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Interval number optimization for household load scheduling with uncertainty
Accepted Manuscript Title: Interval Number Optimization for Household Load Scheduling with Uncertainty Author: Jidong Wang Yinqi Li Yue Zhou PII: DOI: Reference:
S0378-7788(16)30790-3 http://dx.doi.org/doi:10.1016/j.enbuild.2016.08.082 ENB 6981
To appear in:
ENB
Received date: Revised date: Accepted date:
10-1-2016 22-6-2016 27-8-2016
Please cite this article as: Jidong Wang, Yinqi Li, Yue Zhou, Interval Number Optimization for Household Load Scheduling with Uncertainty, Energy and Buildings http://dx.doi.org/10.1016/j.enbuild.2016.08.082 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Interval Number Optimization for Household Load Scheduling with Uncertainty Jidong Wang1,Yinqi Lia, Yue Zhoub Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China
1Corresponding
author. Tel. + 86 13652157798E-mail address: [email protected] Address: School of Electrical Engineering & Automation, Tianjin University, Tianjin, 300072, China a,b
E-mail:[email protected]; [email protected]
Highlights
Household load scheduling with uncertain-but-bounded parameters is analyzed.
Interval number optimization is introduced in the household load scheduling.
Energy cost decreases with the increase of tolerance degrees.
Feasible schemes which are robust to uncertainties could be obtained conveniently.
ABSTRACT An interval number optimization method is proposed in this paper to tackle the household load scheduling problem with uncertain hot water demand and ambient temperature. The household loads considered include residential thermostatically controlled loads such as electric water heater and air conditioner, and interruptible loads such as clothes washer and pool pump. The uncertain-but-bounded parameters are modeled as interval numbers, based on which the uncertain load scheduling problem is formulated and transformed. A binary particle swarm optimization combined with integer linear programming is introduced to solve the transformed problem.
Two schemes, named cost scheme and
trade-off scheme, are contrastively discussed to study the economic impacts of different tolerance 1
degrees for constraint violation. Simulation results demonstrate that the proposed method is flexible to different consumer demands and robust to the uncertainties.
Keywords: residential thermostatically controlled loads; interruptible loads; interval number optimization; household load scheduling; uncertainty; tolerance degree for constraint violation
Nomenclature Variables 𝑋𝑊𝐻,𝑛
Status of electric water heater over period [tn, tn+1] (1―on; 0―off)
𝑋𝐴𝐶,𝑛
Status of air conditioner over period [tn, tn+1] (1―on; 0―off)
𝑋𝐶𝑊,𝑛
Status of clothes washer over period [tn, tn+1] (1―on; 0―off)
𝑋𝑃𝑃,𝑛
Status of pool pump over period [tn, tn+1] (1―on; 0―off)
𝐼 𝐼 𝜃𝑊𝐻,𝑛 , 𝜃𝐴𝐶,𝑛
Interval of hot water temperature and indoor temperature over period [tn, tn+1] (○C)
𝜃𝑊𝐻,𝑐𝑢𝑟
Current hot water temperature (after water draw)
Parameters pn
Real-time price over period [tn, tn+1] (cents/kWh)
𝐼 𝐼 𝜃𝑊𝐻 , 𝜃𝐴𝐶
Comfort setting of hot water in tank and indoor temperature (○C)
tn
Time at nth step(hour)
𝑑𝑛𝐼
Demand interval of hot water drawn over period [tn, tn+1] (gallon)
𝐼 𝜃𝑒𝑛,𝑛
Interval of ambient temperature/inlet water temperature over period [tn, tn+1] (○C)
𝑡𝐶𝑊,𝑏 , 𝑡𝐶𝑊,𝑒
The start time and deadline of clothes washer (hour)
𝑡𝑃𝑃,𝑏 , 𝑡𝑃𝑃,𝑒
The start time and deadline of pool pump (hour)
𝐿𝐶𝑊 , 𝐿𝑃𝑃
Total number of the time steps over load scheduling horizon 2
𝛼
Penalty factor
𝜎
Tolerance degree for constraint violation
𝑃demand
Demand limit of the house (kW)
Ph
Power rating of appliance h (kW)
Pstep,h
Power consumption of appliance h in each step (kW)
Pmustrun,n
Must-run service over period [tn, tn+1] (kW)
𝑄𝑊𝐻 , 𝑄𝐴𝐶
Capacity of electricity water heater and air conditioner (kW)
𝐶𝑊𝐻 , 𝑅𝑊𝐻
Thermal capacitance and resistance of electricity water heater (kWh/○C)
𝐶𝐴𝐶 , 𝑅𝐴𝐶
Thermal capacitance and resistance of air conditioner (○C/kW)
M
The mass of water in full tank (gallon)
𝑟( )
Random number in the interval [0,1]
𝑤
The inertia weight
𝑐1 , 𝑐2
Weighted coefficients of the global and personal best solutions
Subscripts n
Time step index
en
Environment
h
Appliance index
Others H
Set of all appliances
𝐻𝑅𝑇𝐶𝐿
Set of electricity water heater and air conditioner
𝐻𝐼𝐿
Set of clothes washer and pool pump
3
1.
Introduction A home energy management is a component of smart grid that is able to make consumers
participate in the adjustment of power consumption. With the emergence and popularity of various time-varying pricing models and intelligent appliances, consumers may find it difficult to schedule home appliances manually. Considering this reality, home energy management is becoming increasingly necessary for the cost saving and comfortable life in a smart home [1], [2].Based on the development of smart appliances and bi-directional communication technologies, home energy management can effectively reduce household electricity bill and improve consumer power consumption pattern under the premise of satisfying the setting comfort demands [3]. Home energy management makes it possible to schedule home appliances automatically with respect to different pricing mechanisms. One of the most essential parts of home energy management is the household load scheduling [2], [4]. At present, a great amount of studies on the household load scheduling have been done and some valuable achievements have been taken. Among them, many researchers focus on household load scheduling using deterministic optimization methods. In [5], a home energy management algorithm is proposed to schedule household loads based on their load priority. At the same time, the impact of different demand limit levels on demand response potential is also discussed. In [6], a residential energy consumption scheduling framework is introduced to make a trade-off between electricity cost and waiting time. And the pricing mechanisms involved in this paper are real time price combined with inclining block rates. Particularly, a weighted average price prediction filter is proposed to forecast the real-time electricity pricing by assigning different coefficients to prices on some previous days. In [7], to differentiate the importance of desired energy services, different values are assigned to them by 4
consumers firstly. Then, a co-evolutionary particle swarm optimization with stochastic repulsion among particles is proposed to obtain efficient distributed energy resources operation schedules. Furthermore, as various types of uncertainties, such as the uncertainties in model, measurement, communication and forecast, cannot be neglected in real situation, the study of household load scheduling in uncertain context is becoming more and more important. In [8], only electric water heater is studied in the household load scheduling as a representative of thermostatically controlled appliances. Characteristics of an electric water heater are analyzed in detail and a two-step scheduling process is introduced to the electric water heater load scheduling. In [9], in order to achieve the optimal temperature scheduling of air-conditioning, uncertain parameters like outdoor temperature and day-ahead electricity price are transformed into fuzzy parameters. Furthermore, the immune clonal selection programming is applied on the scheduling for air-conditioning. In [10], stochastic optimization is utilized to tackle the uncertain load scheduling involved with the plug-in hybrid vehicle, space heating, pool pumping, water heater and PV generation. Sixteen possibility scenarios are encoded by 8-character codes to describe the uncertainty in the solar insolation, magnitude of service demand, availability of the plug-in hybrid vehicle, and critical peak price status. In [11], a decision-making method is proposed to schedule both interruptible and non-interruptible loads with uncertain electricity price. Relevant statistical knowledge is utilized to reduce the cost. In most aforementioned literatures, when uncertain optimization is transformed into certain optimization, additional function or information need to be introduced into household load scheduling, such as the probability distribution in stochastic programming and fuzzy membership function in fuzzy programming. On one hand, the additional function or information will make the optimization more complex to some extent; on the other hand, both the probability distribution and fuzzy membership 5
function are based on a great amount of history data. In some situations, it is hard to be obtained for new homes or for absence of measurement devices. Faced with these limitations of the existing methods, the methodology of interval number optimization based on tolerance degree for constraint violation is introduced to handle the uncertain optimization in the household load scheduling. After uncertain parameters are expressed as interval numbers, the uncertain household load scheduling is transformed into household load scheduling based on interval number which can be solved with interval number optimization. In this paper, the appliances which are involved in the load scheduling include electric water heater (EWH), air conditioner (AC), clothes washer (CW) and pool pump (PP). The simulation results demonstrate that the interval number optimization is capable of obtaining a viable scheme which could satisfy consumer comfort zone in spite of uncertainties. Considering the diversity of user demand and balance of comfort and electricity bill, the schemes with different tolerance degrees are discussed to give consumer more choices. The main contributions and novelty lie in the following two aspects. 1) As a promising uncertain optimization method, rare works about the theories and applications of interval number optimization are studied in the home energy management. In [12], the uncertain
outdoor temperature and unforeseeable consumption are modelled by interval. Also, multi-parametric programming is introduced to handle this uncertain load scheduling. However it just takes consumer comfort into account, the cost of load scheduling is neglected. In [13], the interval analysis is introduced to the home energy management system (HEMS), and two interval division methods are proposed. However, it is only focused on the impact of optimization time interval. Other parameters like indoor temperature, user demand and so on are considered to be 6
certain. In this paper, an interval number optimization based on tolerance degree for constraint violation is introduced in the household load scheduling with uncertain hot water demand and ambient temperature to make a compromise between the cost and comfort level. 2) Most previous literatures were only focused on the electricity price uncertainty to optimize the cost benefits [14], [15]. Besides, a few works have taken into account the consumer behavior and weather condition with large uncertainty levels [3], [16], in which whether the shower demands will occur at some specific periods and whether the weather will be sunny are considered. But in reality, even when the consumer behavior and weather condition with large uncertainty levels are assumed to be completely determined, some continuous parameters with small uncertainty levels like hot water demand and ambient temperature will also fluctuate in a certain interval, which has a negative influence on user comfort. This proposed method might benefit market players such as HEMS users and producers. For instance, in a new house, the stochastic information about consumers’ habits
and customs is difficult to be obtained as the lack of adequate historical data [17]. As a HEMS user, it is unrealistic to allow HEMS to cost months or even longer to collect related knowledge. However, some uncertain-but-bounded information could be modelled by parameters with small uncertainty levels, for which less uncertainty information is needed [18]. Hence, with the introduction of interval number optimization into HEMS, users’ comfort demand could be satisfied even when only limited information is available. For HEMS producers, to improve competitiveness, it is also important for them to guarantee user comfort for consumers even with only inadequate information. The analysis done in the paper is able to offer a feasible solution to uncertain load scheduling for HEMS users and producers. This paper is organized as follows. The basic load models are introduced in Section 2. The 7
formulation and solution of the load scheduling with uncertain parameters is provided in Section 4. The simulation results are presented in Section 5. This paper is summarized in Section 6. 2. Basic load models 2.1 Traditional load models A typical system structure of smart home is shown in Fig. 1 [19]. The HEMS is the control center of a smart home. It is able to receive external signals and internal information uploaded by smart appliances. Furthermore, the algorithm which is embedded in the control center can tackle these received data and control the power states of smart appliances to improve the power consumption efficiency [19]. Traditionally, smart home devices are divided as controllable services and uncontrollable services [17]. If subdividing them further, EWH and AC are classified as residential thermostatically controlled load (RTCL) but CW and PP belong to interruptible loads. Besides, other uncontrollable devices are lumped together into a must-run service. The uncertain parameters considered in this paper are hot water demand and ambient temperature whose actual values fluctuate in an interval. RTCL in which EWH and AC are two main loads accounts for a great proportion which could reach 40%-50% in domestic load [20], [21], and most peak load demand on the power grid is originated from the RTCL. The schedule of RTCL is aimed at maintaining thermal comfort in house, which is important for consumers [22]. Therefore it is of great significance to take the RTCL scheduling into account. Water heater is a universal appliance in home. Generally speaking, natural gas and electricity are two main components of residential water heating energy use. For instance, in Canada, 51.4% of residential water heating uses natural gas heating; 44.1% uses electricity heating and 4.5% uses other 8
sources [23].This paper is focused on the electricity water heater and its basic working principles are as follows: it turns on to heat the water when water temperature drops to the lower temperature limit, and it turns off when water temperature increases to upper temperature limit. The heat loss of an EWH results from two aspects: the main loss is the cold water inflows with the use of hot water; the other type of loss is the heat exchange between hot water in tank and the environment.
