Virtual Battery Modeling of Air Conditioning Loads in the Presence of Unknown Heat Disturbances

Abstract

Air conditioning loads (ACLs) are potential flexible resources that can provide various grid services to the power system. Recent studies have attempted to represent their flexibility using a virtual battery (VB) model for quantification, but the modeling process requires information on thermal parameters and heat disturbances (e.g., solar irradiation and internal heat load) that are difficult to measure. In this paper, we present a new method that models a VB without prior knowledge of such information. First, we construct a thermal dynamic model of an individual ACL using historical input-output data. The linear regression model parameters are identified without using the measurements of disturbances. Second, we derive a VB model from the linear regression parameters using a change of variable technique. We show that the VB can be directly modeled from the regression model of thermal dynamics without estimating the exact thermal parameters and heat disturbances. Third, aggregation of the VB models is implemented. The energy limits of aggregate VB models are designed considering the baseline load prediction error caused by disturbance uncertainty. Finally, simulation results verify the accuracy and effectiveness of the proposed VB modeling strategy.

1. Introduction

The growing integration of renewable energy sources into power systems brings large fluctuations in the net load, which requires more flexibility provision for generation-load balancing. Traditionally, flexibility has been mainly provided by conventional power plants on the generation side of power systems [1]. However, the power plants are being replaced by renewable energy sources that increase the demand for flexibility. Therefore, it is necessary to find alternative flexibility resources in other sectors. It has been widely recognized that the building sector has great potential to provide additional flexibility to the power system [2]. With the development of building energy management systems and communication technologies, many electrical appliances can be utilized to provide operational flexibility [3].

Among the building loads, air-conditioning loads (ACLs) have attracted the most attention due to their high thermal inertia that can be used for storing energy [4,5]. Because each residential ACL has small flexibility, an aggregator is required to gather and mediate them with a system operator [6]. The crucial role of the aggregator is to design an aggregated model of a large number of resources. However, due to the heterogeneous system parameters, modeling and controlling an ensemble of ACLs is challenging. Therefore, a modeling strategy is required to characterize their aggregate flexibility, and many studies have addressed virtual battery (VB) modeling strategies, which approximate thermal dynamics of aggregate buildings into a first-order linear dynamic model. Similar to a Li-ion battery model, the VB model is also identified by parameters on self-dissipation, energy capacity, and operating power limits.

Several efforts have been made to characterize the aggregate flexibility of ACLs. The modeling strategies can be roughly classified into two categories: top-down and bottom-up approaches. The top-down approaches have attempted to identify the VB model directly without quantifying the individual ACL’s flexibility [7,8,9]. For instance, the authors in [7] proposed an identification technique for the VB model based on a stress-testing procedure. Furthermore, the authors in [8] extended this technique by identifying time-varying power limits using the binary search algorithm. The authors in [9] proposed a VB modeling method based on a variational autoencoder. However, these top-down approaches may stress the system and violate the user’s thermal satisfaction with the identification process.

In the bottom-up approaches, each individual flexibility of ACL is modeled first and aggregated into a VB model in the next step. The flexibility of an ON/OFF controlled ACL was characterized in [10] and aggregated into a stochastic battery model. A priority stack-based control was adopted to track frequency regulation signal accurately. Wang et al. [11] extended this work by taking into account the lock time constraints of ON/OFF controlled ACL, and the modified priority stack-based control was proposed for it. The authors in [12] captured the flexibility of an inverter-type ACL as a thermal battery and aggregated them as an ensemble of batteries. In [13], ACLs are aggregated as a virtual generator and two batteries in a multi time-scale operation. They are used to mitigate the wind power generation variability at the level of power systems. In [14], the flexibility of each individual ACL is approximated with a homothetic polytope, and the aggregate model is calculated by Minkowski Sum. This modeling strategy is called the geometric approach, and some extended works have been studied in [15,16,17].

The above studies designed VB models by converting the physical modeling parameters of its building, which is implemented by a first-order equivalent thermal parameters (ETP) model [18], under the assumption that the physical modeling parameters and entire environmental data are known. However, actual modeling parameters of a building are usually unknown and are difficult to estimate [19,20]. In addition, measuring and estimating heat disturbance, such as solar irradiation and internal heat gain, requires a lot of sensors and sophisticated estimation techniques. Because indoor temperature variation in a building depends on the heat disturbances, ignoring the heat disturbances during the VB modeling process should result in degradation of the modeling accuracy.

