Renewable Energy Output Tracking Control Algorithm Based ... - MDPI

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Renewable Energy Output Tracking Control Algorithm Based on the Temperature Control Load State-Queuing Model Xin Wu, Kaixin Liang * and Xiao Han School of Electric and Electronic Engineering, North China Electric Power University, Changping District, Beijing 102206, China; [email protected] (X.W.); [email protected] (X.H.) * Correspondence: [email protected]; Tel.: +86-132-642-42979





Received: 15 May 2018; Accepted: 4 July 2018; Published: 6 July 2018

Abstract: With poor peak load regulating capacity, renewable energy generation is intermittent and fluctuating, which results in the insufficient acceptance capacity of the power grid. Based on the state-queuing model of aggregate air conditioning loads, this paper develops a control algorithm to achieve renewable energy consumption and output tracking. The load curves of the aggregate air conditioning loads can be controlled by changing the initial temperature distribution. Under different temperature distributions, the load curves represent a fixed fluctuation, which is the basis of output tracking. A virtual load curve set is established based on the state-queuing model. Regarding the load curves as basic signals, the expected renewable energy output can be tracked via an optimal combination of the basic load curves. The validity of the algorithm is testified by numerical emulation data. Keywords: renewable energy output tracking; weighted least squares; immune algorithm

1. Introduction Insufficient peak load regulating capacity of the power grid leads to low utilization of renewable energy. The reason is that the diversity of the power supply is insufficient in some areas, so an amount of renewable energy cannot be consumed efficiently. The demand-side load has strong scheduling potential due to its very large quantity. Therefore, the problem of renewable energy consumption can be solved if the demand-side loads consume renewable energy actively. On the one hand, air conditioning loads, electric water heaters, and other thermostatically-controlled loads have great potential for scheduling [1–4], and they can temporarily store electrical energy in the form of thermal energy [5–10]. On the other hand, for the users, fine-tuning the working time and temperature, the thermal control loads have little impact on user comfort; on the power supply side, they can quickly respond to power dispatching [11]. Recently, there have been a large amount of studies on renewable energy consumption by thermostatically-controlled loads [12–17]. Pourmousavi et al. proposed a tracking method that does not need to provide status information of thermostatically-controlled loads to the central control system in real-time, but it is unsuitable for high-precision tracking [8]. In addition, demand-side loads can also provide a variety of ancillary services for the power supply side [18]. A two-level optimization dispatching model has been established in Gao [19]. This paper aims to maximize the benefits of ancillary services and minimize the dispatching costs. Before the demand-side loads are used for renewable energy consumption or ancillary services, they need to be aggregated firstly. At present, there are two main kinds of control methods for aggregating loads. One is switch control, which changes the power consumption by controlling

