Line Loss Interval Algorithm for Distribution Network with DG Based on Linear Optimization under Abnormal or Missing Measurement Data

Abstract

Data collection is more difficult in distribution network than transmission networks since the structure of distribution networks is more complex. As a result, data could be partly abnormal or missing, which cannot completely describe the operation status of distribution network. In addition, access of distributed generation (DG) to distribution network further aggravates the variability of power flow in distribution network. The traditional deterministic line loss calculation method has some limitations in accurately estimating the line loss of distribution network with DG. A line loss interval calculation method based on power flow calculation and linear optimization is proposed, considering abnormal data collection and distribution network power flow variability. The linear optimization model is established according to sensitivity of line loss to the injected power and sensitivity of transmission power of first branch to the injected power. Introducing the scheduling information into the optimization model, a reliable line loss fluctuation interval can be obtained which actual line loss locates. The effectiveness of the proposed algorithm is verified in IEEE 33-bus distribution network system.

1. Introduction

Theoretical line loss is the theoretical line loss electricity calculated according to the power system parameters and real-time power flow data. By the theoretical line loss calculation, the operator can master the active and reactive power losses of transformers, lines and other components in the power system, find the problems existing in the operation of the power system, so as to guide the loss reduction and improve the economic benefits of the power system [1].

The measurement data obtained by the line loss calculation system include power, electric quantity, voltage, etc., which are essential data for line loss calculation. Although the management of power enterprises requires the efficient maintenance of measurement information, there still exists various missing data or errors in the high-frequency calculation of theoretical line loss. Reference [2] introduces the possible abnormal measurement data in the power system, including data missing, reverse direction, abnormal power data caused by data mutation, as well as the lack and error of meter value in power measurement.

For solving abnormal measurement problem of the line loss calculation system in the process of obtaining data, a lot of methods are presented, of which state estimation [3,4,5,6,7,8] is a research hotspot. Reference [3] analyzes the possibility and necessity of using state estimation for theoretical line loss calculation in 35 kV and above power system, showing that using state estimation real-time data for theoretical line loss calculation can reduce errors caused by operators. Reference [4] introduces the power system state estimation method for 35 kV and above transmission networks while analyzing the real-time wiring of the power network. Reference [4] identifies the wrong data of the metering data of the SCADA system to provide more accurate data for the theoretical line loss calculation. Reference [5] proposes an improved method for the calculation of theoretical line loss of transmission network. Reference [6] proposes to use part of the power system state estimation to avoid the inconvenience in the whole network estimation. Reference [7] uses fuzzy clustering algorithm to identify topological errors. In [7], the power system state estimation model based on direct neural dynamic programming is established to calculate the theoretical line loss to obtain the line loss data close to the real situation of the power system. Based on the analysis of line loss calculation data, reference [8] applies data mining technology to line loss calculation to simulate the original line loss data and eliminate heterogeneous data.

Although a large number of studies on state estimation have been presented, providing theoretical support for solving the abnormal measurement of line loss calculation, there are still some issues in application. Most studies mainly focus on transmission networks above 35 kV, and there are few studies on state estimation of distribution networks below 10 kV. Due to the complex structure of distribution network, data collection in distribution network is more difficult than transmission network. There are a lot of errors in the distribution network topology, operation data, and maintenance information obtained by the line loss calculation system, which is difficult to identify. These errors are one of the main reasons for the poor practicability of the 10 kV distribution network line loss calculation system.

On the other hand, the line loss calculation method adopted by the line loss calculation system is also affected by data collection. At present, traditional equivalent model methods, such as the root mean square current method, power factor method (maximum current method), electric quantity method, and equivalent resistance method, are widely used in the theoretical line loss calculation methods of distribution networks below 10 kV [9,10,11]. The above methods adopt different approximate processing methods in terms of current value, resistance and running time, which makes data acquisition easy and calculation simple. However, since the above methods ignore the influence of load curve and power output variability, error of result is around −23~29% [2]. Especially when multiple distributed generations (DG) are connected to the distribution network, the operation of the distribution network is more complex and variable. There are some differences between the actual load curve and the representative daily load curve, which leads to the increase of the calculation error of the load shape coefficient in the traditional equivalent model method. Therefore, these methods have some limitations in calculating the line loss of the distribution network with DGs.

With the improvement of the power system, the intelligence of the distribution network is also gradually strengthened. All kinds of high-precision power monitoring equipment are more popular and can collect a wider range and higher-precision operation data, which provides the possibility for the application of power flow calculation and artificial intelligence method to line loss calculation. As a calculation method with high calculation accuracy, the power flow calculation obtains the accurate solution of the power flow of the section at a certain time by solving nonlinear equations. However, the power flow calculation method cannot be used when data are incomplete or missing. Abnormal data often leads to large calculation error and even non convergence of power flow. At present, the researches on power flow method to calculate distribution network line loss still focus on correcting abnormal data or supplementing missing data combined by state estimation. References [12,13] adopted power flow and estimation technology, using real-time measurement data to correct the load data to improve the calculation accuracy of distribution network line loss.