The behavior of an EWH could be formulated by thermal dynamic model [8] .When only the heat exchange is considered, the thermal dynamic of an EWH is computed by
WH ,n1 en,n X WH ,nQWH RWH (en,n X WH , nQWH RWH WH ,n ) exp[(tn 1 tn ) / ( RWH CWH )]
(1)
If the hot water demand is taken into account, the thermal dynamic model of EWH is modified as
WH ,n1 [WH ,cur (M dn ) en,n dn ] / M
(2)
where 𝜃𝑊𝐻,𝑐𝑢𝑟 is the 𝜃𝑊𝐻,𝑛+1 calculated in (1). When the above two aspects are considered synthetically, (1) and (2) are combined to be formulated as
WH ,n1 f (WH ,n , tn , QWH , CWH , RWH , dn , XWH ,n ,en,n ).
(3)
At the same time, AC is a RTCL which is installed extensively in home besides EWH [24]. Its basic working principles share many basic similarities in spite of many specific differences compared with EWH. The main difference is that AC load scheduling just takes the heat loss resulted from heat exchange with the environment into account. The thermal dynamic model of AC is expressed as
AC ,n1 en,n X AC ,nQAC RAC (en,n X AC , nQAC RAC AC ,n ) exp[(tn 1 tn ) / ( RAC C AC )]
(4)
Besides the above RTCL, some interruptible loads like CW and PP are also taken into account. Different from uninterruptible loads, interruptible loads are allowed to be interrupted and resume 9
operation in the next scheduling horizon with little or no consequence. Specifically, the mathematical model of CW is expressed as tCW ,e
n tCW ,b
X CW ,n LCW
(5)
The mathematical model of PP is expressed as tPP ,e
n tPP ,b
X PP ,n LPP
(6)
2.2. Load models with interval number Essentially, interval number is a real number set composed of a closed interval which can be described with its upper limit and lower limit. First of all, basic concepts of interval number are introduced as follows. For arbitrary which are subject to bound and
a
a a , AI [a, a]
a, a R
is named as an interval number in which a is the upper
is the lower bound.
Then, when it comes to the basic mathematical operations of interval number, which include the operations of addition, subtraction, multiplication and division, [25-27] have discussed them in detail. In addition, it is necessary to quantitatively compare an interval number with the other one in practical situations. Accordingly, a new mathematical tool named possibility degree analysis of interval is proposed to quantitatively describe the comparison of two interval numbers. The possibility degree P(a B I ) can be described as
1, a B Ba P(a B I ) ,B a B B B 0, a B I
where 𝑎 denotes a real number and B
is an interval number. 10
(7)
Among the aforementioned four devices, the loads which are involved with uncertain parameters (hot water demand and ambient temperature) are EWH and AC. For EWH, water temperature in tank may deviate from the expected value due to the fluctuations of hot water demand and ambient temperature. As for AC, room temperature may also violate the comfort setting due to the error in outdoor temperature forecast. Sometimes, the deviations of hot water temperature in tank and room temperature are so large that it is unacceptable for most consumers. To handle deviation caused by uncertain information, the interval number analysis is introduced. At first, the thermal dynamic models of EWH and AC are transformed into interval number models. I I I I WH , n 1 f (WH , n , tn , QWH , CWH , RWH , d n , X WH , n , en , n ) I I AC , n 1 en , n X AC , n QAC RAC I (enI ,n X AC ,nQAC RAC AC , n ) exp[(tn 1 tn ) / ( RAC C AC )]
(8)
(9)
3. Formulation and solution of the load scheduling with uncertain parameters 3.1 Formulation of the uncertain load scheduling Through the above transformation, the models with uncertain parameters are changed to interval number models in order to analyze the uncertain load scheduling with interval number optimization. The objective function (10) is to minimize the daily electricity payment. Formula (11) and (12) are the comfort constraints of EWH which are involved with interval number, hence being uncertain constraints. Formula (13) and (14) are uncertain comfort constraints just like (11) and (12). Formula (15) and (16) describe the operation constraints of CW and PP. As there are no interval numbers involved in (15) and (16), they are both certain constraints. Formula (17) is the home demand limit constraint. As shown in [5], if home demand limit is ignored in the process of load scheduling, the high load compensation will occur after the DR events end. That is, in some periods which have cheaper electricity price, many home appliances will be 11
turned on synchronously. Obviously, it is unfavourable for peak shaving and load shifting. Hence, the household load scheduling in this paper is combined with demand limit Pdemand , which means the total power consumption in each period should not exceed the demand limit Pdemand
, otherwise the
schedule is deemed to be invalid. N
minimize
H
p n 1 h
subject to
n
X h,n Pstep ,h
(10)
I I I I WH , n 1 f (WH , n , tn , QWH , CWH , RWH , d n , X WH , n , en , n )
(11)
I WH WH , n WH
(12)
I I I AC , n 1 f ( AC ,n , tn , QAC , CAC , RAC , X AC ,n , en ,n )
(13)
I AC AC , n AC
(14)
tCW ,e
n tCW ,b tPP ,e
n tPP ,b
X CW ,n LCW
(15)
X PP ,n LPP
(16)
H
P X h
h,n
Pmustrun,n Pdemand
(17)
h
Uncertain constraints cannot be tackled directly, so they need to be relaxed. This process is completed with possibility degree analysis of interval. The uncertain constraints from (12) to (14) are dealt with by the possibility degree analysis of interval described in (7). This approach is suggested to transform an uncertain equality constraint into two deterministic constraints. Hence, (12) and (14) are relaxed into I P(WH ,n WH ) 1 1 I P(WH ,n WH ) 2
(18)
I P( AC ,n AC ) 1 3 . I P ( ) AC , n AC 4
(19)
12
Here, 1 and
2
denote the tolerance degrees for constraint violation on upper and lower
limits of water temperature respectively. Similarly, 3 and
4
denote the tolerance degrees for
constraint violation on upper and lower limits of indoor temperature respectively. In substance, the tolerance degree for constraint violation is a parameter utilized to weigh the tolerance degree of consumer’s attitude to comfort violation. The larger their values are, the more tolerable the consumer is, and vice versa. 3.2 Solution of the uncertain load scheduling Through mathematical transformation model based on possibility degree of interval, the uncertain load scheduling is transformed into certain optimization problem. It lays a foundation for the further solution. However, as four controllable loads are involved in the scheduling, there are seven constraints in each scheduling step. The simulation results demonstrate that neither heuristic algorithm nor traditional linear programming algorithm is capable of solving it effectively. If heuristic algorithm is adopted, not only the operation time is difficult to meet the qualification, but also the satisfactory solution is hard to obtain. Moreover, if the traditional linear programming algorithm is adopted, this load scheduling is unsolvable due to the massive variable dimensions and constraints. To this end, a binary particle swarm optimization (BPSO) combined with integer linear programming is introduced in this paper to tackle the above load scheduling. Particle swarm optimization (PSO) is a basic heuristic algorithm in which each candidate solution called a particle is treated as a point in the solution space. The fitness value and movement of particles are respectively determined by the objective function and velocity. The trajectory of a particle is influenced by the best location they have visited and the best performing particle. The speed and position are calculated by 13
t t t t vit,k1 wvit,k c1r ()( pGb ,i pi , k ) c2 r ()( pPb,i pi , k )
(20)
pit,k1 pit,k vit,k1
(21)
where Vk (v1,k , , vn,k ) and Pk ( p1,k , , pn,k ) denote the speed and position of the
i th particle [28].
As a modification of basic PSO, the BPSO is introduced to handle discrete optimization problem. In BPSO, sigmoid function (16) is applied to get a possibility which is mapped by the speed of a particle, and the coordinate of each particle is decided by the resulting possibility. If rand () S (vit 1 ) , then pit 1 0 ; otherwise, pit 1 1 [29].
S (vit 1 )
1 1 exp(vit 1 )
(22)
Generally, BPSO is good at solving the programming problem without constraints. With such background, a penalty function method is applied to unify the objective function and constraints to transform the constrained programming problem into unconstrained programming problem.