To overcome these challenges, in this paper, a new VB modeling strategy is proposed which utilizes an operating dataset only. First, a thermal dynamic model is identified using the optimization-based method modified from [21]. To reflect a practical situation, the operating dataset does not include heat disturbance data. Case studies confirm that the effect of heat disturbance on the indoor temperature variation can be inferred and reflected on VB model parameters by using the existing operating dataset. The main contributions of this paper can be summarized as follows.

• An identification method for a thermal dynamic model of a residential AC system is proposed that considers time-varying disturbances. The proposed method does not require any effort for measuring or predicting disturbances in the system.
• From the thermal dynamic model, full derivation of the VB model for individual ACL is given. The VB modeling process does not require a building’s thermal parameters such as thermal resistance and capacitance.
• The energy limits of aggregate VB models are designed considering the baseline load prediction error caused by disturbance uncertainty.

The remainder of the paper is organized as follows. Section 2 describes a single ACL model and flexibility. In Section 3, we obtain an aggregated VB model of ACLs using a change of variable technique. Related case studies are presented in Section 4. In Section 5, we conclude the paper.

2. Modeling of a Single ACL’s Flexibility

2.1. Thermal Dynamic Model of an AC System

To quantify an ACL’s flexibility, it is essential to characterize the thermal dynamic process of an AC system. This paper adopts a first-order ETP model as shown in Figure 1 [22]. Based on the heat balance equation, the indoor temperature dynamics is described as follows:

$$\frac{d{T}_{\mathrm{in}}}{dt}=\frac{1}{R_{\mathrm{th}}{C}_{\mathrm{th}}}(T_{\mathrm{out}}-T_{\mathrm{in}}+R_{\mathrm{th}}{Q}_{\mathrm{ac}}+R_{\mathrm{th}}{Q}_{\mathrm{d}})$$

(1)

where $T_{\mathrm{in}}$ is the indoor temperature; $T_{\mathrm{out}}$ is the outdoor temperature; $Q_{\mathrm{ac}}$ is the heating/cooling capacity; $Q_i$ is the internal heat load; $Q_s$ is the solar heat gain; $R_{\mathrm{th}}$ and $C_{\mathrm{th}}$ are the thermal resistance and thermal capacitance, respectively. In (1), $Q_{\mathrm{d}}=Q_i+Q_s$ denotes the total heat disturbance; and $Q_s=A_s\times I_s$ where $A_s$ is solar heat gain factor and $I_s$ is the solar irradiation [20].

In this work, we assume that an ON/OFF type AC unit operates in the cooling mode. The cooling capacity and power consumption of the ON/OFF type AC unit have the following relationship:

$$Q_{\mathrm{ac}}=-\eta R_{\mathrm{th}}{P}_{\mathrm{rate}}s\left(t\right)$$

(2)

$$s(t)=\{\begin{array}{ll}0,& \mathrm{if} s(t-\epsilon )=1 \mathrm{and} T_{\mathrm{in}}(t)\le T_{\mathrm{set}}-\delta \\ 1,& \mathrm{if} s(t-\epsilon )=0 \mathrm{and} T_{\mathrm{in}}(t)\ge T_{\mathrm{set}}+\delta \\ s(t-\epsilon ),& \mathrm{otherwise}\end{array}$$

(3)

where η is the coefficient of performance (COP), $P_{\mathrm{rate}}$ is the rated power of the AC unit, s(t) is the switching status, which indicates whether the AC unit is turned on or off. Because of the binary variable s(t), the above thermal dynamic model has nonlinear characteristics. For simplicity, we linearize this model by introducing continuous power variable P $\in$ [0, $P_{\mathrm{rate}}$] rather than $P_{\mathrm{rate}}$s(t). The P(k) can be considered as the average of the binary power input $P_{\mathrm{rate}}$s(t) over the scheduling time interval. Then, the model (1)–(3) can be represented as follows:

$$\frac{d{T}_{\mathrm{in}}}{dt}=\frac{1}{R_{\mathrm{th}}{C}_{\mathrm{th}}}(T_{\mathrm{out}}-T_{\mathrm{in}}-\eta R_{\mathrm{th}}P+R_{\mathrm{th}}{Q}_{\mathrm{d}})$$

(4)

It is shown in [10,23] that the aggregate dynamics of a large population of ACLs with (1)–(3) can be approximated by (4) with acceptable accuracy. Also, linear system model Equation (4) can be discretized as follows:

$$T_{\mathrm{in}}(k+1)=a{T}_{\mathrm{in}}(k)-bP(k)+c{T}_{\mathrm{out}}(k)+d^k$$

(5)

where k is the discrete-time index and $\mathsf{\Delta }t$ is the time step interval. The mathematical expression for the parameters a, b, c, and d^k can be seen in

Table 1. Mathematical expression of the thermal dynamic model parameters.