Artificial intelligence method is a kind of method based on statistical model. Reference [14] improves the accuracy of line loss calculation by improving the k-means method to optimize the neural network model. Reference [15] establishes a neural network model based on gate recurrent unit (GRU) model to predict the line loss of distribution network. Reference [16] proposed a deep learning model based on long-term and short-term memory, which calculates the line loss through parameters such as generator power, power factor and load shape coefficient. Reference [17] proposes a tensor-based data completion method for distribution network, which reduces the impact of missing distribution network measurement data on the line loss calculation model. The advantage of line loss calculation method based on artificial intelligence method is that it does not need to establish a mathematical model, and can use the learning ability and nonlinear processing ability of neural network to fit the complex nonlinear relationship between distribution line loss and characteristic parameters. The disadvantage is that it needs a lot of data support. Besides, the accuracy of the method cannot be assured since this method is based on a statistical model. As a result, the method does not always meet the requirements of power enterprises for line loss calculation and various analysis.

As mentioned above, although a lot of research on the calculation of distribution line loss have been carried out, there still remains some issues in the accurate calculation of distribution line loss. Therefore, some studies propose to calculate the line loss interval instead of calculating the line loss. Reference [18] combines interval arithmetic with forward and backward substitution method to calculate line loss. The voltage and power values as initial values are no longer fixed-point values but are replaced by interval values, which is more realistic in power system. However, interval arithmetic does not consider the correlation of uncertain variables, resulting in more conservative results. Based on convolutional neural network, reference [19] calculates the line loss interval according to the operation data of transformers in different distribution networks for line loss management.

Compared with the power transmission network, the data collection of the distribution network is more difficult, and the power flow state is more variable in a short time, especially when the DG is connected to the distribution network. In addition, due to the acquisition density of automatic metering device, the accuracy of traditional distribution network line loss calculation cannot be guaranteed as well as a reasonable error range. Therefore, calculating the line loss interval is a more reasonable and practical method. In this paper, based on power flow calculation and linear optimization, a calculation method of line loss interval of distribution network with DG is proposed, aiming at giving a reliable line loss fluctuation interval in the case of abnormal or missing data. The actual line loss is within this interval, which could provide a basic information for improving the level of line loss management. The contributions of this paper are summarized as follows.

(1) Considering the variable characteristics of power flow in distribution network with DG, multiple variables are set to represent the power flow state at different time points in a period to improve the accuracy of line loss calculation.

(2) The proposed method establishes a non-linear optimization model, which contains power constraints for buses with normal data collection and upper and lower power constraints for buses with abnormal or missing data. By sensitivity analyze, the non-linear optimization is linearized to a linear optimization, to obtain line loss interval.

(3) The proposed method introduces the dispatching information into the optimization model, establishing the upper and lower limit constraints of the transmission power of the first branch of the distribution network to further compresses the line loss interval and improve the reliability of interval estimation.

2. Optimization Model for Distribution Line Loss Interval Calculation Considering Abnormal or Missing Data

Electric power enterprises require all units to actively build and improve the gateway energy automatic measurement system platform, to achieve the “full coverage” of gateway energy measurement at all levels, and the data are returned “without missing and abnormal numbers”, aiming at meeting the functions of unit energy statistical calculation, line loss theoretical calculation, various gateway management, system parameter management, basic data query, system structure and topology analysis at all levels. In principle, the corresponding basic data should be updated and maintained in the shortest time after the change of power grid equipment (including metering device) and power grid operation mode. Figure 1 is a schematic diagram of a 10 kV distribution network with distributed power supply equipped with an automatic metering device (AMD).

The metering devices can be divided into four categories according to Figure 1.

(1) The transformer of 10 kV distribution network connected to the upper power grid

As shown in AMD1 in Figure 1, in order to meet the dispatching requirements, the metering period is usually about several seconds or minutes. The metering information includes the voltage, current and power of the first branch of the distribution network.

(2) Low voltage side of transformer in 10 kV distribution

As shown in AMD2 in Figure 1, the metering information includes voltage, current, power, electric quantity, etc.

(3) Low voltage user

As shown in AMD3 in Figure 1, the metering information includes voltage, current, power, electricity, etc.

(4) DG access to the power system

As shown in AMD4 in Figure 1, the metering information includes voltage, current, forward and reverse power, electricity, etc.

