme for a HEMS from a practicable implementation perspective and to eval- uate the control performance of the scheme proposed. The scheme intends to maximize householder’s utility only by retrofitting the HEMS to a residential household. The main contributions of this paper follow:

1) We develop a scenario-based SRHC scheme, which minimizes mean-variance based on forecasted energy demand scenarios,

for HEMS. The scheme is composed of multiple energy demand scenario forecasting based on Just-In-Time (JIT) modeling and an SRHC with a mean-variance model extracting Markowitz model naturally.

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3

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634 A. Yoshida et al. / Energy and

) We backtest the scheme using measured energy demand and system parameters validated throughout measurement tests. Numerical experiments demonstrate the control performance of the scheme proposed.

) We show that conservative operational strategy tends to yield better control performance than a non-conservative operational strategy, throughout numerical experiments.

. Residential energy system

As shown in Fig. 1, the energy supply system, that this paper argets, consists of PEFC-CGU and an AC. The system meets electric- ty, domestic hot water (HW) and space heating/cooling demand. he energy demand supplies via these devices. The system obtains nergy input from both electrical and gas grids. The PEFC-CGU has our subunits: fuel cell unit, storage tank (ST), and GH for backup, nd an electric heater (EH) for consuming surplus electricity. The C supplies thermal load, which is constrained within the com-

ortable temperature range calculated by a nonlinear function PMV 31]. The thermal load calculates in the SRHC problem by using the esidential-capacitance thermal model of a building. As the com- ortable temperature range needs to assign to the problem as an xogenous variable, the PMV calculation is performed outside of he problem in advance.

. Stochastic energy management strategy

This section describes a scenario-based SRHC for HEMS. We eal with an energy supply system path selection problem giving

plausible energy demand vector called scenario as an exogenous ariable with constraints that the energy supply system meets the lausible energy demand. As an available system configuration is

given condition, we focus on shorter operational strategy rather han long-term design, such as the optimal investment problem. ince the targeted system also has a ST and fuel reformer requiring everal minutes for warming/cooling, this problem is formulated as

multi-period control problem. Fig. 2 shows its functional compo- ition elements, which are management function, MIP optimizer, redictor, data acquisition function, controller, sensors and devices, nd the sequence at the time window t.

The proposed predictor, which is known as JIT modeling [31], orecasts plausible energy demand scenarios for

[ t, t + Tp

] where

p is a prediction horizon, based on realized energy demand at t − Tp, t − 1

] and weather forecast results for

[ t, t + Tp

] . The SRHC

erives the operational strategy, indicating the on/off status of evice for [t, t + Tu] based on the above forecasted scenarios. Note hat this operational strategy considers when and how much nergy Tu the system consumes/converts on the receding horizon

When shifting to the next time window t + 1, the strategy ˜

i (t|t − 1) of first time window t planned at the last time window − 1 is input to devices as control input via the controller. This man- gement continuously repeats the forecast and the control in every ime window t.

.1. Energy demand forecast

A forecast problem selects plausible time series as forecast sce- arios from past measured time series of energy demand when eceiving the query. The query is the energy demand observed in ast receding horizon. The input-output dataset D(t) in the time t

s composed of input vector x and output vector y.

(t) = {(

xm−1, ym )

; m ∈ {

1, . . ., M }}

(1)

where M is the number of tuples. The input vector xm is omposed of realized domestic hot water demand in the time win-

ings 158 (2018) 1632–1639

dows [ t − (m + 1) Tp, t − 1 − mTp

] and weather forecast results for

the time windows [ t − mTp, t − 1 − (m − 1) Tp

] . Weather forecasts

consist of outlet temperature, humidity, and insolation. The output

vector ym is realized domestic hot water demand Q̇ D,∗ m in the time

windows [ t − mTp, t − 1 − (m − 1) Tp

] .