Parameters	a	b	c	dk
Expression	$e^{-\frac{\mathsf{\Delta }t}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}$	$(1-e^{-\frac{\mathsf{\Delta }t}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}{)\eta R}_{\mathrm{th}}$	$1-e^{-\frac{\mathsf{\Delta }t}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}$	$(1-{\mathrm{e}}^{-\frac{\mathsf{\Delta }t}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}{)R}_{\mathrm{th}}{Q}_{\mathrm{d}}\left(k\right)$

. However, calculations of these parameters require information on R_th, C_th, and $Q_{\mathrm{d}}$(k) which are unknown and unmeasurable in common buildings. This means that the parameters a, b, c, and d^k are unknown values and need to be identified based on historical data. In the paper, we obtain model coefficients through an identification instead of using the true values of Table 1.

2.2. Parameter Identification of the Thermal Dynamic Model

In this subsection, we present a method for identifying the parameters a, b, c, and d^k from the historical input-output data. This method is inspired by the optimization-based method proposed in [21] with the following differences:

(1)

a, b, and c are not time-varying variables in the proposed optimization formulation.

(2)

The model parameters are identified using multiple-day data.

In Table 1, we can observe that a, b, and c are time-invariant values associated with the thermal parameters R_th and C_th, whereas d^k is time-varying due to the heat disturbance term $Q_{\mathrm{d}}$(k). Since the thermal parameters and heat disturbance are unknown for most buildings, the model parameters need to be estimated. Although $Q_{\mathrm{d}}$(k) is hard to predict, it can be assumed that its daily pattern is approximately similar to the recent days. From this assumption, we use a variable d^k for multiple-day at time step k. Then, an optimization problem for the VB model identification is formulated as follows:

$$\underset{a,b,c,{\left\{d^k\right\}}_{k=1}^K}{\min } {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{k=1}^K{(T_{\mathrm{in}}^m(k+1)-(a{T}_{\mathrm{in}}^m(k)-b{P}^m(k)+c{T}_{\mathrm{out}}^m(k)+d^k))}^2}}$$

(6)

$$0<a<1,$$

(7)

$$a+c=1,$$

(8)

where m is the index of days and M is the number of days. The objective function is to minimize the regression error of the actual and predicted indoor temperature. It should be noted that the proposed identification method aims to model the thermal dynamic process of a building with scheduling time intervals (e.g., 15 min). Therefore, the proposed method does not require real-time measurement of the historical input-output data.

2.3. Operational Constraints of the AC System

The electrical power consumption of an AC unit is physically constrained by its rated power. Accordingly, we express the power constraints of an ACL as follows:

$$0\le P(k)\le P_{\mathrm{rate}}$$

(9)

In addition, the indoor temperature should be maintained within the acceptable range as follows:

$$T_{\mathrm{set}}-\delta \le T_{\mathrm{in}}(k)\le T_{\mathrm{set}}+\delta$$

(10)

where $T_{\mathrm{set}}$ and δ are the set temperature and half of the dead-band, respectively. In (10), $T_{\mathrm{set}}$ − δ and $T_{\mathrm{set}}$ + δ indicate the minimum and maximum values of their indoor temperature limits, respectively. These parameters can be adjusted by building operators according to their preference.

2.4. Virtual Battery Modeling of an ACL

The above thermal dynamic model and operational constraints can be rewritten as a VB model using a change of variable technique. First, we define the state of charge (SOC) of the AC system from $T_{\mathrm{in}}$(k), as follows [24]:

$$soc(k)=\frac{T_{\mathrm{set}}-T_{\mathrm{in}}(k)}{\delta }$$

(11)

where soc(k) indicates the SOC of the VB at time step k. We can observe that soc(k) = 0 when $T_{\mathrm{in}}$(k) is $T_{\mathrm{set}}$, which means the VB has zero energy at the set temperature. From (11), the temperature constraint (10) can be rewritten as:

$$-1\le soc(k)\le 1$$

(12)