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the switching of the aggregate load. The other is temperature control, i.e., changing the power consumption by the adjustment of temperature settings. Lu has developed a state-queuing model of consumption by the adjustment of temperature settings. Lu has developed a state-queuing model thermostatically-controlled loads (TCLs) [20]. Based on the state-queuing model, it controls the of thermostatically-controlled loads (TCLs) [20]. Based on the state-queuing model, it controls the aggregate loads by switching TCLs. An improved state-queuing model is proposed in [21], enhancing aggregate loads by switching TCLs. An improved state-queuing model is proposed in [21], enhancing the stability of the model. Switch control is easy to operate and has great potential to participate in the stability of the model. Switch control is easy to operate and has great potential to participate power dispatching. However, the method has many drawbacks, notably its load fluctuation. The in power dispatching. However, the method has many drawbacks, notably its load fluctuation. different temperature settings will cause the load fluctuation. In a steady state, a small variation of The different temperature settings will cause the load fluctuation. In a steady state, a small variation of the temperature settings will cause load state diversity loss and cause a significant load fluctuation. the temperature settings will cause load state diversity loss and cause a significant load fluctuation. Additionally, switch control has a large influence on the users’ comfort. Thus, the stability of this Additionally, switch control has a large influence on the users’ comfort. Thus, the stability of this control method needs to be improved and the impact on the users should be taken into account as an control method needs to be improved and the impact on the users should be taken into account as an important factor. Callaway proposed an approach that adjusts the air conditioning loads’ important factor. Callaway proposed an approach that adjusts the air conditioning loads’ temperature temperature settings in real-time to change the load with an application to wind energy. It realizes settings in real-time to change the load with an application to wind energy. It realizes renewable energy renewable energy consumption and tracking ancillary services in a new way [22]. Based on [22], consumption and tracking ancillary services in a new way [22]. Based on [22], Bashash established a Bashash established a bilinear differential equation model for the aggregate loads and designed a bilinear differential equation model for the aggregate loads and designed a sliding mode controller sliding mode controller to realize wind power tracking based on the existing load model [23]. This to realize wind power tracking based on the existing load model [23]. This control method is highly control method is highly accurate and has little effect on user comfort. However, it needs to obtain accurate and has little effect on user comfort. However, it needs to obtain information extracted from information extracted from the states of loads and constantly adjust the deviation. This is a closedthe states of loads and constantly adjust the deviation. This is a closed-loop control and the method is loop control and the method is too complicated [21]. too complicated [21]. This paper develops a control algorithm based on state-queuing model of aggregate air This paper develops a control algorithm based on state-queuing model of aggregate air conditioning loads on the power grid, with the goal of consuming renewable energy and tracking conditioning loads on the power grid, with the goal of consuming renewable energy and tracking renewable energy output. We establish a virtual load curves set based on the state-queuing model. renewable energy output. We establish a virtual load curves set based on the state-queuing model. Then we regard the load curves as basic signals and the renewable energy output as the target signal, Then we regard the load curves as basic signals and the renewable energy output as the target signal, we select the optimal load curves for linear combination by a control algorithm to track renewable we select the optimal load curves for linear combination by a control algorithm to track renewable energy output. energy output. This paper introduces the state-queuing model in Section 2 and presents the control algorithm This paper introduces the state-queuing model in Section 2 and presents the control algorithm for for providing tracking service in Section 3. The modeling results are discussed in Section 4. The providing tracking service in Section 3. The modeling results are discussed in Section 4. The discussion discussion and conclusion are summarized in Section 5. and conclusion are summarized in Section 5.

State-QueuingModel Modelof ofThermostatically-Controlled Thermostatically-ControlledLoads Loads 2.2.State-Queuing Inthis thissection, section,aastate-queuing state-queuingmodel modelof ofthermostatically-controlled thermostatically-controlled appliances appliances isis developed. developed. In Takingairairconditionings conditionings example, Section 2.1 introduces the equivalent thermal parameters Taking as as an an example, Section 2.1 introduces the equivalent thermal parameters (ETP) (ETP) model of a single air conditioning; in Section 2.2, a state-queuing model of air conditioning model of a single air conditioning; in Section 2.2, a state-queuing model of air conditioning loads is loads is established based2.1, on inSection 2.1,relationship in which the relationship load and setting established based on Section which the between load andbetween setting temperature range temperature range be simulated; in Section 2.3,ofthe the state-queuing are can be simulated; incan Section 2.3, the characteristics thecharacteristics state-queuingofmodel are analyzedmodel in detail. analyzed in detail. Section 2 will then use the characteristics to develop a tracking control algorithm. Section 2 will then use the characteristics to develop a tracking control algorithm. 2.1.Thermodynamic ThermodynamicModel ModelofofSingle SingleAir AirConditioning Conditioning 2.1. Anequivalent equivalentthermal thermalparameters parameters(ETP) (ETP)model modelofofair airconditioning conditioningisisshown shownininFigure Figure11[24]. [24]. An P Ca

T

R1

T out

in

R2 Tm Cm

Figure 1. Air conditioning equivalent thermal parameter model. Figure 1. Air conditioning equivalent thermal parameter model.

C : air heat capacity ( J /  C ); Ca :a air heat capacity (J/◦ C); Cm : mass heat capacity ( J /  C );

P : rated power for air conditioning ( W );

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Cm : mass heat capacity (J/◦ C); P: rated power for air conditioning (W); η: energy efficiency ratio of air conditioning; R1 : ambient thermal resistance (◦ C/W); R2 : mass thermal resistance (◦ C/W); Tout : ambient temperature (◦ C); Tin : air temperature inside the house (◦ C); Tm : mass temperature inside the house (◦ C); Based on the Newton’s cooling law [18,19], the ETP model is simplified to construct a thermodynamic model of a single air conditioning [18,19]. Equation (1) is the thermodynamic model of an air conditioning in a closed state and Equation (2) is the thermodynamic model of an air conditioning in an operating state: t +1 t +1 t t+1 ∆t Tin = Tout + ( Tin − Tout )ε (1) t t +1 t +1 t +1 Tin = Tout − ηP/U A + [ Tin − ( Tout − ηP/U A)]ε∆t