Let xq (t) be an input query in the time t, and it is composed of the same combination of realized xm energy demand and weather fore- casting as xm. An input query xq (t) is used for deriving K plausible forecast results from the dataset D(t) based on data dissimilar- ity. Refer to [13] for more detail., Here, we xm adopt the Euclidean distance, dist, between input vectors xq (t) and

dist (

xq (t) , xm )

= √(

xq − xm ) (

xq − xm )�

(2)

Finally, several output vector yk in the dataset D(t) correspond- ing to K-NN inputs xk, from the query xq (t) are returned as forecast results. In other words, this forecast scheme ranks the index m based on data dissimilarity according to input query xq (t) and then returns output vectors until the K-th closest ones. Plausible

electricity demands Ė D,∗ k are also derived by using this forecasting

problem.

3.2. Stochastic receding horizon control problem

This section formulates the scenario-based SRHC problem for minimizing the expected value of energy consumption. This study considers the on/off status sequence for the PEFC unit as an opera- tional strategy. As the residential energy system needs to estimate energy demand for next control horizon Tu, the aforementioned K energy demand scenarios are used for forecast as exogenous vari- able.

3.2.1. Constraints for scenario-based SRHC problem Constraint equations consist of energy and mass balances based

on Fig. 1, and the device characteristics are described in Section 4.1.

s.t.u− k,i

(t|t) = ak,i (t|t) u+k,i (t|t) + bk,i (t|t) ıi (t|t) (3)

s.t.U + k,i (t|t) ıi (t|t) ≤ u

+ k,i

(t|t) ≤ Ū +k,iıi (t|t) (4)

s.t.sk,i (t|t) − lk,i (t − 1|t) sk,i (t − 1|t)

= (

dk,i (t|t) u+k,i (t|t) − f k,i (t|t) u − k,i

(t|t) )

�t (5)

s.t.Sk,i (t|t) ≤ sk,i (t|t) ≤ S̄k,i (t|t) , (6) where

t = [t, t + 1, . . ., t + Tu] , i ∈ {

1, . . ., I }

, k ∈ {

1, . . ., K }

,

ı ∈ {

0, 1 }

, uk,i (t|t) ≡ [ uk,i (t|t) , uk,i (t + 1|t) , . . ., uk,i (t + Tu|t)

] ,

sk,i (t|t) ≡ [ sk,i (t|t) , sk,i (t + 1|t) , . . ., sk,i (t + Tu|t)

] ,

ıi (t|t) ≡ [ ıi (t|t) , ıi (t + 1|t) , . . ., ıi (t + Tu|t)

] ,

Decision variables of u+, u−, s, and ı are the energy inflow to the device, the energy outflow from the device, the amount of energy in the energy storage device, and the binary variable, which mainly represents the on/off status of the device. Excepting ı, all param- eters and decision variables have three indexes, k, t and i. Index

i stands for device kind. The parameter a, b, d, f and l is slopes, intercepts, coefficients for energy inflow, for energy outflow and for amount of energy, respectively. As these parameters mainly repre- sent conversion characteristics between input and output energy

A. Yoshida et al. / Energy and Buildings 158 (2018) 1632–1639 1635

Fig. 1. Schematic diagram of residential energy system targeted.

fram

s p i b o s b

i ı

s

a d

Fig. 2. Conceptual HEMS

uch as the COP of the room air conditioner, these are time-variant arameters. The non-linearity, such as the relationship between

nput and output, of the energy conversion unit, is approximated y a piece-wise-linear function. Moreover, the operational modes f the device have different characteristics, such as starting-up and hutting-down mode, that are separated into operating status by inaries.

The strategy planned at the last calculation loop, ı̃i (t − 1|t − 1), s used for control input in the first time window of control horizon, i (t|t).

