1. Introduction
Due to the penetration of a large number of distributed generation (DG) in the distribution system, together with demand-side interaction enhancement, the traditional distribution network is gradually evolving into an active distribution network (ADN), and its topology and power flow are moving towards complexity and diversification [
1]. In order to reduce the complexity of the distribution network model, together with the electricity market reformation, a single distribution network may be divided into interconnected distribution networks that are operated independently by different owners [
2,
3]. Due to technical issues or business privacy, information between regions cannot be exchanged in a timely and efficient manner. In order to give the regional management centers the required responsiveness, it is necessary to establish an equivalent method for the three-phase unbalanced interconnected distribution network. The multi-microgrid (MMG), defined as multiple interconnected small-scale power grid that is comprised of distributed energy resource systems, storage devices, and local demands, also has similar characteristics as the interconnected ADN [
4,
5]. Therefore, the established equivalent method of the interconnection ADN could also be applicable for MMG.
Various equivalent models have been proposed. The Ward equivalent model, REI equivalent model, and their expansion have been widely used [
6,
7,
8,
9,
10]. However, detailed grid information of the external system is needed by these methods, which cannot make an equivalent for an unknown external system. In practice, to deal with this circumstance, the connected parts of boundary nodes were treated as loads, generators, or a combination of the two when constructing the equivalent model [
11,
12,
13]. However, these methods ignore the coupling relationship among the interconnected external networks, which would lead to large errors when a strong coupling relationship among interconnected grids exists.
Therefore, more accurate equivalent models are needed to reflect the external network information. In [
14,
15,
16,
17], the external network equivalent models were built by acquiring the load data of each port, and good results were obtained. However, these models can be used for a single-phase system only, and cannot be applied to the equivalent modeling of three-phase systems. In [
18,
19,
20], Thevenin’s theorem was used to make external network equivalence of multiphase transmission systems to study the transmission performance under different operating conditions. However, when the measurement interval is small and the number of conditions is large, the output may be unstable. The previously mentioned defects were well solved by the deployment of phasor measurement units (PMUs) in [
21,
22,
23,
24,
25], but the proposed methods are only applicable to power transmission systems, and cannot be used for distribution networks with DGs. As for active distribution systems, equivalent models for the external system were established based on the data acquired by PMUs in [
26,
27]. A gray-box equivalent model of the external distribution network was conducted in [
28]. In [
29], the accuracy of the equivalent model was further optimized from the perspective of the reactive power response factor. However, the mentioned equivalent methods are appropriate for three-phase balanced distribution networks with a single port, but not applicable to interconnected three-phase unbalanced distribution networks.
Therefore, an equivalent model for interconnected ADN is proposed in this paper, which is suitable for three-phase balanced and three-phase unbalanced ADN. The contributions of this paper are as follows: (1) In order to enhance the adaptability of the method, a multi-port Norton equivalent model considering the coupling relationship of interconnected networks is proposed. (2) Regarding ADN with unknown external network information, a multivariable linear regression analysis-based method is developed. The multivariable regression model is strictly derived, and the parameters of the multi-port Norton equivalent model of interconnected ADN are estimated using boundary node measurement data. Furthermore, the external system can be a three-phase unbalanced system with unknown topology and state information. (3) For the equivalent accuracy in the operation process of ADN, the boundary information (i.e., complex current, complex voltage) flowing into the external system are selected as regression variables, which are fitted based on the maximum likelihood estimation. The fitting results are used as the external equivalent network parameters, which give the constructed equivalent model a certain dynamic adaptation characteristic.
The rest of the paper is organized as follows: In
Section 2, the multivariable regression equivalent model for interconnected ADN is proposed. In
Section 3, the estimation procedures of the regression model parameters are introduced. In
Section 4, simulations are conducted and analyzed. Finally, conclusions are drawn in
Section 5.
2. Multivariable Regression Equivalent Model
According to circuit principles, external networks connected with the internal network can be equivalent to multi-port Norton models, as shown in
Figure 1. And the external Norton equivalent model consists of multiple current sources and admittance matrices [
30].
UE and
UB represent external node voltage and boundary node voltage matrix, respectively.
YEE and
YEB represent the external admittance matrix and the external-boundary admittance matrix, respectively.
IE and
IBE respectively represent the Norton equivalent current sources and the outputting current matrix from boundary nodes to external nodes. When the interconnected external grid is a three-phase system, the dimension of these matrixes would be expanded to three times that of the original matrix. Besides, the node number of external systems established by the equivalent model equals the number of boundary nodes.