( P(M hI NhI ) h ) (min(0, P(M hI NhI ) h ))2 m
where
m
m
m
m
m
(23)
I I I I I I I M hI1 WH , n , N h WH , h1 1 1 ；M h2 WH , n ，N h WH ， h2 2 ；M h3 AC , n , 2
1
I I N hI3 AC , h3 1 3 and M h4 AC ,n ， NhI4 AC , h4 4 。
The unconstrained load scheduling can be described as N H RTCL
p n 1
n
h
X h,n Pstep ,h ( P(M hIm N hIm ) hm ) .
(24)
where H RTCL is the load set which composes of EWH and AC, is a penalty factor which is generally assigned to a big prime number. Then, (24) is suitable to be optimized with BPSO. After a feasible solution, that is the penalty function is equal to zero, is found, the integer linear programming is utilized to tackle the load scheduling including CW and PP. The objective function and constraints of the CW and PP loads scheduling are expressed in (25-28). 14
N H IL
minimize
p n 1
n
tCW ,e
subject to
X h,n Pstep ,h
(25)
h
n tCW ,b tPP ,e
n tPP ,b
X CW ,n LCW
(26)
X PP ,n LPP
(27)
H IL
PWH X WH ,n PAC X AC ,n Ph X h ,n Pmustrun ,n Pdemand
(28)
h
where
X WH ,n
and
X AC ,n
are the feasible solution of (24) obtained by BPSO. Next step, the solving
results of integer linear programming are brought into (10) together with the feasible solution of (24). Then the value of (10) is taken as the fitness value of BPSO.
In theory, the aforementioned method is suitable for a joint optimization with other household loads, which originates from the combination of BPSO and linear programming algorithm. For instance, RTCL such as space heater could be scheduled by BPSO; and interruptible loads such as dishwashers and electric vehicles could be scheduled by linear programming algorithm. Therefore, the household load scheduling in which extra RTCL and interruptible loads are added could also be solved by this proposed method. However with the increase of dimensionality, the actual computation time and operation efficiency are still remained to be verified in future works. 4. Simulation results 4.1 Basic parameters The simulation is divided into two parts: at first, a sensitivity analysis for the uncertainties of hot water demand and ambient temperature is conducted to investigate the impacts of their disturbances on optimization results. Then, the BPSO combined with integer linear programming is introduced to handle the load scheduling with data uncertainties in parameters. Moreover, the electricity bill and 15
power consumption profile of schemes with different tolerance degrees are also discussed to make a comparison between consumer comfort and electricity bill. The scheduling horizon starts from 0a.m until 12p.m and the simulation time step is 30min. The home demand limit Pdemand is set to 7kW. Appliances involved in the household load scheduling are EWH, AC, CW and PP. Their specific parameters are shown in Table I-III. As to the equivalent thermal parameters of EWH and AC which are deemed as certain values in line with the usual practice, many works have been done in references [30], [31], [32].The parameters of Table II- III are chosen from the 2012 ASHRAE Handbook [33]. In addition to aforementioned controllable loads, there are some uncontrollable loads in smart appliance such as lighting, TV and so on. For convenience, different uncontrollable loads are lumped together into a must-run service. The must-run service profile is shown in Fig. 2. In this paper, interval number is utilized to describe the uncertain information in hot water demand and ambient temperature. The interval curves of hot water demand and ambient temperature are shown in Fig. 3 and Fig. 4 respectively. The electricity price involved in this paper is real-time price used by the Illinois Power Company [34] and its profile is shown in Fig. 5.
At present, there is a lack of universally applicable mathematical analysis method to determine PSO parameters. Even so, several valuable works have been done in references [35-38] to provide some guidelines for the selection of the parameters. Generally speaking, the w is advised to be set at [0.9,1.2] and the sum of 𝑐1 and 𝑐2 is suggested to be not less than 4. According to the convergence domain proposed by the convergence analysis in reference [39], the inertia weight w is assigned to be 1 16
and the 𝑐1 , 𝑐2 are assigned to be 1.5 and 2.5 respectively. The swarms in BPSO have 300 particles and the simulation has 500 iterations. Moreover, the penalty factor
is assigned to 10000. Lp_solve, as an open source mixed integer linear
programming system, is utilized to solve the load scheduling of CW and PP. 4.2 Sensitivity analysis In reality, hot water demand has a significant impact in the scheduling result of EWH load. Due to the deviation between actual value and forecast value of hot water demand, actual water temperature in tank will be different from the scheduling result to a certain extent. Besides, the uncertain ambient temperature will have a negative effect on EWH and AC loads scheduling. To evaluate the effect of uncertain hot water demand and ambient temperature，a sensitivity analysis is put forward in this section. Ignoring the uncertainties, the EWH and AC loads scheduling formulated by (24) is calculated based on day-ahead forecast values of water demand and ambient
temperature. Here, the lower bound of water demand interval d n and upper bound of ambient temperature interval
en
are taken as the forecast values. Unfortunately, due to the inevitable forecast
errors, actual values will fluctuate in the intervals as shown in Fig.3 and Fig.4. As the difference between forecast values and actual values, violations of comfort setting will occur frequently as shown in Fig. 6 and Fig. 7.
Obviously, due to the uncertainties in hot water demand and ambient temperature, both the hot water temperature in tank and the room temperature are highly likely to violate the comfort zone. In Fig. 6, the lower bound of actual water temperature is 51.72℃, so the largest water temperature deviation between EWH comfort setting is ∆𝜃𝑊𝐻 = 4.28℃ , and the degree of discrepancy is 17
WH , percent
WH 35.67% . Similarly, the lower bound of actual room temperature is WH WH
21.82℃, and hence the largest water temperature deviation from AC comfort setting is ∆𝜃𝐴𝐶 = 1.18℃, and the degree of discrepancy is AC , percent
AC 39.33% . Such a great violation on AC AC
comfort zone is unacceptable for most consumers [40]. Therefore it is necessary to solve the uncertain load scheduling with interval number optimization to obtain a robust schedule for uncertain hot water demand and ambient temperature. 4.4 An analysis of cost scheme and trade-off scheme When consumers have an absolutely strict requirement for comfort, any violation is deemed to be inacceptable. At this moment, the tolerance degrees for constraint violation are set as and
1 2 0
3 4 0 . For simplicity, the schedule which indicates an absolutely conservative attitude for
constraint violation is named as cost scheme. Under the cost scheme, the interval curves of water temperature in tank and the room temperature are shown in Fig. 8 and Fig. 9. Furthermore, the power consumption in home is shown in Fig.10.The interval curves result from the uncertainties of water demand and ambient temperature. Various water demand and ambient temperature result in different temperature profiles. The set of different profiles comprise an interval curve. As shown in Fig. 3 and Fig. 4, there is a great incertitude degree in hot water demand and ambient temperature, which results in that both the intervals of water temperature and room temperature are large as shown in the blue zone of Fig. 8 and Fig. 9. But in spite of this, both the hot water temperature and room temperature are strictly controlled in the comfort zone. It indicates that interval number optimization proposed in this paper is able to obtain a robust scheduling scheme which will be suitable for all possible scenarios, even though the uncertainties of some parameters are great on some level. 18
This simulation results demonstrate that this method is valid for uncertain optimization and able to satisfy the comfort demand in uncertain context.
As a method of scheduling uncertain household loads, the proposed interval number optimization based on tolerance degree for constraint violation could benefit market players such as HEMS producers and users. For a HEMS producer, obtaining robust load schedules with limited uncertainty knowledge is essential to improve their competitiveness; for a HEMS user, this analysis help to make load schedules that minimize the electricity payment at the premise of user comfort, even inevitable prediction errors exist. In practical application, this proposed method can be embedded in the HEMS to control the power states of smart appliances to satisfy users’ comfort demand in uncertain context. However, as the high electricity bill of cost scheme, not all consumers are willing to choose the cost scheme. Many of them are inclined to take the trade-off scheme, indicating a non-absolutely conservative attitude for constraint violation as their preference. Not all values of tolerance degree in a trade-off scheme need to be set to zero, but different values are assigned to tolerance degree according to actual situation. The scheme in which 1 2 0.2 , 3 4 0.2 is discussed as a representative of trade-off schemes. Under the trade-off scheme, interval curves of hot water demand and ambient temperature are plotted in Fig. 11 and Fig. 12. Moreover, the corresponding power consumption in home is shown in Fig. 13.
As shown in Fig. 8 and Fig. 9, both the comfort setting of hot water temperature and room temperature are violated to a certain extent. But fortunately, the degree of constraint violation is controllable. The actual hot water temperature interval is [55.27,68.89] , and the actual room 19
temperature interval is [22.90,25.57] . In addition to this discussed trade-off scheme, the scheduler is also able to adjust the tolerance degree for constraint violation according to user specific requirements to obtain more schemes with non-absolutely conservative attitude for constraint violation. Finally, a comparison between the costs of cost scheme and trade-off scheme are expressed in Table IV. It can be observed that the costs of EWH and AC loads are reduced along with the decrease in conservative degree. As the consumer chooses to sacrifice partial comfort demand and is more tolerable for constraint violation, the electricity bill is reduced naturally. However, at the same time, there is a tiny increase on the costs of CW and PP load. This is because that with the rise of tolerance degree for constraint violation, the scheduler would turn EWH and AC loads on during some valley price periods. Hence more CW and PP loads have to be turned off during the low-price periods due to the demand limit, and are turned on during high-price periods instead. But compared to the reduction in the costs of EWH and AC loads, this increase is insignificant for total cost.