If no external signal is applied, an AC system attempts to maintain the indoor temperature at the initial set temperature, $T_{\mathrm{set}}$. The baseline power, $P_{\mathrm{base}}$(k) maintains the indoor temperature at $T_{\mathrm{set}}$ in time step k. The baseline power of the system can be derived from (5) as follows:

$$P_{\mathrm{base}}(k)=\frac{(a-1)T_{\mathrm{set}}+c{T}_{\mathrm{out}}(k)+d^k}{b}$$

(13)

If an AC system consumes more power than the baseline power, this can be regarded as a “charge” operation because the SOC increases. Conversely, if the AC system consumes less power than the baseline power, this can be regarded as a “discharge” operation because the SOC decreases. Accordingly, we can define the charge/discharge power of the VB model as the deviation from the baseline power as follows:

$$P^b(k)=P(k)-P_{\mathrm{base}}$$

(14)

where P^b(k) is the charge/discharge power at time slot k. From (14), the power constraint (9) can be represented using P^b(k), as follows:

$$-P^{\mathrm{dch}}(k)\le P^b(k)\le P^{\mathrm{ch}}(k)$$

(15)

where $P^{\mathrm{dch}}\left(k\right)=P_{\mathrm{base}}$ and $P^{\mathrm{ch}}\left(\mathrm{k}\right)=P_{\mathrm{base}}-P^{\max }$.

Also, the linear regression model of thermal dynamics can be rewritten using the changed variables. First, subtracting (5) from $T_{\mathrm{set}}$ and dividing by $\delta$, then we have:

$$\frac{T_{\mathrm{set}}-T_{\mathrm{in}}(k+1)}{\delta }=\frac{T_{\mathrm{set}}-(a{T}_{\mathrm{in}}(k)+bP(k)+c{T}_{\mathrm{out}}(k)+d^k)}{\delta }$$

(16)

Using the definition in (11) and (13), we have the following equation:

$$soc(k+1)=asoc(k)+\frac{b}{\delta }{P}^b(k)$$

(17)

Then, the dynamics of the VB model with respect to the SOC is described as follows:

$$soc(k+1)=asoc(k)+\beta P^b(k)$$

(18)

where $\beta =b/\delta$. The above equation can be interpreted in correspondence with the SOC dynamics equation of electrical energy storage system. We can derive the capacity and energy of the virtual battery model from this SOC dynamics equation. First, the capacity of the VB model can be defined as follows:

$$C=\frac{\mathsf{\Delta }t}{\beta }$$

(19)

Since the range of soc(k) is −1 to 1, the above capacity means both upward and downward VB capacities. Additionally, the stored energy of the VB can be expressed as follows:

$$E(k)=C\cdot soc(k)=\frac{\mathsf{\Delta }t}{\beta }soc(k)$$

(20)

If estimations of R_th, C_th, and Q_d(k) are possible, the VB model parameters can be directly obtained using the expression in

Table 2. Comparisons of the VB model parameters.

VB Model Parameters	Modeling w/ Information	Modeling w/o Information
a	$e^{-\frac{\mathsf{\Delta }t}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}$	Estimate
β	$\frac{(1-e^{-\frac{\mathsf{\Delta }t}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}{)\eta R}_{\mathrm{th}}}{\delta }$	Estimate
C	$\frac{C_{\mathrm{th}}\delta }{\eta }$	$-\frac{\mathsf{\Delta }t}{\beta }$
Pbase(k)	$\frac{T_{\mathrm{out}}(k)-T_{\mathrm{set}}+R_{\mathrm{th}}{Q}_{\mathrm{d}}(k)}{\eta R_{\mathrm{th}}}$	$\frac{(a-1)T_{\mathrm{set}}+c{T}_{\mathrm{out}}(k)+d^k}{b}$

without going through the identification process. However, most buildings have equipped insufficient sensing systems to estimate them. Table 2 compares the VB model parameters obtained from different modeling strategies. Building operators can choose one of them depending on their information availability.

