Robust optimization for load scheduling of a smart home with photovoltaic system

Energy Conversion and Management 102 (2015) 247–257

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Robust optimization for load scheduling of a smart home with photovoltaic system q Chengshan Wang a, Yue Zhou a,⇑, Bingqi Jiao a, Yamin Wang b, Wenjian Liu a, Dan Wang a a b

Key Laboratory of Smart Grid of Ministry of Education (Tianjin University), Tianjin 300072, China Electrical and Computer Engineering Department, Clarkson University, Postdam, NY 13699, USA

a r t i c l e

i n f o

Article history: Available online 15 February 2015 Keywords: Robust optimization Smart home Load scheduling Photovoltaic system Uncertainty Energy management Demand response

a b s t r a c t In this paper, a robust approach is developed to tackle the uncertainty of PV power output for load scheduling of smart homes integrated with household PV system. Specifically, a robust formulation is proposed and further transformed to an equivalent quadratic programming problem. Day-ahead load schedules with different robustness can be generated by solving the proposed robust formulation with different predefined parameters. The validity and advantage of the proposed approach has been verified by simulation results. Also, the effects of feed-in tariff and PV output have been evaluated. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Residential homes are getting smarter and smarter with wider use of smart appliances and integration of information and communication technology. On one hand, traditional home appliances are being replaced by smart appliances with communication module and automatic control function; on the other hand, home area networks are established in smart homes, which connect all household appliances and sensors together for easier monitoring and smarter control. Besides, smart homes are being faced with diverse pricing mechanisms where flexible pricing schemes such as time of use, real time pricing and critical peak pricing are being implemented in many countries all around the world. Therefore, there is a great opportunity for residential users to improve their life quality through smart home techniques under flexible pricing mechanisms. But general residential users do not have sufficient time and knowledge to manage all the devices of smart homes by their own. Thus, home energy management systems are often expected to be installed in smart homes to help users manage all the devices and data, and load scheduling is often running in home energy management systems to arrange the work of home appliances optimally.

q This article is based on a four-page proceedings paper in Energy Procedia Volume 61 (2015). It has been substantially modified and extended, and has been subject to the normal peer review and revision process of the journal. ⇑ Corresponding author. Tel.: +86 15822455817; fax: +86 22 27892810. E-mail address: [email protected] (Y. Zhou).

http://dx.doi.org/10.1016/j.enconman.2015.01.053 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

In the future, energy sources of smart homes will be more diverse as well. Besides electricity from the bulk power grid, more and more distributed generation using renewable energy will be encouraged to be installed in smart homes. Household photovoltaic (PV) system is an important type of renewable distributed generation, which converts solar energy to electrical power for residential users. However, the integration of PV system presents new challenges to smart home energy management because of the randomness of solar energy. Researchers have done some work to tackle PV integration issues in household load scheduling: [1–3] involved PV system in the optimization of smart home energy services, but did not consider the forecast uncertainty of its power output. Pedrasa et al. [4] concluded that there is no value in making accurate solar insolation forecasts when the feed-in tariff equals to the time-of-use tariff exactly at every minute of the day, whereas that is a quite special scenario and the real situations are far more diverse and complicated. Chen et al. [5] tackled the forecast error of PV system output by online adapting the operation schedule during the execution, but only heuristic rules are referred when making the adaption, which is not the optimal way. Hubert and Grijalva [6] took the lower limit of the 95% confidence interval for the solar irradiance forecast in the optimization, through which the schedule was robust but only the worst situation was considered. These existing researches have made positive attempts to digest the integration of household PV system, but due to their respective limitations, further research is still needed to deal with the challenge of PV output uncertainties in household load scheduling.

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

Nomenclature Variables x power of an appliance (kW) z, q, m, n, u, v, r, s auxiliary variables Parameters p electricity price ($/kWh) b allowed beginning time of a task (h) L time length of a task (h) d demand of hot water drawn (kg) h temperature (°C) a constant (1/3,600,000) for unit conversion P appliance/PV power (kW) e deadline of a task (h) E energy demand of a task (kWh) cwater specific heat of water (J/kg/°C) M mass of water in full storage (kg) C parameter that controls the robustness Subscripts i, j, n time step index buy buy electricity from the power grid NL noninterruptible loads TCL thermostatically controlled loads

Robust optimization is a promising method to deal with data uncertainty in optimization problems, thus naturally to be a potential solution to PV output uncertainties in household load scheduling. Till now, three major steps have been made in the field of robust optimization: the first step by Soyster [7]; the second step by Ben-Tal and Nemirovski [8–10], El-Ghaoui and Lebret [11], and EI-Ghaoui et al. [12]; and another step by Bertsimas and Sim [13,14]. Robust optimization method proposed by Soyster keeps the linearity of linear programming but proves to be too conservative. Method proposed by Ben-Tal et al. considers linear programming with ellipsoidal uncertainties and solves the robust counterpart in the form of conic quadratic programming, being less conservative. Method proposed by Bertsimas and Sim is capable of keeping the linearity of linear programming and is characterized by introducing a predefined number C to flexibly control the conservative level of final solutions. In terms of the application of robust optimization methods in smart homes, Chen et al. [15] and Conejo et al. [16] used the method proposed by Bertsimas and Sim to tackle the real-time demand response issue with price uncertainty. However, to the best of our knowledge, there was little work done in using robust optimization methods to tackle the PV output uncertainty challenge in household load scheduling. In this paper, robust load scheduling is studied for smart homes integrated with PV systems. The main contribution of this paper is to use robust optimization method to solve the PV output uncertainty problem in smart homes for the first time. Load scheduling problem in smart homes with PV systems has a special nonlinear form, and thus conventional robust optimization method proposed by Bertsimas and Sim cannot apply directly. To tackle this challenge, a robust formulation of load scheduling considering PV output uncertainty is established in this paper, and further transformed to an equivalent quadratic programming problem that can be solved by existing tools easily. By doing so, the uncertainty problem brought by the integration of household PV system is effectively tackled by the robust optimization method proposed in this paper. The work of this paper is novel because it uses a method (robust optimization method) that has not been used in household load

e 0 day PV sell IL req up group

environment initial state day index household photovoltaic system sell electricity to the power grid interruptible loads requested by users upperlimit index of high/medium/low group