In all honesty, the differences within the two schemes in the paper are not large. In existing literatures like [15] and [18], the difference between the most conservative case and the least conservative case is indeed more obvious, but for many other cases with medium conservativeness, the costs of the cases are similar or even identical as well. Therefore, the results presented in this paper are consistent with those of other similar existing research. The conservative scheme discussed in this paper is just a representative of the multiple trade-off schemes, which is used to showcase the relationship between electricity bill and tolerance degree. If lower energy cost is required, consumer could choose to reduce tolerance degrees to a more severe extent. This might lead to a less “marginal” result but the consumers are less likely to hold that extreme attitude, 20
being too conservative or optimistic. Therefore, a temperate case with medium conservativeness (tolerance degree) is chosen in this paper to showcase the results. 5. Conclusion In this paper, an interval number optimization based on tolerance degree for constraint violation is introduced to tackle the uncertain household load scheduling in which both the hot water demand and ambient temperature are uncertain and are described as interval numbers. The proposed method is capable of obtaining a cost scheme which strictly subjects to the comfort constraints. Based on this, a trade-off scheme is put forward to be compared with the cost scheme, in which a trade-off between electricity bill and comfort is made. The simulation results demonstrate that the proposed interval number optimization is able to obtain schemes robust to the uncertain hot water demand and ambient temperature and flexible to the different consumer demands. The
proposed
method
is
aimed
at
tackling
the
household
load
scheduling
with
uncertain-but-bounded parameters, which is ignored by major previous optimization methods. Moreover, as a promising methodology, the interval number optimization is rarely applied in the analysis of home energy management. This paper introduces another possible solution to uncertain household load scheduling. Besides, through this proposed method, an active interaction between consumer and HEMS can be achieved by adjusting the tolerance degree according to consumer preferences. The presented method does not take distributed energy resources such as solar and wind into account. More relative works will be undertaken in future reaches. Furthermore, it is just suitable for the day-ahead load scheduling at present. As for real-time scheduling, the model predictive control [41] will be considered. 21
Acknowledgements This work was supported by National Natural Science Foundation of China (NSFC) (51477111). References [1] Rastegar M, Fotuhi-Firuzabad M, Aminifar F. Load commitment in a smart home. Applied Energy, 2012, 96: 45-54. [2] Ha D L, Joumaa H, Ploix S, et al. An optimal approach for electrical management problem in dwellings. Energy and Buildings, 2012, 45: 1-14. [3] Samadi P, Mohsenian-Rad H, Wong V W S, et al. Tackling the load uncertainty challenges for energy consumption scheduling in smart grid. Smart Grid, IEEE Transactions on, 2013, 4(2): 1007-1016. [4] Sun H C, Huang Y C. Optimization of power scheduling for energy management in smart homes. Procedia Engineering, 2012, 38: 1822-1827. [5] Pipattanasomporn M, Kuzlu M, Rahman S. An algorithm for intelligent home energy management and demand response analysis. Smart Grid, IEEE Transactions on, 2012, 3(4): 2166-2173. [6] Mohsenian-Rad A H, Leon-Garcia A. Optimal residential load control with price prediction in real-time electricity pricing environments. Smart Grid, IEEE Transactions on, 2010, 1(2): 120-133. [7] Pedrasa M A A, Spooner T D, MacGill I F. Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. Smart Grid, IEEE Transactions on, 2010, 1(2): 134-143. [8] Du P, Lu N. Appliance commitment for household load scheduling. Smart Grid, IEEE Transactions on, 2011, 2(2): 411-419. [9] Hong Y Y, Lin J K, Wu C P, et al. Multi-objective air-conditioning control considering fuzzy parameters using immune clonal selection programming. Smart Grid, IEEE Transactions on, 2012, 3(4): 1603-1610. [10] 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 [11] Kim T T, Poor H V. Scheduling power consumption with price uncertainty. Smart Grid, IEEE Transactions on, 2011, 2(3): 519-527. [12] Ha D L, Le M H, Ploix S. An approach for home load energy management problem in uncertain context//Industrial Engineering and Engineering Management, 2008. IEEM 2008. IEEE International Conference on. IEEE, 2008: 336-339. [13] Pan Z, Guo Q, Sun H. Impacts of optimization interval on home energy scheduling for thermostatically controlled appliances. Power and Energy Systems, CSEE Journal of, 2015, 1(2): 90-100. [14] Chen Z, Wu L, Fu Y. Real-time price-based demand response management for residential appliances via stochastic optimization and robust optimization. Smart grid, IEEE transactions on, 2012, 3(4): 1822-1831. [15] Conejo A J, Morales J M, Baringo L. Real-time demand response model. Smart Grid, IEEE Transactions on, 2010, 1(3): 236-242. 22
[16] Pedrasa M A, Spooner E D, MacGill I F. Robust scheduling of residential distributed energy resources using a novel energy service decision-support tool. Innovative Smart Grid Technologies (ISGT), 2011 IEEE PES. IEEE, 2011: 1-8. [17] Wang C, Zhou Y, Wu J, et al. Robust-Index Method for Household Load Scheduling Considering Uncertainties of Consumer Behavior. Smart Grid, IEEE Transactions on, 2015, 6(4): 1806-1818. [18] Jiang C, Han X, Guan F J, et al. An uncertain structural optimization method based on nonlinear interval number programming and interval analysis method. Engineering Structures, 2007, 29(11): 3168-3177. [19] Wang C, Zhou Y, Jiao B, et al. Robust optimization for load scheduling of a smart home with photovoltaic system. Energy Conversion and Management, 2015, 102: 247-257. [20] Iwafune Y, Yagita Y. High-resolution determinant analysis of Japanese residential electricity consumption
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[34] Effective June 1, 2008 day ahead pricing used for billing RTP and HSS service [Online]. Available: https://www2.ameren.com/RetailEnergy/ realtimeprices.aspx Jan. 2010? [35] Shi Y, Eberhart R C. Parameter selection in particle swarm optimization//Evolutionary programming VII. Springer Berlin Heidelberg, 1998: 591-600. [36] Shi Y, Eberhart R C. Empirical study of particle swarm optimization//Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on. IEEE, 1999, 3. [37] Eberhart R, Simpson P, Dobbins R. Computational intelligence PC tools. Academic Press Professional, Inc., 1996. [38] Eberhart R C, Shi Y. Comparing inertia weights and constriction factors in particle swarm optimization//Evolutionary Computation, 2000. Proceedings of the 2000 Congress on. IEEE, 2000, 1: 84-88. [39] Trelea I C. The particle swarm optimization algorithm: convergence analysis and parameter selection. Information processing letters, 2003, 85(6): 317-325. [40] Nicol J F, Humphreys M A. Adaptive thermal comfort and sustainable thermal standards for buildings. Energy and buildings, 2002, 34(6): 563-572. [41] Chen C, Wang J, Heo Y, et al. MPC-based appliance scheduling for residential building energy management controller. Smart Grid, IEEE Transactions on, 2013, 4(3): 1401-1410.
24
Lighting
Water Heater
Home Area Network Control Center– Home Energy Management System Gateway
Sensors
Smart Socket Clothes Washer Pool Pump
Fig. 1.
Air Conditioning
TV
System structure of a typical smart home
Fig.2. Must-run service throughout the day
10 9
Waterdemand(gallon)
8 7 6 5 4 3 2 1 0
4
8
12
16
20
Time(h)
Fig. 3. Hot water demand interval throughout the day
25
24
Fig. 4. Ambient temperature interval throughout the day
4 3.5
Price(cents/kwh)
3 2.5 2
1.5 1
0.5 0
8
4
0
12
24
20
16
Time(h)
Fig. 5. Real-time price throughout the day
70 68 66
Temperature(○C)
64 62 60 58 56 54 52 50
5
10
15
20
25
30
35
40
Periods
Fig. 6.Actual hot water temperature in tank
26
45
27
Temperature(○C)
26
25
24
23
22
21
5
10
15
20
25
30
35
40
45
40
45
Periods
Fig. 7.Actual room temperature
70 68 66
Temperature(○C)
64 62 60 58 56 54 52 50
5
10
15
20
25
30
35
Periods
Fig. 8. Hot water temperature under cost scheme
27
Temperature(○C)
26
25
24
23
22
21
5
10
15
20
25
30
35
40
Periods
Fig. 9. Room temperature under cost scheme
27
45
9
Mustrun Water heater air conditioning clothes washer pool pump
8
Load Power(kW)
7 6 5 4 3 2 1 00
5
10
15
20
25
30
35
40
45
Periods
Fig. 10.Power consumption in home under cost scheme
70 68 66
Temperature(○C)
64 62 60 58 56 54 52 50
5
10
1 5
20
25
30
35
40
45
Periods
Fig. 11. Hot water temperature under trade-off scheme
27
Temperature(○C)
26
25
24
23
22
21
5
10
15
25
20
30
35
40
Periods
Fig. 12. Room temperature under trade-off scheme
28
45
9 Mustrun Water heater air conditioning clothes washer pool pump
8
Load Power(kW)
7 6 5 4 3 2 1 0
0
5
10
15
20
25
30
35
40
45
Periods
Fig. 13.Power consumption in home under trade-off scheme
29
Table I Parameters of Clothes Washer and Pool Pump Appliance
𝑃ℎ (𝑘𝑊)
𝑡𝑏
𝑡𝑒
L
Clothes Washer
1
13:00
22:00
3
Pool Pump
1.2
13:00
22:00
5
Table II Parameters of EWH PWH (kW )
QWH ( kW )
RWH ( C / kW )
M ( gallon)
WH
(kWh / C )
WH
( C )
( C )
3.6
120
0.7623
431.7012
40
56
68
CWH
Table III Parameters of AC PAC (kW )
QAC ( kW )
RAC ( C / kW )
C AC
AC
(kWh / C )
AC
( C )
( C )
1.8
1.8
18
0.525
23
26
EWH
AC
CW
PP
Total cost
（cents）
（cents）
（cents）
（cents）
（cents）
（cents）
Cost scheme
26.6100
192.2400
80.7300
3.3000
6.8400
309.7200
Trade-off scheme
26.6100
182.8800
78.5700
3.5000
7.4400
299
Table IV Comparison of schemes Schemes
Must-run
30
S0378-7788(16)30790-3 http://dx.doi.org/doi:10.1016/j.enbuild.2016.08.082 ENB 6981
To appear in:
ENB
Received date: Revised date: Accepted date:
10-1-2016 22-6-2016 27-8-2016
Please cite this article as: Jidong Wang, Yinqi Li, Yue Zhou, Interval Number Optimization for Household Load Scheduling with Uncertainty, Energy and Buildings http://dx.doi.org/10.1016/j.enbuild.2016.08.082 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Interval Number Optimization for Household Load Scheduling with Uncertainty Jidong Wang1,Yinqi Lia, Yue Zhoub Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China
1Corresponding
author. Tel. + 86 13652157798E-mail address: [email protected] Address: School of Electrical Engineering & Automation, Tianjin University, Tianjin, 300072, China a,b
E-mail:[email protected]; [email protected]
Highlights
Household load scheduling with uncertain-but-bounded parameters is analyzed.