(2)

where: −1 ε∆t : heat dissipation function, ε= e RC ; t: time (minute); ∆t: time step (1 min); U A: standby heat loss coefficient (W/◦ C); C: equivalent heat capacity (J/◦ C); 2.2. A State-Queuing Model of Aggregate Air Conditionings When demand-side resources participate in tracking renewable energy output, user comfort should not be affected. Thus, we convert the user’s usage requirements into parameters and incorporate them into the model [25]. We set the temperature range required by users to be [ Tmin , Tmax ]. Combined with the temperature range, Equations (3) and (4) can be derived from Equations (1) and (2) if an air conditioning is in the heating mode. P is the average power of an air conditioning. Tmin = Tout + ( Tmax − Tout )ετo f f

(3)


Tmin = Tout − ηP/U A + Tmax − Tout + ηP/U A ετon

(4)

τ = τon + τo f f

(5)

Equation (5) represents a cycling time of aggregate air conditionings, where τon is an “on” period and τo f f is an “off” period. There are n air conditionings divided into τ groups, one of which operates without interference, as shown in Figure 2. The state of the air conditioning changes once per minute. The air conditionings are turned on in the first τon minutes and turned off in the next τo f f minutes. Each state of the air conditionings corresponds to a temperature range. Assuming that the control period is 15 min, the “on” period is five minutes and the “off” is 10 min. The state distribution of the τ groups of loads in one control period is shown in Table 1. τon /τ represents the proportion of air conditionings in the “on” state. It can be found that the proportion is constant if the temperature range is not changed during the control period. The load capacity of air conditionings during the x-period is shown by Equation (6). Px =

τon nP τ

(6)

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Temperature ( C ) 6 7 5

8 9

4

10 11

3

12 13

2 On temp Tmin

14 15

1 1

2

3

4

5

 on

6

7

8

9

10

11 12

13 14

15



Time(min)

Figure 2. Load operation state diagram. Figure 2. Load operation state diagram. Table 1. Fifteen groups of load temperature state distributions. Table 1. Fifteen groups of load temperature state distributions. 1 2 1 32 43 14 on 15 on State 45 5 6 6 7 7 88 9 1010 1111 12 1213 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 5 57 6 8 7 9 8 109 10 15 11 52 3 2 4 2 53 64 11 11 12 1213 13 1414 15 5 4 3 5 3 64 75 12 12 13 1314 14 1515 11 5 68 7 9 8 10 9 11 10 11 22 53 5 4 6 4 75 86 9 10 11 12 13 14 15 1 2 3 4 5 7 8 9 10 11 12 13 14 15 1 2 3 5 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 7 811 9 12 10 1311 14 12 13 45 56 7 5 8 5 9 6 10 15 141 15 2 1 3 2 43 5 8 6 9 6 107 11 1 5 8 912 1013 11 1412 15 13 14 152 1 3 2 4 3 54 56 57 9 7 10 7 118 12 13 1114 12 1513 114 15 2 5 9 10 13 2 4 3 5 4 65 67 58 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 5 10 11 14 25 3 6 4 7 5 86 79 510 11 8 12 8 139 14 15 12 1 13 2 14 315 5 11 121 13 2 14 3 15 41 25 36 4 7 5 8 6 97 8 511 12 9 13 9 1410 15 10 5 1310 14 10 1511 12 1 2 3 4 5 6 7 8 9 10 11 12 5 13 14 15 1 2 3 4 5 6 7 8 9 5 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 5 11 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 514 15 1 2 3 4 5 6 7 8 9 10 11 12 13 5 155 1 6 2 7 3 84 59 610 7 11 8 12 9 13 10 11 515 1 12 2 12 313 14 4 14 5 13 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 5 14 14of the 15State-Queuing 1 2 3 Model 4 5 6 7 8 9 10 11 12 13 5 2.3. Characteristics 15 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 5 The state-queuing aggregation model of the air conditionings per minute is constant without 16 1 2setting 3 changing. 4 5 6However, 7 8 the9temperature 10 11 12 13 range 14 sometimes 15 5 the temperature range setting has to State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