Others N Dt J AP D SDAP A S K EP EAP

number of all the time steps of a day length of a time step (h) set of all the N time steps average electricity payment ($) total days of a group standard deviation of average payment ($) set of all the household appliances subset of J number of time steps with solar radiation practical electricity payment of a day ($) expectation of average electricity payment ($)

scheduling with PV integration before and tackles the uncertainty of household PV output to an extent that existing researches have not reached. Specifically, Pedrasa et al. [1,2] and Lujano-Rojas et al. [3] did not consider the household PV output uncertainty, but this paper does; Pedrasa et al. [4] only considered the PV output uncertainty under a special pricing mechanism, but this paper considers much general and diverse pricing mechanisms; Chen et al. [5] dealt with PV output uncertainty by adopting heuristic rules in the execution stage, but this paper dealt with the problem in an optimization way in the day-ahead scheduling stage; Hubert and Grijalva [6] only considered the worst situation, but the method proposed in this paper can control the robustness of final load schedules flexibly. The rest of the paper is organized as follows. Problem description and modeling are presented in Section 2. Robust counterpart establishment and transformation are described in Section 3. Case study is presented in Section 4. Conclusions are summarized in Section 5. 2. Formulation of household load scheduling with the integration of PV system 2.1. System structure and problem description Typical smart home with household PV system is shown in Fig. 1. Electricity of the smart home is supplied by two sources: local PV system and bulk power grid. Household devices such as smart appliances, sensors and smart plugs are connected to each other to form a home area network. Home energy management system uses the network to collect operation data of all the devices and send control signals back to control them. Load scheduling algorithm is running in the home energy management system to generate optimal load schedules based on device information, user settings and pricing mechanism. The core of load scheduling is to utilize load flexibility according to PV output and pricing mechanism to minimize electricity payment under the premise of user comfort. The complexity of this problem is threefold: firstly, different types of loads have different

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

  !   X N X Dt   min ðpbuy;i  psell;i Þ xd;i  PPV;i    d2A 2 i¼1 ! !# X þ ðpbuy;i þ psell;i Þ xd;i  PPV;i "

ð3Þ

d2A

Detailed proof refers to Appendix A. 2.3. Constraints

Fig. 1. Typical smart home with household PV system.

types of flexibility, subjecting to different characteristics and dynamics; secondly, power output of household PV system has strong randomness that is very difficult to be accurately estimated beforehand; thirdly, price of electricity from the bulk power grid is time-varying under time-of-use pricing and real-time pricing, and different feed-in tariff mechanisms for PV power make the problem more complex. Therefore, optimization method is needed for smart home load scheduling to deal with the complexity. Moreover, because of the strong uncertainty involved by household PV system, robust optimization method needs to be applied instead of conventional deterministic optimization. Optimization problem and its robust counterpart for household load scheduling with PV system are established in the following parts of the paper.

The objective function of day-ahead household load scheduling is to minimize the daily electricity payment of the next day:

i¼1

X xd;i

!

!  PPV;i Dt

ð1Þ

d2A

8 > > > > p > < buy;i

X xd;i

> > > > > : psell;i

X xd;i

d2A

ð4Þ

and e X xNL;i ¼ LNL P NL

i 2 Nþ

ð5Þ

i¼b

where N represents total time steps throughout the scheduling horizon, A is the set of all home appliances, xd,i represents the power of appliance d at the ith time step, PPV,i represents the power output of PV system at the ith time step, Dt represents the length of each time step, and pi represents the electricity price at the ith time step. Power of controllable appliances is the decision variables to be optimized. Also, note that the prices to buy and sell electricity can be different, that is,

pi ¼

2.3.1. Constraints of noninterruptible appliances A noninterruptible appliance is allowed to begin to work after tb, and its task is required to be finished no later than te. The power consumption of noninterruptible appliances is assumed to be constant, and the duration of the task lasts for as long as LNL time steps. Given the above, the power status of a noninterruptible appliance throughout the scheduling horizon should satisfy

xNL;i ¼ 0 8 i 2 ½1; bÞ [ ðe; N; i 2 Nþ

2.2. Objective function

N X min pi

The constraints of household load scheduling are about operation limits and user comfort demand for different types of home appliances. According to the controllability, home appliances are divided into controllable appliances and noncontrollable appliances. According to the operational characteristics, controllable appliances are further divided into three classes: noninterruptible appliances, interruptible appliances and thermostatically controlled appliances. It is worth noting that whether an appliance is interruptible is not absolute but depends on its specific hardware design. For example, for some types of cloth washers, the washing tasks are not allowed to be stopped once they start, and thus they are noninterruptible. But for others, the pause of washing tasks may be feasible, and hence they are taken as interruptible appliances in this case. For noncontrollable appliances, they participate in the calculation of the objective function as fixed values but would not appear in the constraints. For controllable appliances, different types of appliances provide their specific constraints for household load scheduling problem.

!

where PNL is the rated power of the noninterruptible appliance, xNL,i is the power of the noninterruptible appliance at the ith time step (which equals PNL when it’s on, and 0 when off), N+ represents the set of all positive natural numbers, b and e are the allowed beginning time and deadline of the task respectively. Note that (e–b) should be no less than LNL. Besides, another set of constraints are needed to satisfy the noninterruptibility [15]: jþL NL 1 X

xNL;i P ðxNL;j  xNL;j1 Þ  LNL

8 j 2 ðb; e  LNL þ 1; i; j 2 Nþ

i¼j

P PPV;i

!

ð6Þ ð2Þ

< PPV;i

d2A

where pbuy,i and psell,i are the prices for users to buy and sell electricity respectively. Note that both of them are positive. Although straightforward, the form of the objective function composed of (1) and (2) is not convenient for further robust formulation. Therefore, an equivalent form of the objective function is presented as follows:

2.3.2. Constraints of interruptible appliances Typical interruptible appliances include plug-in electric vehicles (note that in this paper we do not consider the situation that the plug-in electric vehicles feed the energy back to the power grid, that is, we only consider them as charging load), pool pumps, etc. Similar to noninterruptible appliances, interruptible appliances are assumed to be allowed to work between tb and te as well. The total needed energy of the task is as much as EIL. The power of

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interruptible appliances is assumed to be capable of changing in a range [0, Pmax] continuously (e.g., the charging power of electric vehicles can be changed by power electronic converters; the power of electrical machinery can be changed with variable frequency drive technology). Given the above, the power status of an interruptible appliance throughout the scheduling horizon should satisfy

xIL;i ¼ 0 8 i 2 ½1; bÞ [ ðe; N; i 2 Nþ

data. If the parameters are accurately known before optimization, the problem can be solved by normal optimization techniques. But if the parameters suffer from uncertainties, that is, we only know the range or some statistic information about the parameters, the problem comes to be an uncertain optimization problem. The uncertain household load scheduling problem with PV output uncertainty is the focus of this paper.