Interval number optimization is introduced in the household load scheduling.
Energy cost decreases with the increase of tolerance degrees.
Feasible schemes which are robust to uncertainties could be obtained conveniently.
ABSTRACT An interval number optimization method is proposed in this paper to tackle the household load scheduling problem with uncertain hot water demand and ambient temperature. The household loads considered include residential thermostatically controlled loads such as electric water heater and air conditioner, and interruptible loads such as clothes washer and pool pump. The uncertain-but-bounded parameters are modeled as interval numbers, based on which the uncertain load scheduling problem is formulated and transformed. A binary particle swarm optimization combined with integer linear programming is introduced to solve the transformed problem.
Two schemes, named cost scheme and
trade-off scheme, are contrastively discussed to study the economic impacts of different tolerance 1
degrees for constraint violation. Simulation results demonstrate that the proposed method is flexible to different consumer demands and robust to the uncertainties.
Keywords: residential thermostatically controlled loads; interruptible loads; interval number optimization; household load scheduling; uncertainty; tolerance degree for constraint violation
Nomenclature Variables 𝑋𝑊𝐻,𝑛
Status of electric water heater over period [tn, tn+1] (1―on; 0―off)
𝑋𝐴𝐶,𝑛
Status of air conditioner over period [tn, tn+1] (1―on; 0―off)
𝑋𝐶𝑊,𝑛
Status of clothes washer over period [tn, tn+1] (1―on; 0―off)
𝑋𝑃𝑃,𝑛
Status of pool pump over period [tn, tn+1] (1―on; 0―off)
𝐼 𝐼 𝜃𝑊𝐻,𝑛 , 𝜃𝐴𝐶,𝑛
Interval of hot water temperature and indoor temperature over period [tn, tn+1] (○C)
𝜃𝑊𝐻,𝑐𝑢𝑟
Current hot water temperature (after water draw)
Parameters pn
Real-time price over period [tn, tn+1] (cents/kWh)
𝐼 𝐼 𝜃𝑊𝐻 , 𝜃𝐴𝐶
Comfort setting of hot water in tank and indoor temperature (○C)
tn
Time at nth step(hour)
𝑑𝑛𝐼
Demand interval of hot water drawn over period [tn, tn+1] (gallon)
𝐼 𝜃𝑒𝑛,𝑛
Interval of ambient temperature/inlet water temperature over period [tn, tn+1] (○C)
𝑡𝐶𝑊,𝑏 , 𝑡𝐶𝑊,𝑒
The start time and deadline of clothes washer (hour)
𝑡𝑃𝑃,𝑏 , 𝑡𝑃𝑃,𝑒
The start time and deadline of pool pump (hour)
𝐿𝐶𝑊 , 𝐿𝑃𝑃
Total number of the time steps over load scheduling horizon 2
𝛼
Penalty factor
𝜎
Tolerance degree for constraint violation
𝑃demand
Demand limit of the house (kW)
Ph
Power rating of appliance h (kW)
Pstep,h
Power consumption of appliance h in each step (kW)
Pmustrun,n
Must-run service over period [tn, tn+1] (kW)
𝑄𝑊𝐻 , 𝑄𝐴𝐶
Capacity of electricity water heater and air conditioner (kW)
𝐶𝑊𝐻 , 𝑅𝑊𝐻
Thermal capacitance and resistance of electricity water heater (kWh/○C)
𝐶𝐴𝐶 , 𝑅𝐴𝐶
Thermal capacitance and resistance of air conditioner (○C/kW)
M
The mass of water in full tank (gallon)
𝑟( )
Random number in the interval [0,1]
𝑤
The inertia weight
𝑐1 , 𝑐2
Weighted coefficients of the global and personal best solutions
Subscripts n
Time step index
en
Environment
h
Appliance index
Others H
Set of all appliances
𝐻𝑅𝑇𝐶𝐿
Set of electricity water heater and air conditioner
𝐻𝐼𝐿
Set of clothes washer and pool pump
3
1.
Introduction A home energy management is a component of smart grid that is able to make consumers
participate in the adjustment of power consumption. With the emergence and popularity of various time-varying pricing models and intelligent appliances, consumers may find it difficult to schedule home appliances manually. Considering this reality, home energy management is becoming increasingly necessary for the cost saving and comfortable life in a smart home [1], [2].Based on the development of smart appliances and bi-directional communication technologies, home energy management can effectively reduce household electricity bill and improve consumer power consumption pattern under the premise of satisfying the setting comfort demands [3]. Home energy management makes it possible to schedule home appliances automatically with respect to different pricing mechanisms. One of the most essential parts of home energy management is the household load scheduling [2], [4]. At present, a great amount of studies on the household load scheduling have been done and some valuable achievements have been taken. Among them, many researchers focus on household load scheduling using deterministic optimization methods. In [5], a home energy management algorithm is proposed to schedule household loads based on their load priority. At the same time, the impact of different demand limit levels on demand response potential is also discussed. In [6], a residential energy consumption scheduling framework is introduced to make a trade-off between electricity cost and waiting time. And the pricing mechanisms involved in this paper are real time price combined with inclining block rates. Particularly, a weighted average price prediction filter is proposed to forecast the real-time electricity pricing by assigning different coefficients to prices on some previous days. In [7], to differentiate the importance of desired energy services, different values are assigned to them by 4
consumers firstly. Then, a co-evolutionary particle swarm optimization with stochastic repulsion among particles is proposed to obtain efficient distributed energy resources operation schedules. Furthermore, as various types of uncertainties, such as the uncertainties in model, measurement, communication and forecast, cannot be neglected in real situation, the study of household load scheduling in uncertain context is becoming more and more important. In [8], only electric water heater is studied in the household load scheduling as a representative of thermostatically controlled appliances. Characteristics of an electric water heater are analyzed in detail and a two-step scheduling process is introduced to the electric water heater load scheduling. In [9], in order to achieve the optimal temperature scheduling of air-conditioning, uncertain parameters like outdoor temperature and day-ahead electricity price are transformed into fuzzy parameters. Furthermore, the immune clonal selection programming is applied on the scheduling for air-conditioning. In [10], stochastic optimization is utilized to tackle the uncertain load scheduling involved with the plug-in hybrid vehicle, space heating, pool pumping, water heater and PV generation. Sixteen possibility scenarios are encoded by 8-character codes to describe the uncertainty in the solar insolation, magnitude of service demand, availability of the plug-in hybrid vehicle, and critical peak price status. In [11], a decision-making method is proposed to schedule both interruptible and non-interruptible loads with uncertain electricity price. Relevant statistical knowledge is utilized to reduce the cost. In most aforementioned literatures, when uncertain optimization is transformed into certain optimization, additional function or information need to be introduced into household load scheduling, such as the probability distribution in stochastic programming and fuzzy membership function in fuzzy programming. On one hand, the additional function or information will make the optimization more complex to some extent; on the other hand, both the probability distribution and fuzzy membership 5
function are based on a great amount of history data. In some situations, it is hard to be obtained for new homes or for absence of measurement devices. Faced with these limitations of the existing methods, the methodology of interval number optimization based on tolerance degree for constraint violation is introduced to handle the uncertain optimization in the household load scheduling. After uncertain parameters are expressed as interval numbers, the uncertain household load scheduling is transformed into household load scheduling based on interval number which can be solved with interval number optimization. In this paper, the appliances which are involved in the load scheduling include electric water heater (EWH), air conditioner (AC), clothes washer (CW) and pool pump (PP). The simulation results demonstrate that the interval number optimization is capable of obtaining a viable scheme which could satisfy consumer comfort zone in spite of uncertainties. Considering the diversity of user demand and balance of comfort and electricity bill, the schemes with different tolerance degrees are discussed to give consumer more choices. The main contributions and novelty lie in the following two aspects. 1) As a promising uncertain optimization method, rare works about the theories and applications of interval number optimization are studied in the home energy management. In [12], the uncertain
outdoor temperature and unforeseeable consumption are modelled by interval. Also, multi-parametric programming is introduced to handle this uncertain load scheduling. However it just takes consumer comfort into account, the cost of load scheduling is neglected. In [13], the interval analysis is introduced to the home energy management system (HEMS), and two interval division methods are proposed. However, it is only focused on the impact of optimization time interval. Other parameters like indoor temperature, user demand and so on are considered to be 6
certain. In this paper, an interval number optimization based on tolerance degree for constraint violation is introduced in the household load scheduling with uncertain hot water demand and ambient temperature to make a compromise between the cost and comfort level. 2) Most previous literatures were only focused on the electricity price uncertainty to optimize the cost benefits [14], [15]. Besides, a few works have taken into account the consumer behavior and weather condition with large uncertainty levels [3], [16], in which whether the shower demands will occur at some specific periods and whether the weather will be sunny are considered. But in reality, even when the consumer behavior and weather condition with large uncertainty levels are assumed to be completely determined, some continuous parameters with small uncertainty levels like hot water demand and ambient temperature will also fluctuate in a certain interval, which has a negative influence on user comfort. This proposed method might benefit market players such as HEMS users and producers. For instance, in a new house, the stochastic information about consumers’ habits
and customs is difficult to be obtained as the lack of adequate historical data [17]. As a HEMS user, it is unrealistic to allow HEMS to cost months or even longer to collect related knowledge. However, some uncertain-but-bounded information could be modelled by parameters with small uncertainty levels, for which less uncertainty information is needed [18]. Hence, with the introduction of interval number optimization into HEMS, users’ comfort demand could be satisfied even when only limited information is available. For HEMS producers, to improve competitiveness, it is also important for them to guarantee user comfort for consumers even with only inadequate information. The analysis done in the paper is able to offer a feasible solution to uncertain load scheduling for HEMS users and producers. This paper is organized as follows. The basic load models are introduced in Section 2. The 7
formulation and solution of the load scheduling with uncertain parameters is provided in Section 4. The simulation results are presented in Section 5. This paper is summarized in Section 6. 2. Basic load models 2.1 Traditional load models A typical system structure of smart home is shown in Fig. 1 [19]. The HEMS is the control center of a smart home. It is able to receive external signals and internal information uploaded by smart appliances. Furthermore, the algorithm which is embedded in the control center can tackle these received data and control the power states of smart appliances to improve the power consumption efficiency [19]. Traditionally, smart home devices are divided as controllable services and uncontrollable services [17]. If subdividing them further, EWH and AC are classified as residential thermostatically controlled load (RTCL) but CW and PP belong to interruptible loads. Besides, other uncontrollable devices are lumped together into a must-run service. The uncertain parameters considered in this paper are hot water demand and ambient temperature whose actual values fluctuate in an interval. RTCL in which EWH and AC are two main loads accounts for a great proportion which could reach 40%-50% in domestic load [20], [21], and most peak load demand on the power grid is originated from the RTCL. The schedule of RTCL is aimed at maintaining thermal comfort in house, which is important for consumers [22]. Therefore it is of great significance to take the RTCL scheduling into account. Water heater is a universal appliance in home. Generally speaking, natural gas and electricity are two main components of residential water heating energy use. For instance, in Canada, 51.4% of residential water heating uses natural gas heating; 44.1% uses electricity heating and 4.5% uses other 8
sources [23].This paper is focused on the electricity water heater and its basic working principles are as follows: it turns on to heat the water when water temperature drops to the lower temperature limit, and it turns off when water temperature increases to upper temperature limit. The heat loss of an EWH results from two aspects: the main loss is the cold water inflows with the use of hot water; the other type of loss is the heat exchange between hot water in tank and the environment.