According to circuit principles, the network equations described by an admittance matrix of the power system can be represented as:
The network is divided into the internal node set {
N}, the boundary node set {
B}, and the external node set {
E}, then Equation (1) can be re-expressed as:
The voltage sub-matrix
UE corresponding to the external network in (2) can be eliminated through Gaussian elimination method, and the following linear equation is obtained:
Equation (3) contains the information of the internal network and external unknown network. Specifically,
YBEYEE−1YEB contains admittance information of the external network and
IE contains the external network current information, which is unknown. Besides, the remaining parameters in the equation are known to be associated with the internal network, the following equation can be obtained according to (3):
where
YEB is the mutual admittance matrix between boundary nodes and external nodes. Since the number of external nodes of the established multi-port Norton model is equal to the boundary node number, and the phase loss portion of the boundary node of the unbalanced system can be further deleted,
YEB is a nonsingular matrix. Therefore, (4) can be transformed into:
Let (
YEB−1IE), (
YBEYEE−1YEB)
−1 and (−
IB + YBNUN +
YBBUB) be represented by
UEeq,
β, and
K, then Equations (3) and (5) can be expressed as:
Equation (7) is the mathematical equivalent model of the external grid with unknown parameters. It is vital to estimate the unknown parameters to obtain the real operation state.
The following equation is obtained through expanding (6):
where
b is the number of boundary nodes.
In order to accurately estimate the unknown parameters in Equation (8), the following model is constructed according to the multivariable linear regression analysis method:
Equation (9) is the multivariable linear regression model of the interconnected active distribution network. By solving the model, the external network equivalent parameters β and UEeq can be obtained to estimate the operating state of the system.
3. Algorithm for Regression Model
The algorithm for the multivariable linear regression model of the interconnected ADN is described in this section. Firstly, the boundary system data are collected by measuring equipment placed at boundary nodes. From the theoretical point of view, the equivalent parameters estimation method of the external network is then derived based on the maximum likelihood estimation. Finally, the estimation procedures are introduced.
3.1. Collection Boundary Nodes Information
Measuring equipment, such as PMUs and micro-PMUs, are placed at boundary nodes to synchronously collect the complex current flowing into the external ADN and the complex voltage of the boundary bus [
31,
32], as shown in
Figure 2. The collected boundary information is transmitted to state estimator to constructed Equation (8).
3.2. Maximum Likelihood Estimation of Equivalent Parameters
Regression vertical outliers and poor leverage points can be suppressed by maximum likelihood estimation to achieve the highest statistical efficiency. When the system has a mean Gaussian distribution measurement noise, the Equation (9) can be expressed as follows:
where
U is the set of voltage values collected by the internal system measuring equipment,
k is the current flowing into the external ADN,
β is the constructed external network information matrix, which is called regression coefficient, and
ε is the error vector.
The specific form of
ε is shown as follows:
The density function of error is as follows:
where the constant
σ2 represents the variance.
The maximum likelihood function is the joint density function of
, or
, and its specific form is as follows:
Bring Equation (10) into (13), the likelihood function can be transformed as follows:
For a constant value
σ, to maximize the likelihood function, the optimal estimation matrix
β is required to minimizing the term
. And the residual squared function
S(
β) is introduced:
It is noted that
and
are the same matrix due to the symmetry of the system admittance matrix, and
S(
β) can be further expressed as the following equation:
Since all columns of the regression variable
k obtained according to multiple sets of power flow are not linearly combined with other columns,
k is a nonsingular matrix. Hence, the estimated vector of
is as follows:
The obtained
is the maximum likelihood estimation parameter matrix, and the detailed solution of the multivariable linear regression model can be found in [
33]. The matrix
UEeq can be obtained by bringing the estimated matrix
into Equation (6). Bring
and
UEeq into Equation (7) to establish the multi-port Norton equivalent model of the external system. Thus, the operation state of the interconnected ADN can be effectively estimated accordingly.
3.3. Equivalent Procedures
The external network equivalent method proposed in this paper consists of two phases. In the first phase, the measuring devices are used to synchronously collect the boundary node data (complex voltage and complex current, see
Figure 1). In the second phase, the multiple regression equivalent models are established and solved to obtain the external network equivalent parameters. The external equivalent network is constructed by using the equivalent parameters to reflect the real operating state of the system.
The proposed external network equivalent algorithm can be summarized by the following steps:
Step 1: Set up the measuring devices to synchronously measure the complex voltage UB and complex current K flowing into the external network.
Step 2: The collected boundary information is used as the regression variable in Equation (8). In order to estimate the external network parameters more accurately, the number of regression variables collected can be referenced to [
34].
Step 3: Standardize boundary data according to the following equation:
where
is the average of the collected data, and N is the number of collected data.
Step 4: A multivariate linear regression model is established based on Equation (9), and then the maximum likelihood estimation is used to solve the model based on Equations (10)–(17) to obtain external network equivalent parameters β and UEeq.
The external network equivalent procedures are shown in
Figure 3.