ð7Þ 3.1. Data uncertainty of PV output

and e X xIL;i  Dt ¼ EIL

xIL;i 2 ½0; Pmax ; i 2 Nþ

ð8Þ

i¼b

where xIL,i is the power status of the interruptible appliance at the ith time step. Note that in (8), the parameters should satisfy e X Pmax  Dt P EIL

i 2 Nþ

ð9Þ

PV power output is affected by solar radiation significantly, and thus involves strong randomness which is quite hard to be forecast exactly beforehand at household level. To model the uncertain PV output, it is assumed that before the day for which we make the load schedule, we only know that the PV output will range in intermax vals [Pmin PV, i, PPV, i]. The intervals can be obtained in a set of ways such as:

i¼b

2.3.3. Constraints of thermostatically controlled appliances Thermostatically controlled appliances are interruptible but with unique characteristics. Water heaters, air conditioners and refrigerators are three typical types of thermostatically controlled appliances. In the following, a water heater with hot water storage is chosen to demonstrate the modeling process. The heating power of the water heater is assumed to be capable of changing in a range [0, Pmax] continuously. In the first place, the heat energy should satisfy customers’ demand at each time step: i i X X xTCL;n  Dt P Cn n¼1

xTCL;i 2 ½0; Pmax ; 8 i 2 ½1; N; i 2 Nþ

ð10Þ

n¼1

where xTCL,n is the power status of the water heater at the nth time step, and Cn is the heat consumption of the nth time step. Ci is calculated by

C i ¼ a  di  cwater  ðhreq  he;i Þ 8 i 2 ½1; N; i 2 Nþ

ð11Þ

where di is the demand of hot water drawn during the ith time step, cwater is the specific heat of water, hreq is the desired the water temperature and he,i is the environmental temperature at the ith time step. In the second place, the heat storage at each time step must not exceed the maximum limit of the water storage, that is, i i X X xTCL;n  Dt 6 a  M  cwater  ðhup  h0 Þ þ Cn n¼1

xTCL;i

n¼1

2 ½0; Pmax ;

8 i 2 ½1; N; i 2 Nþ

ð12Þ

where M is the mass of water in full storage, hup is the upper limit of the water temperature in storage, and h0 is the initial water temperature in storage. a is a constant coefficient for unit conversion between J and kWh, equaling to 1/3,600,000. Note that the standby heat loss is assumed to be 0 both in (10) and (12) because it is very small for commercially available heaters [17]. 3. Robust formulation of household load scheduling considering PV output uncertainty Household load scheduling is an optimization problem composed of objective function (3) and device/comfort constraints (4)–(12). In the problem, the power of all controllable home appliances are decision variables to be optimized, while other parameters, such as electricity price, PV output, deadline of device tasks, temperature comfort band, etc., are considered as pre-known

(a) If the forecast modules in home energy management systems or other forecast service providers offer detailed statistic information about the PV output of the next day, the intervals can be confidence intervals of the forecast PV output at certain confidence level. (b) If the forecast values are available without confidence interval information, the PV output intervals can be chosen around the forecast value within certain ranges, such as [(1  a)P0 PV,i, (1 + a)P0 PV,i] in which P0 PV,i represents the forecast values and a is a percentage ranged from 0% to 100%. (c) If no forecast modules or services are available but there is historical PV output data, the intervals can be simply decided like this: Pmax PV,i equals to the maximum PV output that ever appeared in history, and Pmin PV,i equals to the minimum PV output that ever appeared in history. (d) If no sufficient historical data is available, the PV output intervals can refer to relative data or choices of nearby homes, or just give a rough estimation based on experience. Note that many well-developed methods and tools [18] can be used by forecast modules in home energy management systems or forecast service providers to make PV output forecast that is needed in the above (a) and (b). Detailed forecast models and algorithms will not be discussed in this paper because they are not the main focus of the paper. It can be expected that the more information there is to model the intervals, the better results there should be. Inaccurate estimation of the intervals may make the load schedules either too optimistic or too conservative. But even though the predefined intervals are much larger than the practical, the generated schedules are still of value because the proposed robust optimization formulation is capable of adjusting the conservatism flexibly. 3.2. Robust counterpart establishment max With PV output ranged in [Pmin PV, i, PPV, i], robust counterpart of household load scheduling needs to be formulated. Because the PV output appears only in the objective function (3), only the objective function (3) should be transformed. Although the typical robust approach proposed by Bertsimas and Sim focuses on the uncertainty of cost vector in a linear programming problem, their basic idea can be applied to formulate robust counterpart of household load scheduling problem with PV output uncertainty as well. Specifically, the objective function of the robust counterpart is

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

8   " ! ! !#   X N X X > >  max  max D t > >  P  P ðp  p Þ x þ ðp þ p Þ x   d;i d;i buy;i sell;i buy;i sell;i > PV;i PV;i 2 >   d2A > d2A > i¼1   > " ! 8 >   < >   X > Dt > min xd;j  Pmin > PV;j  þ ðpbuy;j þ psell;j Þ < 2 ðpbuy;j  psell;j Þ  X > > d2A >   " ! > þmaxfSjSJ;jSj6bCcg >   > X > >  max  j2J > > Dt > > >  ðp  p Þ x  P  > d;j buy;j sell;j : 2 PV;j  þ ðpbuy;j þ psell;j Þ :   d2A

where J is the set of all the N time steps (that is, |J| = N), and C is a predefined parameter to control the conservatism of the solutions. The constraints of the robust counterpart remain the same, to be (4)–(12). By solving the robust counterpart composed of objective function (13) and constraints (4)–(12), load schedules with different robust level can be obtained. The robustness of load schedules varies with the parameter C. The larger the C is, the more conservative load schedules would be. To demonstrate this fact, two extreme cases are analyzed in the following. When C = 0, the objective function (13) becomes

  ( " !   X N X Dt   min ðpbuy;i  psell;i Þ xd;i  Pmax PV;i    2 d2A i¼1 ! !#) X  þ pbuy;i þ psell;i Þ xd;i  P max PV;i d2A

ð13Þ

d2A

subjecting to additional constraints

X zj 6 bCc

ð17Þ

j2J

0 6 zj 6 1 j 2 J

ð18Þ

where zj is auxiliary variables. According to strong duality, the objective function (16) and additional constraints (17) and (18) can be further transformed equivalently as

  !   X  max  min ðpbuy;i  psell;i Þ xd;i  PPV;i    d2A 2 i¼1 ! !# ) X X max þ ðpbuy;i þ psell;i Þ þCzþ xd;i  PPV;i qj ( " N X Dt

ð14Þ

ð19Þ

j2J

d2A

subjecting to additional constraints

where PV output at all the time steps is considered to reach the maximum level. The load schedules obtained under this consideration minimize the electricity payment of the most optimistic scenario, thus to be the least conservative schedules. On the contrary, when C = |J|, the objective function (13) becomes