The behavior of an EWH could be formulated by thermal dynamic model [8] .When only the heat exchange is considered, the thermal dynamic of an EWH is computed by
WH ,n1 en,n X WH ,nQWH RWH (en,n X WH , nQWH RWH WH ,n ) exp[(tn 1 tn ) / ( RWH CWH )]
(1)
If the hot water demand is taken into account, the thermal dynamic model of EWH is modified as
WH ,n1 [WH ,cur (M dn ) en,n dn ] / M
(2)
where 𝜃𝑊𝐻,𝑐𝑢𝑟 is the 𝜃𝑊𝐻,𝑛+1 calculated in (1). When the above two aspects are considered synthetically, (1) and (2) are combined to be formulated as
WH ,n1 f (WH ,n , tn , QWH , CWH , RWH , dn , XWH ,n ,en,n ).
(3)
At the same time, AC is a RTCL which is installed extensively in home besides EWH [24]. Its basic working principles share many basic similarities in spite of many specific differences compared with EWH. The main difference is that AC load scheduling just takes the heat loss resulted from heat exchange with the environment into account. The thermal dynamic model of AC is expressed as
AC ,n1 en,n X AC ,nQAC RAC (en,n X AC , nQAC RAC AC ,n ) exp[(tn 1 tn ) / ( RAC C AC )]
(4)
Besides the above RTCL, some interruptible loads like CW and PP are also taken into account. Different from uninterruptible loads, interruptible loads are allowed to be interrupted and resume 9
operation in the next scheduling horizon with little or no consequence. Specifically, the mathematical model of CW is expressed as tCW ,e
n tCW ,b
X CW ,n LCW
(5)
The mathematical model of PP is expressed as tPP ,e
n tPP ,b
X PP ,n LPP
(6)
2.2. Load models with interval number Essentially, interval number is a real number set composed of a closed interval which can be described with its upper limit and lower limit. First of all, basic concepts of interval number are introduced as follows. For arbitrary which are subject to bound and
a
a a , AI [a, a]
a, a R
is named as an interval number in which a is the upper
is the lower bound.
Then, when it comes to the basic mathematical operations of interval number, which include the operations of addition, subtraction, multiplication and division, [25-27] have discussed them in detail. In addition, it is necessary to quantitatively compare an interval number with the other one in practical situations. Accordingly, a new mathematical tool named possibility degree analysis of interval is proposed to quantitatively describe the comparison of two interval numbers. The possibility degree P(a B I ) can be described as
1, a B Ba P(a B I ) ,B a B B B 0, a B I
where 𝑎 denotes a real number and B
is an interval number. 10
(7)
Among the aforementioned four devices, the loads which are involved with uncertain parameters (hot water demand and ambient temperature) are EWH and AC. For EWH, water temperature in tank may deviate from the expected value due to the fluctuations of hot water demand and ambient temperature. As for AC, room temperature may also violate the comfort setting due to the error in outdoor temperature forecast. Sometimes, the deviations of hot water temperature in tank and room temperature are so large that it is unacceptable for most consumers. To handle deviation caused by uncertain information, the interval number analysis is introduced. At first, the thermal dynamic models of EWH and AC are transformed into interval number models. I I I I WH , n 1 f (WH , n , tn , QWH , CWH , RWH , d n , X WH , n , en , n ) I I AC , n 1 en , n X AC , n QAC RAC I (enI ,n X AC ,nQAC RAC AC , n ) exp[(tn 1 tn ) / ( RAC C AC )]
(8)
(9)
3. Formulation and solution of the load scheduling with uncertain parameters 3.1 Formulation of the uncertain load scheduling Through the above transformation, the models with uncertain parameters are changed to interval number models in order to analyze the uncertain load scheduling with interval number optimization. The objective function (10) is to minimize the daily electricity payment. Formula (11) and (12) are the comfort constraints of EWH which are involved with interval number, hence being uncertain constraints. Formula (13) and (14) are uncertain comfort constraints just like (11) and (12). Formula (15) and (16) describe the operation constraints of CW and PP. As there are no interval numbers involved in (15) and (16), they are both certain constraints. Formula (17) is the home demand limit constraint. As shown in [5], if home demand limit is ignored in the process of load scheduling, the high load compensation will occur after the DR events end. That is, in some periods which have cheaper electricity price, many home appliances will be 11
turned on synchronously. Obviously, it is unfavourable for peak shaving and load shifting. Hence, the household load scheduling in this paper is combined with demand limit Pdemand , which means the total power consumption in each period should not exceed the demand limit Pdemand
, otherwise the
schedule is deemed to be invalid. N
minimize
H
p n 1 h
subject to
n
X h,n Pstep ,h
(10)
I I I I WH , n 1 f (WH , n , tn , QWH , CWH , RWH , d n , X WH , n , en , n )
(11)
I WH WH , n WH
(12)
I I I AC , n 1 f ( AC ,n , tn , QAC , CAC , RAC , X AC ,n , en ,n )
(13)
I AC AC , n AC
(14)
tCW ,e
n tCW ,b tPP ,e
n tPP ,b
X CW ,n LCW
(15)
X PP ,n LPP
(16)
H
P X h
h,n
Pmustrun,n Pdemand
(17)
h
Uncertain constraints cannot be tackled directly, so they need to be relaxed. This process is completed with possibility degree analysis of interval. The uncertain constraints from (12) to (14) are dealt with by the possibility degree analysis of interval described in (7). This approach is suggested to transform an uncertain equality constraint into two deterministic constraints. Hence, (12) and (14) are relaxed into I P(WH ,n WH ) 1 1 I P(WH ,n WH ) 2
(18)
I P( AC ,n AC ) 1 3 . I P ( ) AC , n AC 4
(19)
12
Here, 1 and
2
denote the tolerance degrees for constraint violation on upper and lower
limits of water temperature respectively. Similarly, 3 and
4
denote the tolerance degrees for
constraint violation on upper and lower limits of indoor temperature respectively. In substance, the tolerance degree for constraint violation is a parameter utilized to weigh the tolerance degree of consumer’s attitude to comfort violation. The larger their values are, the more tolerable the consumer is, and vice versa. 3.2 Solution of the uncertain load scheduling Through mathematical transformation model based on possibility degree of interval, the uncertain load scheduling is transformed into certain optimization problem. It lays a foundation for the further solution. However, as four controllable loads are involved in the scheduling, there are seven constraints in each scheduling step. The simulation results demonstrate that neither heuristic algorithm nor traditional linear programming algorithm is capable of solving it effectively. If heuristic algorithm is adopted, not only the operation time is difficult to meet the qualification, but also the satisfactory solution is hard to obtain. Moreover, if the traditional linear programming algorithm is adopted, this load scheduling is unsolvable due to the massive variable dimensions and constraints. To this end, a binary particle swarm optimization (BPSO) combined with integer linear programming is introduced in this paper to tackle the above load scheduling. Particle swarm optimization (PSO) is a basic heuristic algorithm in which each candidate solution called a particle is treated as a point in the solution space. The fitness value and movement of particles are respectively determined by the objective function and velocity. The trajectory of a particle is influenced by the best location they have visited and the best performing particle. The speed and position are calculated by 13
t t t t vit,k1 wvit,k c1r ()( pGb ,i pi , k ) c2 r ()( pPb,i pi , k )
(20)
pit,k1 pit,k vit,k1
(21)
where Vk (v1,k , , vn,k ) and Pk ( p1,k , , pn,k ) denote the speed and position of the
i th particle [28].
As a modification of basic PSO, the BPSO is introduced to handle discrete optimization problem. In BPSO, sigmoid function (16) is applied to get a possibility which is mapped by the speed of a particle, and the coordinate of each particle is decided by the resulting possibility. If rand () S (vit 1 ) , then pit 1 0 ; otherwise, pit 1 1 [29].
S (vit 1 )
1 1 exp(vit 1 )
(22)
Generally, BPSO is good at solving the programming problem without constraints. With such background, a penalty function method is applied to unify the objective function and constraints to transform the constrained programming problem into unconstrained programming problem.
( P(M hI NhI ) h ) (min(0, P(M hI NhI ) h ))2 m
where
m
m
m
m
m
(23)
I I I I I I I M hI1 WH , n , N h WH , h1 1 1 ；M h2 WH , n ，N h WH ， h2 2 ；M h3 AC , n , 2
1
I I N hI3 AC , h3 1 3 and M h4 AC ,n ， NhI4 AC , h4 4 。
The unconstrained load scheduling can be described as N H RTCL
p n 1
n
h
X h,n Pstep ,h ( P(M hIm N hIm ) hm ) .
(24)
where H RTCL is the load set which composes of EWH and AC, is a penalty factor which is generally assigned to a big prime number. Then, (24) is suitable to be optimized with BPSO. After a feasible solution, that is the penalty function is equal to zero, is found, the integer linear programming is utilized to tackle the load scheduling including CW and PP. The objective function and constraints of the CW and PP loads scheduling are expressed in (25-28). 14
N H IL
minimize
p n 1
n
tCW ,e
subject to
X h,n Pstep ,h
(25)
h
n tCW ,b tPP ,e
n tPP ,b
X CW ,n LCW
(26)
X PP ,n LPP
(27)
H IL
PWH X WH ,n PAC X AC ,n Ph X h ,n Pmustrun ,n Pdemand
(28)
h
where
X WH ,n
and
X AC ,n
are the feasible solution of (24) obtained by BPSO. Next step, the solving
results of integer linear programming are brought into (10) together with the feasible solution of (24). Then the value of (10) is taken as the fitness value of BPSO.