3. Aggregate Virtual Battery Model

In this section, we derive an aggregate virtual battery (AVB) model that captures the overall VBs’ flexibility as a single battery model. Let us consider a population of N ACLs. Each ACL is parameterized by index i. We can then define the capacity and energy of the AVB as follows:

$$C_{\mathrm{agg}}={\displaystyle \sum _{i\in \mathcal{N}}{C}_i}, E_{\mathrm{agg}}(k)={\displaystyle \sum _{i\in \mathcal{N}}{E}_i}(k)$$

(21)

where $𝒩$≔{1,…,N} is the set of ACLs; $C_i$ and $E_i$(k) are the capacity and energy of the ith VB, respectively. From (19), the SOC of the AVB is defined as follows:

$$so{c}_{\mathrm{agg}}(k)=\frac{E_{\mathrm{agg}}(k)}{C_{\mathrm{agg}}}=\frac{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(so{c}_i(k)/{\beta }_i\right)}}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}}$$

(22)

3.1. ${\mathit{soc}}_{\mathrm{agg}}$ Dynamics of the AVB Model

To describe the energy state dynamics of the AVB, we need to derive the ${\mathit{soc}}_{\mathrm{agg}}$ dynamics equation with respect to the aggregated VB power, which is defined by $P_{\mathrm{agg}}^b$(k)$={\displaystyle \sum }_i{P}_i^b$(k). First, SOC update equation of the ith VB is expressed as follows:

$$so{c}_i(k+1)=a_{i}so{c}_i(k)+{\beta }_i{P}_i^b(k)$$

(23)

We multiply each equation of (22) by $\frac{(1/{\beta }_i)}{{{\displaystyle \sum }}_i(1/{\beta }_i)}$ and sum the results to obtain:

$$\frac{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(so{c}_i(k+1)/{\beta }_i\right)}}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}}=\frac{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(a_{i}so{c}_i(k)/{\beta }_i\right)}}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}}+\frac{{\displaystyle {\sum }_{i\in \mathcal{N}}{P}_i^b(k)}}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}}$$

(24)

By definition in (22), the left-hand side term is represented by soc_agg(k + 1). However, the right-hand side of (24) is not expressed by soc_agg(k). To obtain soc_agg(k), we approximate it by assuming all soc_i(k) are equal. Then we have the following equation:

$$so{c}_{\mathrm{agg}}(k+1)=\frac{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(a_{i}so{c}_i(k)/{\beta }_i\right)}}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}}+{\beta }_{\mathrm{agg}}{P}_{\mathrm{agg}}^b(k)\approx a_{\mathrm{agg}}so{c}_{\mathrm{agg}}(k)+{\beta }_{\mathrm{agg}}{P}_{\mathrm{agg}}^b(k)$$

(25)

where $a_{\mathrm{agg}}=\frac{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(a_i/{\beta }_i\right)}}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}} , {\beta }_{\mathrm{agg}}=\frac{1}{{\displaystyle {\sum }_{i\in \mathcal{N}}\left(1/{\beta }_i\right)}}$.

3.2. Charge/Discharge Power Limits of the AVB Model

When the indoor temperature constraints (or, equivalently SOC constraints) are not taken into account, aggregate VBs can freely charge up to ${\displaystyle \sum }_i{P}_i^{\mathrm{ch}}$ or discharge down to $-{\displaystyle \sum }_i{P}_i^{\mathrm{dch}}$. However, when the temperature constraints are taken into account, their charge/discharge operation are forced according to their indoor temperature conditions. According to (8), AC units cannot be turned on when the indoor temperature reaches their minimum temperature, while AC units cannot be turned off when the indoor temperature reaches their maximum temperature. The VB models with small capacity and high charge/discharge power limits reach their temperature limits quickly. Considering the SOC constraints of all VBs, the charge and discharge power limits of the AVB model need to be limited to the smallest charge and discharge rate, respectively, among the VBs, as follows:

$$-P_{\mathrm{agg}}^{\mathrm{dch}}(k)\le P_{\mathrm{agg}}^b(k)\le P_{\mathrm{agg}}^{\mathrm{ch}}(k)$$

(26)

$$P_{\mathrm{agg}}^{\mathrm{dch}}(k)=\underset{i}{\min }\left\{\frac{P_i^{\mathrm{dch}}(k)}{C_i}\right\}\times \left({\displaystyle \sum _{i\in \mathcal{N}}{C}_i}\right)$$

(27)

$$P_{\mathrm{agg}}^{\mathrm{ch}}(k)=\underset{i}{\min }\left\{\frac{P_i^{\mathrm{ch}}(k)}{C_i}\right\}\times \left({\displaystyle \sum _{i\in \mathcal{N}}{C}_i}\right)$$

(28)

where $P_{\mathrm{agg}}^{\mathrm{dch}}$ and $P_{\mathrm{agg}}^{\mathrm{ch}}$ are the maximum discharge and charge power limit of the AVB, respectively.