  ( " !   X N X Dt   ðpbuy;i  psell;i Þ xd;i  Pmin PV;i    2 d2A i¼1 ! !#) X þ ðpbuy;i þ psell;i Þ xd;i  Pmin PV;i

9 > > > > > > > > ! !# 9 > > = X > min > > xd;j  PPV;j > => d2A > ! !# > > > > X > > > max > > > xd;j  PPV;j > ;> ;

min

ð15Þ

d2A

where PV output at all the time steps is considered to be the least. Under this consideration, load schedules minimizes the payment of the worst scenario, thus to be the most conservative. If C takes values between 0 and |J|, load schedules with medium conservative level would be obtained. 3.3. Transformation of the robust counterpart Observing (13), it can be found that the robust counterpart include strong nonlinearity such as min–max optimization and many absolute value calculations. Therefore, a series of transformation is presented in this part to transform the robust counterpart into a form that is easier to solve. First of all, the min–max problem is tackled by transforming the objective function (13) into the following equivalent form:

 9  ! 8 "  >  X >  min  > D t > 2 ðpbuy;j psell;j Þ > xd;j PPV;j  > > > >  >  > > > > d2A > > > > ! !# > > > > X > > > > min > > > > þðp þp Þ x P d;j buy;j sell;j > > PV;j < = d2A  ; qj P 0; z P 0; j 2 J  " ! zþqj P  >  X > >  > > > >  Dt ðpbuy;j psell;j Þ > xd;j Pmax > PV;j  > 2 >  >  d2A > > > > > > > ! !# > > > > > X > > > > max > þðp > > > P þp Þ x d;j : ; buy;j sell;j PV;j d2A

ð20Þ where qj and z are auxiliary variables. So far, the min–max problem has been tackled with the equivalent robust counterpart composed of objective function (19), device/comfort constraints (4)–(12) and additional constraints (20). To further simplify the objective function (19), removing the absolute value calculations, the objective function (19) can be transformed as

min

( " N X Dt i¼1

2

ðpbuy;i  psell;i Þðmi þ ni Þ

þðpbuy;i þ psell;i Þ

X xd;i

!

!#  P max PV;i

d2A

  " ! ! !# 8 9   X N X X > >   > > max max D t > > ðp  P  P  p Þ x þ ðp þ p Þ x   > > d;i d;i buy;i sell;i buy;i sell;i PV;i PV;i 2 > >   > > > > d2A d2A > i¼1 >   > > " ! ! !# 8 9 > > < =   X X > Dt >  min  min > > > > min  p Þ x þ ðp þ p Þ x ðp  P  P   d;j d;j > > buy;j sell;j buy;j sell;j PV;j PV;j 2 < =   > > X > > d2A > þmax > >   d2A " ! ! !# > zj z > > > >   X > > X > > > > > >  j2J max  max > > D t > > > > > >  P  P ðp  p Þ x þ ðp þ p Þ x    d;j d;j : ; buy;j sell;j buy;j sell;j PV;j PV;j : ; 2   d2A d2A

þCzþ

X qj

) ð21Þ

j2J

ð16Þ

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

subjecting to additional constraints

X xd;i

!  Pmax PV;i þ mi  ni ¼ 0

ð22Þ

d2A

mi  ni ¼ 0 mi P 0;

ð23Þ ni P 0;

i ¼ 1; . . . ; N

where mi and ni are auxiliary variables. Also, absolute value calculations in additional constraints (20) should be removed as well. (20) can be transformed as the following (24)–(28):

8 Dt  9 ðpbuy;j  psell;j Þðui þ v i Þ > > 2 > > > > ! !# > > > > X > > > > min > > > þðpbuy;j þ psell;j Þ > xd;j  PPV;j > > < = d2A  z þ qj P ; qj P 0; z P 0; j 2 J D t > > > > >  2 ðpbuy;j  psell;j Þðr i þ si Þ! > > > !# > > > > X > > > > max > þðp > > > þ p Þ x  P d;j buy;j sell;j : ; PV;j d2A

ð24Þ X xd;j

 Pmin PV;j þ uj  v j ¼ 0

Appliance

PNL (W)

tb

te

LNL

Clothes washer Dish washer 1 Dish washer 2

1000 300 300

8:00 14:00 19:00

21:00 18:00 22:00

3 1 1

ð25Þ Table 2 Parameters of interruptible appliances.

!  Pmax PV;j þ r j  sj ¼ 0

ð26Þ

d2A

uj  v j ¼ 0

ð27Þ

r j  sj ¼ 0

ð28Þ

uj P 0;

Table 1 Parameters of noninterruptible appliances.

!

d2A

X xd;j

Fig. 2. Daily load curve of all the uncontrollable appliances.

v j P 0;

r j P 0;

sj P 0;

j2J

where uj, vj, rj and sj are auxiliary variables. Detailed proof process refers to Appendix B. So far, the transformation of the robust counterpart has finished. The final robust formulation consists of objective function (21), device/comfort constraints (4)–(12) and additional constraints (22)–(28). It is worth pointing out that the final robust formulation is in nature mixed integer quadratic programming, and a series of developed methods and tools can be used to solve the problem efficiently.

Appliance

Pmax (W) IL

tb

te

EIL (W h)

Electric vehicle Pool pump

2500 2000

8:00 8:00

17:00 14:00

4  Pmax  Dt 2  Pmax  Dt

Table 3 Parameters of thermostatically controlled appliances: electric water heater. Power (W)

Q (W)

R (°C/kW)

C (kWh/°C)

M (kg)

hreq (°C)

hup (°C)

6000

400

0.7623

431.7

189.25

37

80

4. Case study 4.1. Case design 4.1.1. Device parameter and comfort settings In this section, household load scheduling of a smart home integrated with PV system would be studied to validate the proposed robust formulation and explore the effects of some relative factors. Day-ahead load schedules were made and the length of a time step, Dt, is as long as 1 h. The capacity of household PV system was chosen as 10.5 kWp. Daily load curve of all the uncontrollable appliances were assumed to be known and shown in Fig. 2. Appliance parameters and corresponding comfort settings are listed in Tables 1–3. Daily hot water demand was assumed to be known and shown in Fig. 3. Daily real-time price to buy electricity (pbuy) was assumed to be known as well and shown in Fig. 4, and price to sell electricity was assumed to be 0 throughout the days. 4.1.2. Evaluation environment Day-ahead load schedules were generated for 3 months of summer days (June, July and August) by solving the proposed robust

Fig. 3. Daily hot water demand.

formulations. LINGO14 were used in this case study for solving (LINGOÒ is a comprehensive tool designed for building and solving mathematical optimization models, including a powerful language for expressing optimization models, a full-featured environment for building and editing problems, and a set of fast built-in solvers capable of efficiently solving most classes of optimization models). Then the load schedules were evaluated in ‘‘practical’’ scenarios: ‘‘practical’’ PV output of the three months was simulated using HOMER day by day, as shown in Fig. 5 (HOMERÒ is a product that nests three powerful tools including simulation, optimization and sensitivity analysis, for optimizing microgrid design with multiple

C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

253

the three months’ summer days. According to the magnitude of the practical PV output, the days were classified into three groups: high-PV output group, medium-PV output group and low-PV output group (simplified as high group, medium group and low group in the following). Each group includes 30 days. Average practical electricity payment (AP) of each group was calculated for each C by (29):

P APC;group ¼

Fig. 4. Daily real-time price to buy electricity (pbuy).