.t.ıi (t|t) = ı̃i (t|t − 1) (7)

As these forecast and control problems are solved iteratively t each one-time window with receding horizon, realized energy emands are used for the first time windows of the control hori-

ework and its sequence.

zon. Expecting the first time window t, the problem uses K energy demands scenarios yk.

s.t. I∑

i=1 uk,i (t|t) = y∗ (t) (8)

s.t. I∑

i=1 uk,i ([

t + 1, . . .t + Tp ]

|t )

= yk ([

t + 1, . . .t + Tp ]

|t )

(9)

where Initial status values in the current calculation time window,

ıi (t − 1|t) and si (t − 1|t), inherit the last realized control results, ı∗

i (t − 1|t − 1) and s∗

i (t − 1|t − 1).

s.t.ıi (t − 1|t) = ı∗i (t − 1|t − 1) (10)

1636 A. Yoshida et al. / Energy and Buildings 158 (2018) 1632–1639

Table 1 Numerical conditions.

Delta time meaning one-time window �t hours 0.25

Prediction horizon Tp time windows 96 Number of datasets M 10 Control horizon Tu time windows 96 Event probability Pk 1/K

s

3

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j

ı

t E

o [ W S M

a

s

4

n a t s a F o

e 2 a

Table 2 Specifications of the residential energy system.

Primary energy conversion factor at night time MJ/kWh 9.97

Primary energy conversion factor at day time MJ/kWh 9.28 Rated power output of PEFC unit kW 0.75 Rated thermal output of PEFC unit kW 1.08 Starting-up time of PEFC-CGU hours 0.75 Shutting-down time of PEFC-CGU hours 1.5 Electricity consumption of PEFC-CGU for starting-up kWh 0.98 Gas consumption of PEFC-CGU for starting-up Nm3 0.10 Electricity consumption of PEFC-CGU for shutting-down kWh 0.39 Gas consumption of PEFC-CGU for shutting-down Nm3 0.079 Maximum continuous operating time of PEF-CGU hours 20

Risk aversion parameter � 0, 0.01, 10.0 Number of scenarios K 1, 3, 5, 7, 9

.t.si (t − 1|t) = s∗i (t − 1|t − 1) (11)

.3. Objective function and problem formulation

To plan one schedule against the K plausible energy demand sce- arios, we adopt mean-variance model to planning problem. As the ean-variance minimization problem is a bi-criterion one and the

roblem involves the trade-off relationship between input energy nd its variance, we take a scalarization approach to the bi-criterion roblem. The objective function represents the linear combination f both the expected value of primary energy consumption E [j] and ts variance V [j] through control horizon against scenarios having wn probabilities Pk, as shown in the equation below. This prob- em adopting the mean-variance model becomes the deterministic quivalent one from the stochastic one.

k = I∑

i=1

t+Hp∑ t=1

ck,i (t|t) uk,i (t|t) (12)

i (t|t) = argmin u+,u−,s,ı

⎧⎨ ⎩

K∑ k=1

Pkjk + ˛�Kk=1Pk

( jk −

K∑ k′=1

Pk′ jk′

)2⎫⎬ ⎭ (13)

where � is risk aversion parameter (RAP). In Eq. (13), the first erm and the last term is the expected primary energy consumption [j] and variance V [j], in explicit form, respectively.

As Eq. 13 includes a quadratic term, the problem becomes MIQP ne. To solve the problem by a general MIP optimizer, e.g., Gurobi 32], we convert the MIQP problem to the equivalent MISOCP one.

e introduce the auxiliary variable ω to Eq. (13), to change QP to OCP. Finally, the scenario-based SRHC problem is formulated as ISOCP. The problem is as followed.

rgmin.�K k=1Pkjk + ˛V [j] (14)

.t.�K k=1ω

2 k Pk ≤ V [j] (15)

s.t.ωk = jk − �Kk′=1Pk′ jk′ (∀k ) (16) s. t. Eqs. (3)–(12)

. Model parameters and case setting

This section describes model parameters and the case setting for umerical experiments described later. Parameters for sensitivity nalysis are the RAP � and the number of scenarios K. Table 1 shows he numerical conditions. As the receding horizon is 24 h, the SRHC cheme derives a 24 h ahead operational strategy based on 24-h head energy demand forecasting, and input the strategy to devices. or the next time window, devices input on/off status based on the perational strategy.