3.3. ${\mathit{soc}}_{\mathrm{agg}}\left(k\right)$ Constraint Considering the Disturbance Uncertainty

From the definition in (22), ${\mathit{soc}}_{\mathrm{agg}}\left(k\right)$ must be constrained within −1 to 1 according to the SOC constraint of each VB model. Using $C_{\mathrm{agg}}{\mathit{soc}}_{\mathrm{agg}}\left(k\right)=E_{\mathrm{agg}}\left(k\right)$, we can express this constraint in terms of the aggregate energy of the AVB model, as follows:

$$-C_{\mathrm{agg}}\le E_{\mathrm{agg}}(k)\le C_{\mathrm{agg}}$$

(29)

The above upward and downward capacities are fully available if the designed VB models have no uncertainty. However, the proposed VB models have uncertainty associated with unknown disturbances. Although we assumed that the daily disturbance pattern is approximately the same, the resulting uncertainty needs to be taken into account for reliable operation. In the proposed model, major model uncertainty is correlated with the baseline power due to the term d^k. The baseline power of the AVB model is calculated by the summing of all VBs’ baseline power as follows:

$$P_{\mathrm{agg},\mathrm{base}}(k)={\displaystyle \sum _{i\in \mathcal{N}}\frac{(a_i-1)T_{\mathrm{set},i}+c_i{T}_{\mathrm{out}}(k)+d_i^k}{b_i}}$$

(30)

On the other hand, the actual baseline power of the AVB model at time step k can be expressed by the sum of true baseline power (see Table 2) as follows:

$$P_{\mathrm{agg},\mathrm{base}}^{\mathrm{actual}}(k)={\displaystyle \sum _{i\in \mathcal{N}}\frac{T_{\mathrm{out}}(k)-T_{\mathrm{set},i}+R_{\mathrm{th},i}{Q}_{d,i}(k)}{\eta R_{\mathrm{th},i}}}$$

(31)

Depending on the disturbance differences between the event day and the recent days, the following prediction error may occur:

$$\mathsf{\Delta }{P}_{\mathrm{agg},\mathrm{base}}(k)=P_{\mathrm{agg},\mathrm{base}}^{\mathrm{actual}}(k)-P_{\mathrm{agg},\mathrm{base}}(k)$$

(32)

To cope with the aggregate baseline power uncertainty, limited energy bound is introduced. If the upper bound of the aggregate baseline power, $\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)$, can be estimated from the historical data, then the following constraint can ensure the SOC constraint under uncertainty:

$$-C_{\mathrm{agg}}+\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)\mathsf{\Delta }t\le E_{\mathrm{agg}}(k)\le C_{\mathrm{agg}}-\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)\mathsf{\Delta }t$$

(33)

Dividing (33) by $C_{\mathrm{agg}}$ gives the following SOC constraint that taken into account the baseline power uncertainty.

$$-1+\frac{\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)}{C_{\mathrm{agg}}}\mathsf{\Delta }t\le so{c}_{\mathrm{agg}}(k)\le 1-\frac{\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)}{C_{\mathrm{agg}}}\mathsf{\Delta }t$$

(34)

With the new SOC bounds above, normal VB operation is possible for the aggregate baseline prediction error less than $\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)$.

4. Results and Discussion

4.1. Simulation Setup

To validate our proposed VB modeling method, 1000 AC simulation models were developed using a first-order ETP model, with the parameters listed in

Table 3. ACL simulation parameters.

	Value	Unit
Cth	~U(1.5, 2.5)	kWh/°C
Rth	~U(1.5, 2.5)	°C/kW
Tset	21	°C
δ	1	°C
Prate	~U(4, 6)	kW
η	3	-
$A_s$	~U(3.72, 4.56)	m2

. The environmental data used in this simulation included data on solar irradiation I_s, outdoor temperature T_out, and internal heat gain Q_i. Solar irradiation and the internal heat gain were utilized to determine the total heat disturbance Q_d. The solar irradiation and outdoor temperature were measured in Austin, Texas, from 22–27 July, as shown in Figure 2 and Figure 3. The internal heat gain was generated by adding 10% of white noise randomly to the typical profile of the heat gain in a day. This dataset was obtained from Gridlab-D software [25] which is widely used in AC system research. The unit time step Δt for the ACL schedules was set to 15 min, and the sizes of the historical data to obtain the VB model parameters were 480, or 5 days. To validate the proposed algorithm, the obtained schedules were simulated by the individual ACL models, calculating their indoor temperatures every 4 s, under the assumption that the environmental factors vary little within the unit time step Δt. The initial indoor temperatures were assumed to be uniformly distributed within the desired range of indoor temperature, which is determined by indoor temperature set-point T_set and dead-band δ. The case studies were implemented using MATLAB software on a computer equipped with a 3.7 GHz Intel Core i5 processor and 32 GB memory.