Fig. 5. PV output of the three summer months and the PV output interval max [Pmin PV, i, PPV, i].

day2Dgroup EPC;day

jDgroup j

ð29Þ

where EPC,day represents the practical electricity payment of a day with a specific C, and Dgroup is the set of days of high/medium/ low group. Results are shown in Fig. 6. Observing Fig. 6, three conclusions can be drawn. In the first place, for schedules with any C, the electricity price decreases with the increase of the PV power output. This is a very straightforward conclusion because less electricity needs to be bought from the power grid with more household PV generation. In the second place, load schedules with higher C values cost less than those with lower C values under the worst PV output scenarios (the low group), indicating that schedules made with higher predefined values of C are more robust. For example, load schedules with C = 14 can save up to 0.082 dollars per day averagely in low PV output scenarios. Therefore, one can improve the robustness of the load schedules by predefining a higher C. In the third place, it can be observed that schedules with higher values of C would cost more when there is large PV power output (the high group), indicating that schedules improve their robustness at the cost of missing some opportunities to better utilize the PV output when there is abundant solar energy. For example, load schedules with C = 0 can save at most 0.25 dollars per day averagely in high PV output scenarios. 4.3. Selection of the parameter C

Fig. 6. Evaluation results for load schedules with different values of C.

energy resources). In both schedule generation and evaluation, each day’s user demand and electricity price were assumed to be the same as presented in the part 4.1.1, because the PV output uncertainty is the focus of the evaluation. 4.2. Validation of the proposed robust formulation For each C in the predefined set {0, 1, . . ., K}, day-ahead load schedules for each day of the three summer months were generated. Considering the sunlight of the three months lasts only for 14 h, K took the value of 14 in this case study. The predefined PV max output interval [Pmin PV, i, PPV, i] was assumed to be shown in Fig. 5. For evaluation, the ‘‘practical’’ electricity payment of the generated load schedules was calculated under ‘‘practical’’ PV output of

Before doing robust optimization for household load scheduling, the parameter C should be predefined. According to the simulation results in the previous part 4.2 and theoretical analysis in the part 3.2, users can control the robustness of load schedules by predefining different values of C. Higher C value results in higher robustness, while lower C value is potential to utilize abundant PV energy better. Therefore, users predefines the C value as a tradeoff between loss paid in bad scenarios and benefits reaped in good scenarios, according to their financial situations and risk preferences. In spite of the ability to control the robustness flexibly, the above heuristic rules to predefine C cannot guarantee the optimal selection of C that results in the least total electricity payment. The reason lies in the fact that it is too difficult for users to accurately estimate the optimal robustness of load schedules beforehand with only PV interval information presented in the part 3.1. To demonstrate the relationship between C values and total electricity payment, total electricity payment under load schedules with different C values was calculated using the practical PV output data of the three summer months, as presented in Table 4. In Table 4, it can be observed that in each month load schedules made under different C values result in different monthly total electricity payment in most scenarios. The optimal C values that result in the least total electricity payment can be different for different months: for June, the optimal C is 6; while for July and August, it is 1, 2, 3 or 4. This means that users need to adjust predefined C values dynamically to keep the least payment for each month. Note that in practice the information presented in Table 4 is not available because users do not know actual PV output in advance. But if there is historical data of PV output, a previous

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

Table 4 Total electricity payment under load schedules with different C for the three summer months.

C

Cost ($) Month June

July

August

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

34.62 33.51 33.51 33.51 33.51 33.17 33.09 33.11 33.14 33.17 34.06 34.21 34.41 34.38 34.47

25.75 24.98 24.98 24.98 24.98 25.16 25.47 26.11 26.27 26.45 27.88 28.09 28.48 28.44 28.56

29.88 28.99 28.99 28.99 28.99 29.04 29.29 29.82 29.97 30.11 31.39 31.60 31.95 31.59 31.69

study can be done like the simulation shown in Table 4 to select the optimal or near-optimal values of C (usually it is not exactly the optimal because future PV output may differ from the past to some extent), which is similar to the method described in [16] that deals with the price uncertainty. 4.4. Evaluation of the effects of feed-in tariff and PV output The value of robust optimization in household load scheduling with PV system is affected by specific feed-in tariff and PV output. Therefore, simulation similar to that of the part 4.2 was carried out under a series of feed-in tariff. To evaluate the value of robust optimization under different feed-in tariff, standard deviation of average electricity payment (SDAP) is defined as

SDAPgroup

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 PK  C¼0 APC;group  EAPgroup ¼ K

ð30Þ

where EAP is defined as

PK EAPgroup ¼

C¼0 APC;group

ð31Þ

K

SDAP reflects the difference among load schedules with different robust levels. The larger the SDAP, the larger the difference is, indicating that the effect of robust optimization is more significant. On the contrary, if the SDAP is quite small, the difference made by robust optimization is small, indicating that it does not count whether the robust optimization should be used or not.