change in practice. Therefore, the relationship between the temperature setting range and the output 2.3. Characteristics of the State-Queuing Model characteristics of the state-queuing model needs to be analyzed. The temperature range is divided into The τo f f state-queuing = 10 segmentsaggregation and the temperature interval of each segment is called a temperature model of the air conditionings per minute is constant withoutunit. the When the control range isHowever, lowered by six units, that is, the temperature space becomes temperature rangetemperature setting changing.   the temperature setting range sometimes has to 3 Tmin −in35 (practice. Tmax − TTherefore, ( T − T ) , the state process of range a group shown in change the relationship between thechanging temperature setting andisthe output max min ), Tmax − min 5 Figure 3. FromofFigure 3, it can be seen that sometostates of the airThe conditionings disappeared characteristics the state-queuing model needs be analyzed. temperaturehave range is divided and become There a state transitioninterval phenomenon. 1 is changed state 4; state2 is  off = 10vacant. into segments andisthe temperature of eachState segment is called into a temperature unit. changed into state 5; state 3 is changed into state 6; states 4–11 disappear; state 12 is changed into When the control temperature range is lowered by six units, that is, the temperature space becomes state 6; state 13 is changed into state 7; state 14 is changed into state 8; state 15 is changed into state 9;  3 3  and 10–15 appear  (Tmax  Tmin ),Tmax vacant. (Tmax  Tmin ) , the state changing process of a group is shown in Figure 3. Tminstates 5 5   From Figure 3, it can be seen that some states of the air conditionings have disappeared and become vacant. There is a state transition phenomenon. State 1 is changed into state 4; state2 is changed into state 5; state 3 is changed into state 6; states 4–11 disappear; state 12 is changed into state 6; state 13 is changed into state 7; state 14 is changed into state 8; state 15 is changed into state 9; and states 10–15 appear vacant.

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Figure 3. Load state transition diagram. Figure 3. Load state transition diagram.

Table 2 2shows of load loadstates statesafter after units of temperature down. The Table showsthe the distribution distribution of sixsix units of temperature settlesettle down. The shaded shaded part indicates that the air conditioning’s state is on. It can be seen that the proportion of airair part indicates that the air conditioning’s state is on. It can be seen that the proportion of conditionings in the “on” state is no longer equal in each time period. The load of air-conditioning conditionings in the “on” state is no longer equal in each time period. The load of air-conditioning fluctuates during a control period. If the airair conditionings sustain operation forfor multiple control fluctuates during a control period. If the conditionings sustain operation multiple control periods, it is not difficult to determine that such load fluctuations are periodic. The core of of this periods, it is not difficult to determine that such load fluctuations are periodic. The core this phenomenon is that air conditionings which meet the new temperature range are insufficient, thus, phenomenon is that air conditionings which meet the new temperature range are insufficient, thus, a transition is needed. needed.When When temperature air conditionings in all a transitionperiod period is thethe newnew temperature rangerange has airhas conditionings in all temperature temperature states, it reaches a stable stage. However,  /  in each time period is no longer equal, states, it reaches a stable stage. However, τ /τ in oneach time period is no longer equal, that is, on

that the proportion air conditionings the “on” no longer theis,proportion of airof conditionings in thein“on” state state are noare longer equal.equal. Table 2. Fifteen groups of load state distributions after temperature adjustment. Table 2. Fifteen groups of load state distributions after temperature adjustment.

State1 State 1 1 4 2 2 5 3 6 3 7 4 5 4 8 6 5 9 7 6 10 8 11 9 7 12 10 8 13 11 9 14 12 15 10 1 13 14 11 2 15 12 3 16 13 4 17 5 18 14 6 19 15 7 20 16 8 21 9