4.2. Parameter Identification Results

First, normal operations of AC units were conducted from 12:00 a.m. on 22 July to 12:00 p.m. on 26 July to generate the operating data. We then performed the identification procedure to obtain the model parameters a, b, c, and d^k. Figure 4 shows the comparisons of the true and estimated values of a, b, and c. The true values were calculated according to Table 1 using the knowledge of thermal parameters. Due to the page limit, we showed the parameters of the first 100 AC units in Figure 4. The mean absolute error (MAE) between the estimated and true values was evaluated by 0.002 (0.21 %), 0.00052 °C/kW (0.14 %), and 0.002 (2.94 %) for a, b, and c, respectively. Because of constraint (8), the MSEs of a and c are obtained identically. Since δ was set to 1 °C in our simulation, the value of b was equal to β by definition, $\beta =b/\delta$, in Section 2.3. This means that the estimation error of $\beta$ can be obtained by 0.00052 1/kW (0.14%). As a result of the error analysis, we confirmed that the model parameters a, b, and c were estimated with high accuracy.

Figure 5a compares the true and estimated values of d^k associated with heat disturbances term Q_d(k). The values of d^k in Figure 5a were the average values of d^k over all ACLs. The MAE between the true and estimated values was evaluated by 0.023 °C (17.1%). This means that the estimated indoor temperature at the next time step can differ from the actual indoor temperature by an average of 0.023 °C. Since $\delta$ was set to 1, the SOC value also has the same uncertainty as the indoor temperature. Therefore, the estimated value of the next step, SOC, can be different from the actual value of SOC by an average of 0.023. The estimation errors of d^k were higher than the errors of a, b and c. This is because the solar irradiation on July 27th had a different profile, compared to average of other days. We can see this in Figure 2. Unlike other days, more solar energy was irradiated in the afternoon than in the morning, and this effect can also be seen in Figure 5a. Such an error of d^k can cause an error in VB modeling.

To validate the identification method, normal operations of AC units were conducted on 27 July. The, indoor temperature was estimated using the identified thermal dynamic model and the power consumption data. Figure 5b shows the comparison of estimated and true indoor temperature. The estimation was performed for 96 points at 15 min intervals. The MAE between the true and estimated indoor temperature was evaluated by 0.111 °C (0.53%). The prediction error was obtained by subtracting the estimated value from the true value. Since the true d^k is greater than the estimated d^k until 10:00 a.m., we can observe that the true indoor temperature is relatively higher than the estimated indoor temperature. However, between 8:00 a.m. and 12:00 a.m., the estimated d^k is larger, so it can be seen that the prediction error gradually decreases.

4.3. Virtual Battery Modeling and Tracking Performance

Using the estimated parameters of ACLs, we calculated the AVB model parameters according to the definitions in Section 3.

Table 4. Aggregate virtual battery model parameters.

	True	Estimated	Unit
aagg	0.9378	0.9360	-
βagg	3.6228 × 10−4	3.6180 × 10−4	1/kW
Cagg	690.07	690.98	kWh
$P_{\mathrm{agg}}^{\mathrm{dch}}$	[459, 2110]	[450, 2085]	kW
$P_{\mathrm{agg}}^{\mathrm{ch}}$	[279, 2605]	[313, 2615]	kW

shows the comparisons of the true and estimated values for the AVB model parameters. Because the parameters a, b and c of each VB model were estimated with high accuracies, as shown in Section 4.2, the estimated values of a_agg, b_agg and C_agg were calculated similar to their true values. Figure 6 shows the estimated and actual profiles of the aggregated baseline power. In the figure, the aggregated baseline power had more deviation from its true profile than the model parameters a_agg, b_agg and C_agg, because the estimation error of d^k can affect the estimation of aggregated baseline power, and the value of d^k had a higher estimation error than the other parameters. The maximum baseline power prediction error was calculated to be 269.7 kW at 6:00 a.m.