Specific simulation results are presented in Table 5. One important conclusion is about the effects of feed-in tariff. First of all, it can be observed that when the feed-in tariff equals to the real-time price (|psell| = |pbuy|), the SDAP equals exactly to 0 under all groups, indicating that robust optimization made no difference under this situation. Therefore, there is no need to conduct robust optimization when the feed-in tariff and real-time price have little difference. This conclusion coincides with the previous research of [4]. Furthermore, it can be observed that under all groups the SDAP increases with the increase of the difference between the feed-in tariff and real-time price. The more they differ, the more significant results robust optimization could make. Another important conclusion is about the effect of PV output. It can be figured out that under all feed-in tariff scenarios (except for the scenario of |psell| = |pbuy|), the better the solar energy resource is, the larger the SDAP is (SDAPlow < SDAPmedium < SDAPhigh for any feed-in tariff level), indicating that the more important the robust optimization would be. This fact reveals that for household load scheduling with PV system: (a) robust optimization is more significant in areas/seasons with better solar energy resources; (b) robust optimization is more significant for houses with greater capacity of PV system. 4.5. Comparison to the existing methods As summarized in the Introduction section, to the best of our knowledge, only a few literatures discuss the PV output uncertainty problems in household load scheduling. Among them, only Hubert et al. proposed a method to deal with this problem in the day-ahead scheduling stage [6]. Hubert et al. made robust schedules by taking the lower limit of the PV output intervals throughout the day. In this part, simulation results were presented to compare the robust optimization method proposed in this paper (named as ‘‘New Method’’ in the following for short) and the method proposed by Hubert in [6] (named as Hubert Method for short). Load schedules using the two methods were generated for comparison. For New Method, the least robust case (C = 0), the most robust case (C = 14) and the optimal case (C = 14 for low group, C = 6 for medium group, and C = 0 for high group; all obtained by the optimal selection process described in the part 4.3) were studied. The load schedules were evaluated under the low group, medium group and high group by comparing their practical total electricity payment. The results are presented in Table 6. Observing Table 6, the conclusion is threefold. First of all, it is easy to find that under all the three groups the electricity payment of load schedules made by Hubert Method equals to that of the most robust case (C = 14) of New Method. This indicates that Hubert Method can be seen as a special case (the most robust case) of New Method. Moreover, New Method can generate load schedules that are less conservative than those of Hubert Method.

Table 5 Evaluation results for different feed-in tariffs in days with high/mid/low-level PV output. Table 6 Comparison of the load schedules made by New Method and Hubert Method.

psell pbuy

SDAP Group Low PV output

Medium PV output

High PV output

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

0.0309 0.0284 0.0231 0.0060 0.0000 0.0189 0.0352 0.0358

0.0518 0.0365 0.0251 0.0098 0.0000 0.0340 0.0659 0.0803

0.0861 0.0596 0.0396 0.0167 0.0000 0.0397 0.0896 0.1218

Group

Cost($) Method New Method

Low group Medium group High group *

Hubert Method

Optimal C*

C=0

C = 14

44.56 23.56 16.28

46.41 25.59 16.28

44.56 26.90 21.88

44.56 26.90 21.88

Optimal C: low group C = 14; medium group C = 6; high group C = 0.

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

For example, in Table 6, New Method with C = 0 results in much lower electricity payment ($16.28) than Hubert Method ($21.88) when the PV energy is abundant (high group). Finally, with detailed PV output information, New Method with the optimal C is capable of generating optimal load schedules with the lowest electricity payment, saving 12.4% and 25.6% money compared to Hubert Method in medium group and high group respectively. In sum, compared to Hubert Method, New Method is able to: (a) control the robustness of load schedules flexibly to adjust to different scenarios; (b) generate optimal load schedules if there is sufficient historical data.

At the same time, there is

  " !   X Dt   ðpbuy;i  psell;i Þ xd;i  PPV;i    d2A 2 ! !# X þðpbuy;i þ psell;i Þ xd;i  PPV;i d2A

"

þðpbuy;i þ psell;i Þ

!

!  PPV;i

d2A

X xd;i

!

!#

ðA2Þ

 PPV;i

d2A

"

5. Conclusions

X xd;i

Dt ðpbuy;i  psell;i Þ () 2

! !# X Dt ðpbuy;i  psell;i þ pbuy;i þ psell;i Þ xd;i  P PV;i 2 d2A ! ! X () pbuy;i xd;i  PPV;i Dt 8 i ¼ 1; 2; . . . ; N

()

This paper developed a robust optimization approach for household load scheduling considering power output uncertainty of household PV system. Specifically, a robust formulation that can generate load schedules with different robust levels was established, and further transformed to a quadratic programming problem for easy solving. Through case study, several results were obtained: (a) Load schedules with different robustness were generated: in the case the most robust schedules could save up to 0.082 dollars per day averagely in low PV output scenarios, while the least robust schedules could save up to 0.25 dollars per day averagely in high PV output scenarios. (b) Without historical PV output information, the value of parameter C can be predefined heuristically according to users’ financial situation and risk preference, while with historical PV output information, optimal C values can be predefined, saving 4.42%, 12.5% and 9.26% at most in June, July and August respectively in the case. (c) The proposed robust approach is more significant for areas/seasons with higher solar energy resources or for places with greater difference between feed-in tariff and real-time price. (d) Compared to Hubert Method, the proposed robust approach is capable of controlling the robustness of load schedules flexibly, and can save 12.4% and 25.6% money in medium and high PV output scenarios respectively with optimal C values in the case. Future research directions may include: (a) develop approaches to predefine optimal intervals of PV output with different information levels; (b) combine the proposed robust approach with stochastic methods given that there is abundant statistical data; (c) consider more devices in smart home such as energy storage system, wind turbines, etc.; (d) extend the proposed robust approach to scheduling issues of microgrid. Acknowledgements Y. Zhou, C. Wang, B. Jiao and W. Liu acknowledge the financial supports from NFSC (National Natural Science Foundation of China)/EPSRC project OPEN (51261130473, EP/K006274/1). Appendix A.

d2A

Comparing (A1) and (A2), it can be figured out that when P ð d2A xd;i Þ P PPV;i , there is

! ! ! X X Dt pi xd;i  PPV;i Dt ¼ ½ðpbuy;i  psell;i Þj xd;i 2 d2A d2A ! ! X xd;i  PPV;i ; 8 i ¼ 1; 2; .. .; N  PPV;i jþðpbuy;i þ psell;i Þ

ðA3Þ

d2A

(b) When there is

pi

P



!

!

X xd;i

d2A xd;i

< P PV;i , pi ¼ psell;i according to (2). Therefore,

 PPV;i Dt ¼ psell;i

d2A

X xd;i

!

!  PPV;i Dt;

d2A

8 i ¼ 1; 2; . . . ; N

ðA4Þ

At the same time, there is

  " !   X Dt   ðpbuy;i  psell;i Þ xd;i  PPV;i    d2A 2 ! !# X þðpbuy;i þ psell;i Þ xd;i  PPV;i d2A

" Dt ðpbuy;i  psell;i Þ () 2 þðpbuy;i þ psell;i Þ

! ! X xd;i  PPV;i d2A

! !# X xd;i  PPV;i

ðA5Þ

d2A

" ! !# X Dt ðpbuy;i þ psell;i þ pbuy;i þ psell;i Þ xd;i  PPV;i () 2 d2A ! ! X xd;i  PPV;i Dt; 8 i ¼ 1;2; .. .; N () psell;i d2A

Comparing (A4) and (A5), it can be figured out that (A3) holds as P  well when d2A xd;i < P PV;i . Combining the conclusions drawn in (a) and (b), (A3) holds and thus the proposition is proved to be true.h

Proposition. The objective function (1) with (2) is equivalent to (3). Proof. (a) When Therefore, there is

pi

X d2A

!