In theory, the aforementioned method is suitable for a joint optimization with other household loads, which originates from the combination of BPSO and linear programming algorithm. For instance, RTCL such as space heater could be scheduled by BPSO; and interruptible loads such as dishwashers and electric vehicles could be scheduled by linear programming algorithm. Therefore, the household load scheduling in which extra RTCL and interruptible loads are added could also be solved by this proposed method. However with the increase of dimensionality, the actual computation time and operation efficiency are still remained to be verified in future works. 4. Simulation results 4.1 Basic parameters The simulation is divided into two parts: at first, a sensitivity analysis for the uncertainties of hot water demand and ambient temperature is conducted to investigate the impacts of their disturbances on optimization results. Then, the BPSO combined with integer linear programming is introduced to handle the load scheduling with data uncertainties in parameters. Moreover, the electricity bill and 15
power consumption profile of schemes with different tolerance degrees are also discussed to make a comparison between consumer comfort and electricity bill. The scheduling horizon starts from 0a.m until 12p.m and the simulation time step is 30min. The home demand limit Pdemand is set to 7kW. Appliances involved in the household load scheduling are EWH, AC, CW and PP. Their specific parameters are shown in Table I-III. As to the equivalent thermal parameters of EWH and AC which are deemed as certain values in line with the usual practice, many works have been done in references [30], [31], [32].The parameters of Table II- III are chosen from the 2012 ASHRAE Handbook [33]. In addition to aforementioned controllable loads, there are some uncontrollable loads in smart appliance such as lighting, TV and so on. For convenience, different uncontrollable loads are lumped together into a must-run service. The must-run service profile is shown in Fig. 2. In this paper, interval number is utilized to describe the uncertain information in hot water demand and ambient temperature. The interval curves of hot water demand and ambient temperature are shown in Fig. 3 and Fig. 4 respectively. The electricity price involved in this paper is real-time price used by the Illinois Power Company [34] and its profile is shown in Fig. 5.
At present, there is a lack of universally applicable mathematical analysis method to determine PSO parameters. Even so, several valuable works have been done in references [35-38] to provide some guidelines for the selection of the parameters. Generally speaking, the w is advised to be set at [0.9,1.2] and the sum of 𝑐1 and 𝑐2 is suggested to be not less than 4. According to the convergence domain proposed by the convergence analysis in reference [39], the inertia weight w is assigned to be 1 16
and the 𝑐1 , 𝑐2 are assigned to be 1.5 and 2.5 respectively. The swarms in BPSO have 300 particles and the simulation has 500 iterations. Moreover, the penalty factor
is assigned to 10000. Lp_solve, as an open source mixed integer linear
programming system, is utilized to solve the load scheduling of CW and PP. 4.2 Sensitivity analysis In reality, hot water demand has a significant impact in the scheduling result of EWH load. Due to the deviation between actual value and forecast value of hot water demand, actual water temperature in tank will be different from the scheduling result to a certain extent. Besides, the uncertain ambient temperature will have a negative effect on EWH and AC loads scheduling. To evaluate the effect of uncertain hot water demand and ambient temperature，a sensitivity analysis is put forward in this section. Ignoring the uncertainties, the EWH and AC loads scheduling formulated by (24) is calculated based on day-ahead forecast values of water demand and ambient
temperature. Here, the lower bound of water demand interval d n and upper bound of ambient temperature interval
en
are taken as the forecast values. Unfortunately, due to the inevitable forecast
errors, actual values will fluctuate in the intervals as shown in Fig.3 and Fig.4. As the difference between forecast values and actual values, violations of comfort setting will occur frequently as shown in Fig. 6 and Fig. 7.
Obviously, due to the uncertainties in hot water demand and ambient temperature, both the hot water temperature in tank and the room temperature are highly likely to violate the comfort zone. In Fig. 6, the lower bound of actual water temperature is 51.72℃, so the largest water temperature deviation between EWH comfort setting is ∆𝜃𝑊𝐻 = 4.28℃ , and the degree of discrepancy is 17
WH , percent
WH 35.67% . Similarly, the lower bound of actual room temperature is WH WH
21.82℃, and hence the largest water temperature deviation from AC comfort setting is ∆𝜃𝐴𝐶 = 1.18℃, and the degree of discrepancy is AC , percent
AC 39.33% . Such a great violation on AC AC
comfort zone is unacceptable for most consumers [40]. Therefore it is necessary to solve the uncertain load scheduling with interval number optimization to obtain a robust schedule for uncertain hot water demand and ambient temperature. 4.4 An analysis of cost scheme and trade-off scheme When consumers have an absolutely strict requirement for comfort, any violation is deemed to be inacceptable. At this moment, the tolerance degrees for constraint violation are set as and
1 2 0
3 4 0 . For simplicity, the schedule which indicates an absolutely conservative attitude for
constraint violation is named as cost scheme. Under the cost scheme, the interval curves of water temperature in tank and the room temperature are shown in Fig. 8 and Fig. 9. Furthermore, the power consumption in home is shown in Fig.10.The interval curves result from the uncertainties of water demand and ambient temperature. Various water demand and ambient temperature result in different temperature profiles. The set of different profiles comprise an interval curve. As shown in Fig. 3 and Fig. 4, there is a great incertitude degree in hot water demand and ambient temperature, which results in that both the intervals of water temperature and room temperature are large as shown in the blue zone of Fig. 8 and Fig. 9. But in spite of this, both the hot water temperature and room temperature are strictly controlled in the comfort zone. It indicates that interval number optimization proposed in this paper is able to obtain a robust scheduling scheme which will be suitable for all possible scenarios, even though the uncertainties of some parameters are great on some level. 18
This simulation results demonstrate that this method is valid for uncertain optimization and able to satisfy the comfort demand in uncertain context.
As a method of scheduling uncertain household loads, the proposed interval number optimization based on tolerance degree for constraint violation could benefit market players such as HEMS producers and users. For a HEMS producer, obtaining robust load schedules with limited uncertainty knowledge is essential to improve their competitiveness; for a HEMS user, this analysis help to make load schedules that minimize the electricity payment at the premise of user comfort, even inevitable prediction errors exist. In practical application, this proposed method can be embedded in the HEMS to control the power states of smart appliances to satisfy users’ comfort demand in uncertain context. However, as the high electricity bill of cost scheme, not all consumers are willing to choose the cost scheme. Many of them are inclined to take the trade-off scheme, indicating a non-absolutely conservative attitude for constraint violation as their preference. Not all values of tolerance degree in a trade-off scheme need to be set to zero, but different values are assigned to tolerance degree according to actual situation. The scheme in which 1 2 0.2 , 3 4 0.2 is discussed as a representative of trade-off schemes. Under the trade-off scheme, interval curves of hot water demand and ambient temperature are plotted in Fig. 11 and Fig. 12. Moreover, the corresponding power consumption in home is shown in Fig. 13.
As shown in Fig. 8 and Fig. 9, both the comfort setting of hot water temperature and room temperature are violated to a certain extent. But fortunately, the degree of constraint violation is controllable. The actual hot water temperature interval is [55.27,68.89] , and the actual room 19
temperature interval is [22.90,25.57] . In addition to this discussed trade-off scheme, the scheduler is also able to adjust the tolerance degree for constraint violation according to user specific requirements to obtain more schemes with non-absolutely conservative attitude for constraint violation. Finally, a comparison between the costs of cost scheme and trade-off scheme are expressed in Table IV. It can be observed that the costs of EWH and AC loads are reduced along with the decrease in conservative degree. As the consumer chooses to sacrifice partial comfort demand and is more tolerable for constraint violation, the electricity bill is reduced naturally. However, at the same time, there is a tiny increase on the costs of CW and PP load. This is because that with the rise of tolerance degree for constraint violation, the scheduler would turn EWH and AC loads on during some valley price periods. Hence more CW and PP loads have to be turned off during the low-price periods due to the demand limit, and are turned on during high-price periods instead. But compared to the reduction in the costs of EWH and AC loads, this increase is insignificant for total cost.