The SRHC problem is solved by Gurobi 7.0 on a computer mploying a 2.27 GHz Intel Xeon X7560 CPU of 32 processors with 50 GB RAM. The convergence criteria is 1% gap between the upper nd lower bounds.

Fig. 3. Input-output relationship of PEFC unit.

4.1. Specification of residential energy system and demand

Table 2 shows the residential energy system specifications. Fig. 3 shows the conversion characteristic of the PEFC unit, where the characteristic is formulated by piece-wise-linear function. Model parameters of the PEFC-CGU are validated by Yoshida et al. [6]. The storage tank adopts a perfect mixing model, and the maximum capacity depends on the gap temperature between the supply tem- perature of the PEFC unit and the outlet temperature. The efficiency of the electric heater is set at a 95% constant. The efficiency of the GH is set at a 92% constant. The building specification is based on an insulation level equivalent to 1999 energy conservation standard in Japan as considered a Level 3 [33].

4.2. Energy demand profile

Energy consumption data were obtained from the Architectural Institute of Japan. The Institute had measured approximately 80 households from 2002 to 2004 [34]. Energy consumption data were broken down into specific uses: space heating/cooling, hot water, cooking, cleaning, and other. For the sake of converting to each energy demand, those specific uses were multiplied by the effi- ciency of each device, and then the data were carefully re-classified into four kinds of demand: hot water, space heating/cooling, elec- tricity, and cooking. One measured household as shown in Fig. 4, for which we had completed data treatment, was chosen for this study. As the daily demand is almost located under the heat-to- power ratio of the PEFC-CGU, 1.44, this households is suitable to install the PEFC-CGU. The annual electricity and hot water demand is 2851 kWh/year and 2798 kWh/year, respectively. The average annual residential energy consumption per household in Japan in 2004 was 3726 kWh/year for hot water and 4232 kWh/year for electricity [35]. Note that the electricity demand of 2851 kWh/year

does not include space heating/cooling demand. The SRHC problem adds space heating/cooling demand constrained by comfort tem- perature bounds to make space cooling/heating demand shiftable.

A. Yoshida et al. / Energy and Build

Fig. 4. Daily energy demand of the targeted household.

Fig. 5. Mean absolute error of forecast, (a) electrici

Fig. 6. Standard deviation and expected va

ings 158 (2018) 1632–1639 1637

5. Numerical analysis of SRHC scheme

This section evaluates the proposed SRHC scheme. Numerical experiments focus on the control of only the PEFC-CGU to evaluate the impact of robust operational strategy on control results.

5.1. Energy demand forecasting error

The energy demand forecasting is evaluated over one year. Fig. 5 shows the mean absolute error (MAE) corresponding to the realized value in each time window. Horizontal axis and vertical axis is the

number of scenarios for using the forecast and MAE, respectively. The violin plot, middle bar, box in violin plot and center square is distribution, median, quantile points and mean, respectively. As shown in both cases (a) and (b), increasing the number of scenar-

ty demand, (b) domestic hot water demand.

lues of primary energy consumption.

1638 A. Yoshida et al. / Energy and Buildings 158 (2018) 1632–1639

Fig. 7. Operating time of PEFC-CGU as control input.

i T f t M

5

o s i e a R r c

o c p t c e

p i c c 1

home EMS, we focused on deciding on/off status throughout the

Fig. 8. Computational time of each SRHC problem.

os is linked with shrinking the gap between best and worst cases. hese means and medians stay at the same level in all case. It is ound that domestic hot water demand is more difficult to forecast han electricity demand because hot water demand has a larger

AE.

.2. Solution behavior

Fig. 6 shows optimal mean-variance portfolios except in the case f K:1 because standard deviation is zero in that case. In Fig. 6, the cattered small plot is the expected value E [j] and the variance V [j] n each control loop, and the round point is the realized primary nergy consumption and its variance over 24 h, throughout evalu- tion periods. Increasing the number of scenarios, K, and increasing AP, �, is linked to decreasing the standard deviation, which rep- esents a conservative operational strategy. Fig. 6 shows that RAP an control operational strategy.