P

d2A xd;i



Appendix B. P P PV;i , pi ¼ pbuy;i according to (2).

!

xd;i  PPV;i Dt ¼ pbuy;i

! ! X xd;i  PPV;i Dt 8 i ¼ 1;2;...;N d2A

ðA1Þ

Proposition. If the optimal solution (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; r j ; sj ) exists for the optimization problem a composed of objective function (21) and constraints (22)–(28), (xa;d;i ; za ; qa;j ) is the optimal solution of the optimization problem b composed of objective function (19) and constraints (20).

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C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

Proof. First of all, we prove that (xa;d;i ; za ; qa;j ) is a feasible solution of optimization problem b if (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; r j ; sj ) is a feasible solution of optimization problem a. We assume that (xa;d;i ; za ; qa;j ) is not a feasible solution of the optimization problem b, then there exists j 2 J that leads to

za þ qa;j

8 " 9   !   X > > > >  min  Dt  > > > > ðp  p Þ x  P buy;j sell;j  > > PV;j  2 > >   d2A a;d;j > > > > > > ! !# > > > > > > X > > min > >  > > þðp  P þ p Þ x > > buy;j sell;j PV;j a;d;j < = d2A   " ! <  >  X > >  >  > > > >  D2t ðpbuy;j  psell;j Þ xa;d;j  Pmax > PV;j  > >  >  d2A > > > > > > ! !# > > > > > > X > > max > >  > > þðp þ p Þ x  P > > buy;j sell;j PV;j a;d;j : ;

optimization problem b if (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; r j ; sj ) is the optimal solution of optimization problem a. We assume that      (x a;d;i ; za ; qa;j ) which does not equal to (xa;d;i ; za ; qa;j ) is the optimal solution of optimization problem b. Then we have

ðB1Þ

d2A

za þ qa;j

  ( " !   X N X Dt  max   P ðpbuy;i  psell;i Þ x a ;d;i PV;i    d2A 2 i¼1 ! !# ) X X max   þ ðpbuy;i þ psell;i Þ x C  z þ q  P þ a;d;i a a;j PV;i j2J

d2A

Because (xa;d;i ; za ; qa;j ; mj ; nj ; uj ; v j ; rj ; sj ) is a feasible solution of optimization problem a, we also have

8 " 9 > > > > D t   > > > > ðpbuy;j  psell;j Þðuj þ v j Þ > > 2 > > > > > > > > ! !# > > > > > > X > > min > >  > >  P þ p Þ x þðp > > buy;j sell;j PV;j a;d;j < = d2A " P > > > > > > >  D2t ðpbuy;j  psell;j Þðr j þ sj Þ > > > > > > > > > > > ! !# > > > > > > X > > max > þðp >  > >  P þ p Þ x > > buy;j sell;j PV;j a ;d;j : ;

So far, (B8) and (B9) are contradictory. Thus, the assumption that (xa;d;i ; za ; qa;j ) is not a feasible solution of the optimization problem b is not true. That is, (xa;d;i ; za ; qa;j ) is a feasible solution of the optimization problem b. Further, we prove that (xa;d;i ; za ; qa;j ) is the optimal solution of

  ( " !   X N X Dt  max   ðpbuy;i  psell;i Þ xa;d;i  PPV;i  <   d2A 2 i¼1 ! !# ) X X max    þ ðpbuy;i þ psell;i Þ þ C  za þ ðB10Þ xa;d;i  PPV;i qa;j j2J

d2A

ðB2Þ

  Using (x a;d;i ; za ; qa;j ), we can define a solution for optimization problem a as the following:

x0a;d;i ¼ x a;d;i

ðB11Þ

z0a ¼ z a

ðB12Þ

q0a;j ¼ q a;j

ðB13Þ

d2A

X xa;d;j

! 

Pmin PV;j

þ

uj

v ¼0  j

ðB3Þ

d2A

X xa;d;j

!    Pmax PV;j þ r j  sj ¼ 0

ðB4Þ

m0i

¼

d2A

8 > > > >0 > <

X

> X > max > > x > a;d;i : PPV;i 

d2A

!

X

d2A

uj  v j ¼ 0

ðB5Þ

r j  sj ¼ 0

ðB6Þ

8j 2 J

n0i ¼

8 ! X > >  > > xa;d;i  P max > PV;i < d2A

> > > > > :0

!  Pmax PV;i P 0

x a;d;i ! x a;d;i

ðB14Þ 

Pmax PV;i

<0

d2A

! X  xa;d;i  P max PV;i P 0 d2A

! 8 i ¼ 1;2;. . .; N 8 j 2 J X max x < 0  P PV;i a;d;i d2A

Substituting (B3) and (B4) into (B1), we have

za þ qa;j

8 " 9 > > > > Dt   > > > > ðpbuy;j  psell;j Þjuj  v j j > > 2 > > > > > > > > ! !# > > > > > > X > > min > þðp >  > >  P þ p Þ x > > buy;j sell;j PV;j a;d;j < = d2A " < > > > > > > > >  D2t ðpbuy;j  psell;j Þjrj  sj j > > > > > > > > > ! !# > > > > > > > X > > > > max  > >  P þ p Þ x þðp > > buy;j sell;j PV;j a;d;j : ;

ðB15Þ

ðB7Þ

d2A

Comparing (B2) and (B7), there is

½ðpbuy;j  psell;j Þðuj þ v j Þ  ðpbuy;j  psell;j Þðr j þ sj Þ < ½ðpbuy;j  psell;j Þjuj  v j j  ðpbuy;j  psell;j Þjr j  sj j

ðB8Þ

But because of (B5) and (B6), it is easy to verify that

½ðpbuy;j  psell;j Þðuj þ v j Þ  ðpbuy;j  psell;j Þðr j þ sj Þ ¼ ½ðpbuy;j  psell;j Þjuj  v j j  ðpbuy;j  psell;j Þjr j  sj j