In all honesty, the differences within the two schemes in the paper are not large. In existing literatures like [15] and [18], the difference between the most conservative case and the least conservative case is indeed more obvious, but for many other cases with medium conservativeness, the costs of the cases are similar or even identical as well. Therefore, the results presented in this paper are consistent with those of other similar existing research. The conservative scheme discussed in this paper is just a representative of the multiple trade-off schemes, which is used to showcase the relationship between electricity bill and tolerance degree. If lower energy cost is required, consumer could choose to reduce tolerance degrees to a more severe extent. This might lead to a less “marginal” result but the consumers are less likely to hold that extreme attitude, 20
being too conservative or optimistic. Therefore, a temperate case with medium conservativeness (tolerance degree) is chosen in this paper to showcase the results. 5. Conclusion In this paper, an interval number optimization based on tolerance degree for constraint violation is introduced to tackle the uncertain household load scheduling in which both the hot water demand and ambient temperature are uncertain and are described as interval numbers. The proposed method is capable of obtaining a cost scheme which strictly subjects to the comfort constraints. Based on this, a trade-off scheme is put forward to be compared with the cost scheme, in which a trade-off between electricity bill and comfort is made. The simulation results demonstrate that the proposed interval number optimization is able to obtain schemes robust to the uncertain hot water demand and ambient temperature and flexible to the different consumer demands. The
proposed
method
is
aimed
at
tackling
the
household
load
scheduling
with
uncertain-but-bounded parameters, which is ignored by major previous optimization methods. Moreover, as a promising methodology, the interval number optimization is rarely applied in the analysis of home energy management. This paper introduces another possible solution to uncertain household load scheduling. Besides, through this proposed method, an active interaction between consumer and HEMS can be achieved by adjusting the tolerance degree according to consumer preferences. The presented method does not take distributed energy resources such as solar and wind into account. More relative works will be undertaken in future reaches. Furthermore, it is just suitable for the day-ahead load scheduling at present. As for real-time scheduling, the model predictive control [41] will be considered. 21
Acknowledgements This work was supported by National Natural Science Foundation of China (NSFC) (51477111). References [1] Rastegar M, Fotuhi-Firuzabad M, Aminifar F. Load commitment in a smart home. Applied Energy, 2012, 96: 45-54. [2] Ha D L, Joumaa H, Ploix S, et al. An optimal approach for electrical management problem in dwellings. Energy and Buildings, 2012, 45: 1-14. [3] Samadi P, Mohsenian-Rad H, Wong V W S, et al. Tackling the load uncertainty challenges for energy consumption scheduling in smart grid. Smart Grid, IEEE Transactions on, 2013, 4(2): 1007-1016. [4] Sun H C, Huang Y C. Optimization of power scheduling for energy management in smart homes. Procedia Engineering, 2012, 38: 1822-1827. [5] Pipattanasomporn M, Kuzlu M, Rahman S. An algorithm for intelligent home energy management and demand response analysis. Smart Grid, IEEE Transactions on, 2012, 3(4): 2166-2173. [6] Mohsenian-Rad A H, Leon-Garcia A. Optimal residential load control with price prediction in real-time electricity pricing environments. Smart Grid, IEEE Transactions on, 2010, 1(2): 120-133. [7] Pedrasa M A A, Spooner T D, MacGill I F. Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. Smart Grid, IEEE Transactions on, 2010, 1(2): 134-143. [8] Du P, Lu N. Appliance commitment for household load scheduling. Smart Grid, IEEE Transactions on, 2011, 2(2): 411-419. [9] Hong Y Y, Lin J K, Wu C P, et al. Multi-objective air-conditioning control considering fuzzy parameters using immune clonal selection programming. Smart Grid, IEEE Transactions on, 2012, 3(4): 1603-1610. [10] 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 [11] Kim T T, Poor H V. Scheduling power consumption with price uncertainty. Smart Grid, IEEE Transactions on, 2011, 2(3): 519-527. [12] Ha D L, Le M H, Ploix S. An approach for home load energy management problem in uncertain context//Industrial Engineering and Engineering Management, 2008. IEEM 2008. IEEE International Conference on. IEEE, 2008: 336-339. [13] Pan Z, Guo Q, Sun H. Impacts of optimization interval on home energy scheduling for thermostatically controlled appliances. Power and Energy Systems, CSEE Journal of, 2015, 1(2): 90-100. [14] Chen Z, Wu L, Fu Y. Real-time price-based demand response management for residential appliances via stochastic optimization and robust optimization. Smart grid, IEEE transactions on, 2012, 3(4): 1822-1831. [15] Conejo A J, Morales J M, Baringo L. Real-time demand response model. Smart Grid, IEEE Transactions on, 2010, 1(3): 236-242. 22
[16] Pedrasa M A, Spooner E D, MacGill I F. Robust scheduling of residential distributed energy resources using a novel energy service decision-support tool. Innovative Smart Grid Technologies (ISGT), 2011 IEEE PES. IEEE, 2011: 1-8. [17] Wang C, Zhou Y, Wu J, et al. Robust-Index Method for Household Load Scheduling Considering Uncertainties of Consumer Behavior. Smart Grid, IEEE Transactions on, 2015, 6(4): 1806-1818. [18] Jiang C, Han X, Guan F J, et al. An uncertain structural optimization method based on nonlinear interval number programming and interval analysis method. Engineering Structures, 2007, 29(11): 3168-3177. [19] Wang C, Zhou Y, Jiao B, et al. Robust optimization for load scheduling of a smart home with photovoltaic system. Energy Conversion and Management, 2015, 102: 247-257. [20] Iwafune Y, Yagita Y. High-resolution determinant analysis of Japanese residential electricity consumption
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2016,116:274-284 [21] Heffner G. Loads providing ancillary services: Review of international experience. Lawrence Berkeley National Laboratory, 2008. [22] Cheng Y, Niu J, Gao N. Thermal comfort models: A review and numerical investigation. Building and Environment, 2012, 47: 13-22. [23] NRC. Survey of household energy use 2007. Technical report. Natural Resources Canada; 2010. [24] Pérez-Lombard L, Ortiz J, Pout C. A review on buildings energy consumption information. Energy and buildings, 2008, 40(3): 394-398. [25] Jaulin L. Applied interval analysis: with examples in parameter and state estimation, robust control and robotics. Springer Science & Business Media, 2001. [26] Jiang C, Han X, Liu G R, et al. A nonlinear interval number programming method for uncertain optimization problems. European Journal of Operational Research, 2008, 188(1): 1-13. [27] Jiang C, Han X, Liu G P. Uncertain optimization of composite laminated plates using a nonlinear interval number programming method. Computers & Structures, 2008, 86(17): 1696-1703. [28] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. 1995 IEEE Int. Conf. Neural Netw., vol. 4, pp. 1942–1948. [29] Kennedy J, Eberhart R C. A discrete binary version of the particle swarm algorithm//Systems, Man, and Cybernetics, 1997. Computational Cybernetics and Simulation, 1997 IEEE International Conference on. IEEE, 1997, 5: 4104-4108. [30] Katipamula S, Lu N. Evaluation of residential HVAC control strategies for demand response programs. Transactions-American society of heating refrigerating and air conditioning engineers, 2006, 112(1): 535. [31] Pratt R G, Taylor Z T. Development and testing of an equivalent thermal parameter model of commercial buildings from time-series end-use data. Pacific Northwest Laboratory, Richland, WA, 1994. [32] Taylor Z T, Pratt R G. The effects of model simplifications on equivalent thermal parameters calculated from hourly building performance data//Proceedings of the 1988 ACEEE Summer Study on Energy Efficiency in Buildings. 1988, 10: 268. [33] 2012 ASHRAE Handbook—HVAC Systems and Equipment” Amer. Soc. Heating, Refrigerating and
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[34] Effective June 1, 2008 day ahead pricing used for billing RTP and HSS service [Online]. Available: https://www2.ameren.com/RetailEnergy/ realtimeprices.aspx Jan. 2010? [35] Shi Y, Eberhart R C. Parameter selection in particle swarm optimization//Evolutionary programming VII. Springer Berlin Heidelberg, 1998: 591-600. [36] Shi Y, Eberhart R C. Empirical study of particle swarm optimization//Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on. IEEE, 1999, 3. [37] Eberhart R, Simpson P, Dobbins R. Computational intelligence PC tools. Academic Press Professional, Inc., 1996. [38] Eberhart R C, Shi Y. Comparing inertia weights and constriction factors in particle swarm optimization//Evolutionary Computation, 2000. Proceedings of the 2000 Congress on. IEEE, 2000, 1: 84-88. [39] Trelea I C. The particle swarm optimization algorithm: convergence analysis and parameter selection. Information processing letters, 2003, 85(6): 317-325. [40] Nicol J F, Humphreys M A. Adaptive thermal comfort and sustainable thermal standards for buildings. Energy and buildings, 2002, 34(6): 563-572. [41] Chen C, Wang J, Heo Y, et al. MPC-based appliance scheduling for residential building energy management controller. Smart Grid, IEEE Transactions on, 2013, 4(3): 1401-1410.
24
Lighting
Water Heater
Home Area Network Control Center– Home Energy Management System Gateway
Sensors
Smart Socket Clothes Washer Pool Pump
Fig. 1.
Air Conditioning
TV
System structure of a typical smart home
Fig.2. Must-run service throughout the day
10 9
Waterdemand(gallon)
8 7 6 5 4 3 2 1 0
4
8
12
16
20
Time(h)
Fig. 3. Hot water demand interval throughout the day
25
24
Fig. 4. Ambient temperature interval throughout the day
4 3.5
Price(cents/kwh)
3 2.5 2
1.5 1
0.5 0
8
4
0
12
24
20
16
Time(h)
Fig. 5. Real-time price throughout the day
70 68 66
Temperature(○C)
64 62 60 58 56 54 52 50
5
10
15
20
25
30
35
40
Periods
Fig. 6.Actual hot water temperature in tank
26
45
27
Temperature(○C)
26
25
24
23
22
21
5
10
15
20
25
30
35
40
45
40
45
Periods
Fig. 7.Actual room temperature
70 68 66
Temperature(○C)
64 62 60 58 56 54 52 50
5
10
15
20
25
30
35
Periods
Fig. 8. Hot water temperature under cost scheme
27
Temperature(○C)
26
25
24
23
22
21
5
10
15
20
25
30
35
40
Periods
Fig. 9. Room temperature under cost scheme
27
45
9
Mustrun Water heater air conditioning clothes washer pool pump
8
Load Power(kW)
7 6 5 4 3 2 1 00
5
10
15
20
25
30
35
40
45
Periods
Fig. 10.Power consumption in home under cost scheme
70 68 66
Temperature(○C)
64 62 60 58 56 54 52 50
5
10
1 5
20
25
30
35
40
45
Periods
Fig. 11. Hot water temperature under trade-off scheme
27
Temperature(○C)
26
25
24
23
22
21
5
10
15
25
20
30
35
40
Periods
Fig. 12. Room temperature under trade-off scheme
28
45
9 Mustrun Water heater air conditioning clothes washer pool pump
8
Load Power(kW)
7 6 5 4 3 2 1 0
0
5
10
15
20
25
30
35
40
45
Periods
Fig. 13.Power consumption in home under trade-off scheme
29
Table I Parameters of Clothes Washer and Pool Pump Appliance
𝑃ℎ (𝑘𝑊)
𝑡𝑏
𝑡𝑒
L
Clothes Washer
1
13:00
22:00
3
Pool Pump
1.2
13:00
22:00
5
Table II Parameters of EWH PWH (kW )
QWH ( kW )
RWH ( C / kW )
M ( gallon)
WH
(kWh / C )
WH
( C )
( C )
3.6
120
0.7623
431.7012
40
56
68
CWH
Table III Parameters of AC PAC (kW )
QAC ( kW )
RAC ( C / kW )
C AC
AC
(kWh / C )
AC
( C )
( C )
1.8
1.8
18
0.525
23
26
EWH
AC
CW
PP
Total cost
（cents）
（cents）
（cents）
（cents）
（cents）
（cents）
Cost scheme
26.6100
192.2400
80.7300
3.3000
6.8400
309.7200
Trade-off scheme
26.6100
182.8800
78.5700
3.5000
7.4400
299
Table IV Comparison of schemes Schemes
Must-run
30