Fig. 7 shows the operating time of the PEFC-CGU for 24 h ver evaluation periods. The legends in Fig. 7 represent RAPs. We onfirmed that the operational time is apt to become larger in pro- ortion with an increasing number of scenarios. Operating time in he case of K:1, which is 14 h, is smaller than the other conservative ases, because the operational strategy adjusts to only one of the nergy demand scenarios.

Fig. 8 shows the computational time in each control loop. As roblem size is always caused by the number of scenarios, increas-

ng the number of scenarios gives rise to computational time. The omputational time in the case of K:9 requires 30 min in the worst

ase and is 7 min on average. The calculation time guarantees under 5 min in the case using under five scenarios at most.

Fig. 9. Energy saving ratio in comparison with K:1.

5.3. Energy saving ratio

Fig. 9 shows the statistical trend of the energy saving ratio of control results in comparison with the case of K:1. The energy saving ratio is calculated from the realized primary energy con- sumption for 24 h averaged over evaluation periods. The realized primary energy consumption is the same value as the round point in Fig. 6. As all of the cases have a positive value, the control results in the cases of K:3-9 overperform the case of K:1 despite the intention of a conservative operational strategy. Minimizing the variance of primary energy consumption only has a small impact on improving the energy saving ratio. The case results indicate that a conservative operational strategy exhibiting robustness to uncertainty leads to better control results than a non-conservative operational strategy by approximate 3% from the viewpoint of energy saving ratio. More- over, as the large RAP results in the worse improvement ratio of control result, it is preferred that we select the number of scenarios rather than RAP, to obtain robust operational strategy.

6. Discussion

Sending on/off status to the device can manage the control per- formance of it. The advantage of this scheme is easily practicable implementation via actual communication standard.

The average calculation time in Fig. 8 is under 7 min in all case, and it is fast enough. However, case studies consider only one controllable device. As household can have multiple controllable devices, to guarantee enough short calculation time, it requires faster algorithm to solve the problem and the simplified modeling of a device. For example, as MISOCP requires a longer calculation time than MILP in general, it might be better to drop variance term from the problem to make the calculation time faster. The rea- son is that the conservative operational strategy can be lead by increasing the number of scenarios as well as RAP. Additionally, the effect of variance term in the mean-variance model is to intend to avoid not only the increasing of energy consumption but also the decreasing of energy consumption, from the expected value. From viewpoints of control performance and calculation time, this paper therefore concludes that the recommended parameters for this scenario-based SRHC problem are five scenarios or less and RAP of zero.

7. Conclusion

The objective of this study is to construct a control scheme to achieve the stable control result for home EMSs. As the scheme intends to implement with an actual communication standard for

proposed problem. We evaluated the scheme with measured data and obtained following main results.

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[34] Energy consumption for residential buildings in Japan, Architectural Institute

A. Yoshida et al. / Energy and

) We developed a scenario-based stochastic receding horizon con- trol scheme. The scheme minimizes mean-variance of energy consumption corresponding to forecasted energy demand sce- narios. The scheme includes the plausible energy demand forecasting known as Just-In-Time modeling.

) As the large risk aversion parameter results in the worse improvement ratio of control result, it is preferred that we select the number of scenarios rather than the risk aversion parameter, to obtain robust operational strategy.

) From numerical experiments, it is concluded that conservative operational strategy tends to yield better control performance than a non-conservative operational strategy. The conservative operational strategy can be lead from multiple scenarios and larger risk aversion parameter.

) As increasing the number of scenarios results in bigger the prob- lem size, the calculation time guarantees under 15 minutes in the case using under five scenarios at most.

) Sending on/off status to the device can affect the control perfor- mance of it.

cknowledgements

This work is supported by CREST, the Japan Science and Tech- ology Agency, Grant Number JPMJCR15K5. The authors would like o thank anonymous reviewers for their helpful suggestions.

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