1 2 2 3 3 44 77 88 9 9 1010 11 11 12 1213 13 14 15 55 66 14 on 15 on 4 5 5 6 6 off off off off off 98 2 9 off off off off off off off offoff offoff 6 6 7 78 2 6 7 off off off off off off off 6 7 8 9 10 1 5 6 7 off off off off off off off 6 7 8 9 10 1 7 8 6 off off off off off 6 7 8 9 10 11 0 6 8 7 9 8 7 6 off off off off off off 6 7 8 9 10 11 012 off off off 6 7 8 9 10 11 0 7 9 8 10 9 8 7 off off 6 7 7 8 8 9 9 1010 11 11 12 6 off off off off6 12 013 0 76 off off 6 13 014 0 8 10 9 1110 9 8 off 67 7 8 8 9 9 10 10 1111 12 12 13 8 6 14 0 9 11 10 1211 109 7 off 67 78 8 9 9 10 10 11 11 1212 13 13 14 015 12 13 11 9 7 8 9 10 11 12 13 14 15 1 1 1013 11 1412 1210 108 79 810 911 1012 11 13 12 1413 15 14 151 0 2 86 2 1114 12 1513 1311 119 8 911 1012 1113 12 14 13 1514 15 12 1 3 97 10 1 3 15 1 14 12 10 11 12 13 14 15 1 2 3 4 5 12 13 14 12 10 8 9 10 11 12 13 14 15 1 2 2 1 2 15 13 11 12 13 14 15 1 2 3 4 5 7 13 2 14 3 15 113 14 11 12 9 10 11 1215 13 1 14 2 15 3 1 42 35 3 6 13 14 9 14 3 15 4 1 214 15 12 10 11 46 5 7 13 14 12 15 131 14 2 15 3 1 4 2 53 9 1 14 15 131 142 15 3 1 4 2 5 3 64 10 15 4 1 5 2 315 13 11 12 57 7 8 2 15 1 9 1 5 2 6 3 41 14 12 13 142 153 1 4 2 5 3 6 4 75 68 9 9 6 7 5 3 1 2 3 4 5 6 7 8 9 10 8 2 7 3 8 4 62 15 13 14 154 1 5 2 6 3 7 4 8 5 96 710 911 4 2 3 5 3 8 4 9 5 73 14 15 15 2 6 3 7 4 8 5 9 6 107 811 1012 51 3 4 4 62 4 2 4 9 5 10 6 8 4 15 15 26 3 7 4 8 5 9 6 107 118 912 913 10 11 9 7 5 6 7 8 9 10 11 12 13 14 1 17 5 6 7 5 3 1 2 3 4 5 6 7 8 9 10 8 18 6 7 8 6 4 2 3 4 5 6 7 8 9 10 11 5 Figure load curves up and 19 4 7shows 8 the 9 air 7conditionings’ 5 3 4 5 6 when 7 the 8 temperature 9 10 11range 12 moves 4 down six units, respectively. The red curve represents an upward change in the temperature, and the 20 8 9 10 8 6 4 5 6 7 8 9 10 11 12 13 2 blue curve represents a downward change in the temperature. Different temperature adjustments 21 9 10 11 9 7 5 6 7 8 9 10 11 12 13 14 1 correspond to different load curves. The diversity of load curves can remain abundant. In practical applications, this can only by knowing temperature range instead of adjusting Figure 4 shows the be airachieved conditionings’ load curves the when the temperature range moves up andthe temperature if a certain fluctuating load curve is obtained. The corresponding loads are selected down six units, respectively. The red curve represents an upward change in the temperature, and the according to the loada temperature state distribution table. Different initial temperature distributions blue curve represents downward change in the temperature. Different temperature adjustments correspond to different load curves. The diversity of load curves can remain abundant. In practical

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applications, this can be achieved only by knowing the temperature range instead of adjusting the 6 of 16 certain fluctuating load curve is obtained. The corresponding loads are selected according to the load temperature state distribution table. Different initial temperature distributions correspond to different load curves. If such a load curve is regarded as a basic signal, a target signal correspond to different load curves. If such a load curve is regarded as a basic signal, a target signal can can be obtained by superimposing basic signals of different amplitudes and different frequencies be obtained by superimposing basic signals of different amplitudes and different frequencies according according to the principle of signal decomposition. Therefore, we can make use of the periodic to the principle of signal decomposition. Therefore, we can make use of the periodic fluctuations of the fluctuations of the air conditioning load to achieve renewable energy output tracking. air conditioning load to achieve renewable energy output tracking. The number of units in “on”state

Appl. Sci. 2018, 8, 1099 temperature if a

Time (min)

Figure Load fluctuation curve. Figure 4. 4. Load fluctuation curve.