The charge/discharge power limits are time-varying according to the baseline power change, and these profiles can be seen in Figure 7. The charge power limit is low during the day because of the high baseline power, while the charge power limit is high during the night due to the low baseline power. Conversely, the discharge power limit is high during the day and low during the night.

For the real-time operation of the proposed AVB model, frequency regulation tests were conducted to demonstrate the tracking performance in the presence of disturbance uncertainty. A regulation signal $P_{\mathrm{reg}}^b$ was modeled from the test data provided by the PJM electricity market [26] with modification to ensure the AVB model constraints. To track the regulation signal, a priority-stack based controller proposed in [10] was adopted to control the aggregate ON/OFF-type ACLs. In this control method, ON/OFF priorities are ordered according to the SOC values of the ACLs, and this strategy is suitable for use in situations where unknown disturbances exist.

Figure 7 shows the tracking results when the regulation signal satisfies the power constraints and SOC constraints of the AVB model. The tracking error in Figure 7a is calculated by 0.18% of the maximum magnitude of the regulation signal. Figure 7b shows the SOC evolution of the VB models and the AVB model. All VB models were operated, satisfying their SOC constraints. The black line indicates the SOC of the AVB model and the colored lines indicate the SOC of each VB model. Figure 7c shows the indoor temperature of each AC unit. When the regulation signal violates constraints of the AVB model, a huge tracing error is observed. Figure 8 shows that when the regulation signal exceeds the SOC limits, the ensemble of ACLs cannot track the regulation signal due to the temperature constraints.

This subsection conducted two frequency regulation tests to demonstrate the proposed SOC limit constraints. The initial soc_agg(k) was set to −0.8, and the regulation signals were generated by targeting ${\mathit{soc}}_{\mathrm{agg}}^{\mathrm{est}}$ at the end of the regulation test. We performed this regulation test from 6:00 a.m. to 6:15 a.m. on July 22, when the aggregate baseline power error was maximum. Because the maximum baseline power prediction error was measured by 269.7 kW, $\mathsf{\Delta }{\overline{P}}_{\mathrm{agg},\mathrm{base}}(k)$ was set to 300 kW in this study. The new SOC bounds were then calculated using (34), and the value of the lower bound was obtained by $-0.89$. The following test scenarios were performed and the results can be seen in Figure 8.

Test 1: Tracking of a regulation signal generated with ${\mathit{soc}}_{\mathrm{agg}}^{\mathrm{est}}=-0.95$.

Test 2: Tracking of a regulation signal generated with ${\mathit{soc}}_{\mathrm{agg}}^{\mathrm{est}}=-0.89$.

Although the ${\mathit{soc}}_{\mathrm{agg}}^{\mathrm{est}}$ was set to $-0.95$ for Test 1, the SOC of the AVB model reached −1 at around 372 min in Figure 8a due to the baseline power prediction error. Figure 8c shows that the aggregated power of ACLs did not follow the regulation signal at the same time because of SOC violation. In Figure 8b, ${\mathit{soc}}_{\mathrm{agg}}^{\mathrm{est}}$ reached $-0.981,$ although ${\mathit{soc}}_{\mathrm{agg}}^{\mathrm{est}}$ was set to $-0.89$. However, Figure 8d shows that $P_{\mathrm{agg}}^b$ followed $P_{\mathrm{reg}}^b$ successfully because the actual SOC did not reach $-1$ during the regulation test. From the results, we conclude that the proposed VB model requires stricter SOC lower limit than $-1$ to ensure reliable operation against the disturbance uncertainty.

5. Conclusions

This paper proposed a virtual battery modeling strategy of air-conditioning loads that can be used in the presence of unknown heat disturbances and thermal parameters. First, an identification method of a thermal dynamic model was proposed. The identification of modeling parameters is performed under an insufficient historical dataset, which means that some heat disturbances are unmeasured. This identification aims not to find exact thermal parameters or heat disturbances but to find the model parameters. From the model parameter of the thermal dynamic model, a VB model is implemented by parameter conversions. Multiple VB models are aggregated into one representative model, which aims at maintaining the thermal comfort of occupant in buildings when the aggregator operates their ACLs. Simulation results demonstrated the accuracy and effectiveness of the proposed approaches. Compared with the previous method, the proposed method used less information but has a similar accuracy.