For the solution defined by (B11)–(B15), the corresponding objective function value of optimization problem a equals to ! ! 8 N " X X > max > Dt  > ðp  p Þ x  P > buy;i sell;i PV;i a;d;i 2 > > > d2A i¼1 > > ! !# > > X > > max > > þðpbuy;i þ psell;i Þ x > a;d;i  P PV;i > > d2A > > ! > > > X X > > max   > þC  z þ q when x > a a;j a;d;i  P PV;i P 0 > < d2A j2J " !! N > X X   max > > Dt  > p  p  x P > buy;i sell;i PV;i a;d;i 2 > > > i¼1 d2A > > ! !# > > X > > max  > > þðpbuy;i þ psell;i Þ xa;d;i  P PV;i > > > > d2A > ! > > > X X > > max   > þC  z þ q when x > a a;j a;d;i  P PV;i < 0 : j2J

d2A

ðB16Þ

ðB9Þ which is equivalent to

C. Wang et al. / Energy Conversion and Management 102 (2015) 247–257

  !   X N X Dt  max   xa;d;i  PPV;i  ðpbuy;i  psell;i Þ   d2A 2 i¼1 ! !# X X max þ ðpbuy;i þ psell;i Þ x q þ C  z a;d;i  P PV;i a þ a;j "

ðB17Þ

j2J

d2A

For the optimal solution (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; rj ; sj ) of optimization problem a, the objective function value equals to

( " N X Dt ðpbuy;i  psell;i Þðmi þ ni Þ 2 i¼1 ! !# ) X X   þ xa;d;i  Pmax C  z þ q þðpbuy;i þ psell;i Þ a a;j PV;i

ðB18Þ

j2J

d2A

subjecting to

X xa;d;i

!    Pmax PV;i þ mi  ni ¼ 0

ðB19Þ

d2A

mi  ni ¼ 0 mi P 0;

ðB20Þ

ni P 0;

i ¼ 1; . . . ; N

ðB21Þ

According to (B19)–(B21), it is easy to find out that

mi ¼

8 > > > > <0

X xa;d;i

X > > max > > xa;d;i : PPV;i 

!

d2A

¼

ðB22Þ

 Pmax PV;i < 0

d2A

8 ! X > >  > > xa;d;i  Pmax > PV;i <

X xa;d;i

> > > > > :0

X xa;d;i

d2A

 Pmax PV;i P 0 !

X xa;d;i

d2A

ni

!

d2A

!  Pmax PV;i P 0

8 i ¼ 1; 2; . . . ; N

! 

Pmax PV;i

<0

d2A

ðB23Þ Substituting (B22)–(B24) and (B25), we can get the objection function value of the optimal solution (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; rj ; sj ) as

  !   X  max   ðpbuy;i  psell;i Þ xa;d;i  PPV;i    2 d2A i¼1 ! !# X X þðpbuy;i þ psell;i Þ þ C  za þ xa;d;i  Pmax qa;j PV;i

N X Dt

"

d2A

ðB24Þ

j2J

Because of the fact that (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; r j ; sj ) is the optimal solution, we have

  ( " !   X N X Dt  max   ðpbuy;i  psell;i Þ xa;d;i  PPV;i    d2A 2 i¼1 ! !# ) X X   þðpbuy;i þ psell;i Þ þ xa;d;i  Pmax C  z þ q a a;j PV;i d2A j2J   ( " !   X N X Dt  max    PPV;i  ðpbuy;i  psell;i Þ x <   d2A a;d;i 2 i¼1 ! !# ) X X max    þðpbuy;i þ psell;i Þ þ C  za þ xa;d;i  PPV;i qa;j d2A

j2J

ðB25Þ

257

Comparing (B10) and (B25), it can be found that they are contradic  tory, indicating that the assumption of (x a;d;i ; za ; qa;j ) other than (xa;d;i ; za ; qa;j ) to be the optimal solution of optimization problem b is not true. That is, (xa;d;i ; za ; qa;j ) is the optimal solution of optimization problem b if (xa;d;i ; za ; qa;j ; mi ; ni ; uj ; v j ; r j ; sj ) is the optimal solution of optimization problem a. So far, the proposition has been proved. Therefore, we can obtain the optimal solution of optimization problem b by solving optimization problem a.h References [1] Pedrasa MA, Spooner TD, MacGill IF. Improved energy services provision through the intelligent control of distributed energy resources. In: 2009 Bucharest PowerTech, Bucharest, Romania; 2009. p. 1–8. [2] Pedrasa MA, Spooner TD, MacGill IF. Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. IEEE Trans Smart Grid 2010;1(2):134–43. [3] Lujano-Rojas JM, Monteiro C, Dufo-López R, Bernal-Agustín JL. Optimum residential load management strategy for real time pricing (RTP) demand response programs. Energy Policy 2012;45:671–9. [4] Pedrasa MA, Spooner TD, MacGill IF. The value of accurate forecasts and a probabilistic method for robust scheduling of residential distributed energy resources. In: 2010 IEEE 11th international conference on probabilistic methods applied to power systems, Singapore; 2010. p. 587–92. [5] Chen X, Wei T, Hu S. Uncertainty-aware household appliance scheduling considering dynamic electricity pricing in smart home. IEEE Trans Smart Grid 2013;4(2):932–41. [6] Hubert T, Grijalva S. Modeling for residential electricity optimization in dynamic pricing environments. IEEE Trans Smart Grid 2012;3(4):2224–31. [7] Soyster AL. Convex programming with set inclusive constraints and applications to inexact linear programming. Oper Res 1973;21(5):1154–7. [8] Ben-Tal A, Nemirovski A. Robust convex optimization. Math Oper Res 1998;23(4):769–805. [9] Ben-Tal A, Nemirovski A. Robust solutions to uncertain programs. Oper Res Lett 1999;25(1):1–13. [10] Ben-Tal A, Nemirovski A. Robust solutions of linear programming problems contaminated with uncertain data. Math Program 2000;88(3):411–24. [11] El-Ghaoui L, Lebret H. Robust solutions to least-square problems to uncertain data matrices. SIMA J Matrix Anal Appl 1997;18:1035–64. [12] El-Ghaoui L, Oustry F, Lebret H. Robust solutions to uncertain semidefinite programs. SIMA J Matrix Anal Appl 1998;9(1):33–52. [13] Bertsimas D, Sim M. Robust discrete optimization and network flows. Math Program 2003;98(1–3):49–71. [14] Bertsimas D, Sim M. The price of robustness. Oper Res 2004;52(1):35–53. [15] Chen Z, Wu L, Fu Y. 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. [16] Conejo AJ, Morales JM, Baringo L. Real-time demand response model. IEEE Trans Smart Grid 2010;1(3):236–42. [17] Du P, Lu N. Appliance commitment for household load scheduling. IEEE Trans Smart Grid 2011;2(2):411–9. [18] Diagne M, David M, Lauret P, Boland J, Schmutz N. Review of solar irradiance forecasting methods and a proposition for small-scale insular grids. Renew Sustain Energy Rev 2013;27:65–76.