ControlAlgorithm Algorithm 3.3.Control Basedononthe theabove, above,the theload loadcurves curvesofofthe theaggregate aggregateairairconditionings conditioningscan canbebecontrolled controlledbyby Based changing the initial temperature distribution of the air conditionings. The corresponding load curve changing the initial temperature distribution of the air conditionings. The corresponding load curve can can be extended by one cycle to obtain a periodically fluctuating load curve. Therefore, a virtual load be extended by one cycle to obtain a periodically fluctuating load curve. Therefore, a virtual load curve curve set Q is established based on the controllable air conditioning loads’ temperature states’ set Q is established based on the controllable air conditioning loads’ temperature states’ distribution distribution Theare load curves used as the and basicthe signals, and the renewable table. The loadtable. curves used as theare basic signals, renewable energy outputenergy is usedoutput as the is usedsignal. as theAccording target signal. According to the signalprinciple, decomposition signals with different target to the signal decomposition signalsprinciple, with different amplitudes and amplitudes and frequencies can be superimposed to obtain the desired target signal, that is, frequencies can be superimposed to obtain the desired target signal, that is, in accordance with thein accordance the actualenergy demand of renewable energy select the to appropriate load actual demandwith of renewable consumption, we select consumption, the appropriatewe load curves superimpose curves to superimpose in order to track renewable energy output. This control algorithm tracks the in order to track renewable energy output. This control algorithm tracks the energy output in each energy output in each period. The redistribution of load resources at each stage can effectively period. The redistribution of load resources at each stage can effectively mobilize the load resources mobilize the loadthe resources while maintaining the state diversity of theisload resources. control while maintaining state diversity of the load resources. Load control more accurateLoad because of is more accurate because of adequate load status. adequate load status. Thefirst firstpart partofofthis thissection sectionestablishes establishesthe theobjective objectivefunction functionbased basedononthe theactual actualtracking tracking The requirements; the objective function divided into two parts:the thesolving solvingcoefficients coefficients andthe the target requirements; the objective function is is divided into two parts: and target optimization. The second part describes in detail that how to solve the unknown coefficients ofof the optimization. The second part describes in detail that how to solve the unknown coefficients objective function with fixed load curves combination, so so as as to minimize thethe objective function. The the objective function with fixed load curves combination, to minimize objective function. third part expands the scope of the solution. Under the condition that the load curve combination The third part expands the scope of the solution. Under the condition that the load curve combination is is unknown, unknown,the theoptimal optimalload loadcurve curvecombination combinationisisselected selectedin inthe theQ, Q, so so that that the the objective objective function function is is minimized. minimized. Thus, this paper establishes a renewable energy output tracking control algorithm based the Thus, this paper establishes a renewable energy output tracking control algorithm based onon the state-queuing model of thermostatically controlled loads. state-queuing model of thermostatically controlled loads. 3.1.Objective Objective function 3.1. Function The errorduring duringthe thetracking trackingprocess processisisthe thedifference differenceγ between between renewable energy output The error thethe renewable energy output and aggregate airair conditioning shown inin Equation (7). establish objective andthe the aggregate conditioningloads, loads,asas shown Equation (7).Then Thenwe we establishanan objective function with the goal ofof minimizing overall tracking error, as as described byby Equations (8)(8) and (9):(9): function with the goal minimizing overall tracking error, described Equations and

Yo =YY  ss +γ Y o

(7)(7)

τ

F = min ∑ [Yo ( j) − Ys ( j)]2 j =1

(8)

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m

Ys ( j) =

∑ β(i)Y ( j)(i)

(9)

i =1

m 0). Moreover, the multiplier of the load curve represents the required number of loads for each group and must be non-negative. The constraints can be described as follows: β(i ) ≥ 0, i ∈ [1, m]

(10)

Ys ( j) ≤ Yo ( j), j ∈ [1, τ ]

(11)

3.2. Solution of the Objective Function Since the actual data is heteroskedastic, this paper uses the weighted least squares method to solve the objective function and eliminate the influence of heteroscedasticity. By doing so, the solution parameter β satisfies the Markov estimation [26,27]. Compared with the basic least squares method, the weighted least squares method adds a weighting factor ω to the objective function, which is the reciprocal of the square of the load deviation at the corresponding point. The improved objective function and ω are: 1 ω ( j) = , j ∈ [1, τ ] (12) Y j − ( o ( ) Ys ( j))2  τ   F = min ω ( j)[Yo ( j) − Ys ( j)]2 ∑    j =1    1  ω j = , j ∈ [1, τ ] ( )   (Yo ( j)−Ys ( j))2   m (13) Ys ( j) = ∑ β(i )Y ( j)(i )   i =1    β(i ) ≥ 0, i ∈ [1, m]      Ys ( j) ≤ Yo ( j), j ∈ [1, τ ]    m