Theoretical line loss calculation needs to calculate multiple time sections, such as 24-point calculation (that is, one must select a time section every hour to calculate the theoretical line loss and consider the calculated value of this time as the average value of this hour). In this paper, it is assumed that the metering cycle of equipment at all locations of 10 kV distribution network is one hour.

Although the management of power enterprises requires the efficient maintenance of collected information, in the calculation of high-frequency theoretical line loss, there still are various cases of missing or bad data. The various abnormal acquisition conditions that lead to missing or wrong data of line loss calculation are as follows.

(1) The missing data caused by the problems of the acquisition system and data processing system, such as insufficient collected values and lost information.

(2) Due to the instability of the acquisition system, some sudden wrong data sometimes appear.

(3) There are many metering gates in the distribution network, and the frequent meter changing operation leads to the loss and error of the meter of the switching electric energy meter.

(4) Jitter or poor contact of relay contact; Power interference and fault; Strong electromagnetic interference; The loss or abnormality of remote signaling information caused by the restart of telecontrol system and the corresponding time of protocol.

(5) Errors in basic maintenance information and data loss caused by management.

These situations may lead to abnormal or missing measurement data, which seriously affects the calculation and analysis of line loss. When individual data is lost, the prediction method can be used for filling. However, when missing data is too much, prediction method cannot provide accurate instantaneous power value for line loss calculation since the obtained load curve is incomplete.

In fact, the incomplete measurement data can still reflect some characteristics of the system power flow state in the period. Although it is impossible to give an accurate line loss value by deterministic methods such as power flow calculation, a line loss fluctuation range can be obtained theoretically, which actual line loss locates.

In order to briefly explain the principle of line loss interval calculation, the 5-bus system in Figure 2 is taken as an example to calculate line loss of the distribution network in T period. In Figure 2, bus 1 is the first bus of distribution network, bus 3 and 5 are load buses, and bus 4 is connected to a DG. $U_N(t)$ is the voltage of first bus of the distribution network. $P_{L1}(t),P_{L2}(t)$ is the active load of load bus 3 and 5, respectively, and $P_G(t)$ is the active power output of DG. $U_N(t),P_{L1}(t),P_{L2}(t),P_G(t)$ are all time-varying, which are functions of time t. For simplicity, it is assumed that the reactive power of load and DG is 0, the metering data of bus 5 is missing, and the metering data of other buses are normal.

When $U_N(t)$, $P_{L1}(t)$, $P_{L2}(t)$, $P_G(t)$ are determined, the line loss rate of the distribution network can be accurately solved by power flow calculation, denoted as $loss(t)=F(U_N(t),P_{L1}(t),P_{L2}(t),P_G(t))$. However, the metering device can only collect the current and voltage information of a certain time or the electric quantity information of a period of time. According to the information collected by the metering equipment, it can be obtained that the system state variables in t period are limited by the constraints shown in Formula (1). The first and second constraints represent the upper and lower limits of active load. The third constraint represents the upper and lower limits of active output of DG. The fourth and fifth constraints represent the active power constraints of bus 3 and distributed power generation The sixth constraint represents the voltage constraint of the first bus.

$$\{\begin{array}{c}\begin{array}{l}{P}_{L1}^{\min }\le P_{L1}(t)\le P_{L1}^{\max },\forall t\in [0,T]\\ P_{L2}^{\min }\le P_{L2}(t)\le P_{L2}^{\max },\forall t\in [0,T]\end{array}\\ P_G^{\min }\le P_G(t)\le P_G^{\max },\forall t\in [0,T]\\ {\displaystyle \underset{0}{\overset{T}{\int }}{P}_{L1}(t)dt}=W_{L1}^{(q_s)}-W_{L1}^{(q_1)}\\ \begin{array}{l}{\displaystyle \underset{0}{\overset{T}{\int }}{P}_G(t)dt}=W_G^{(q_s)}-W_G^{(q_1)}\\ U_N(t)=U_N^{(t)},\forall t=q_1,q_{2,}\cdots ,q_s\end{array}\end{array}$$

(1)

where t is the time variable, $q_1,q_{2,}\cdots ,q_s$ is the acquisition time point of the metering device at the transformer connected to the upper grid. $q_1,q_s$ is the beginning and end of the period T. $W_{L1}^{(q)}$ is the value of the active watt-hour meter of bus 3 at time q, and $W_G^{(q)}$ is the value of the active watt-hour meter of DG of bus 4 at time q. $P_G^{\max },P_G^{\min }$ is the maximum and minimum value of active power output of DG. $P_{Li}^{\max },P_{Li}^{\min }$ (i = 1,2) is the maximum and minimum value of active load of bus 3 or 5.

Most of the traditional line loss calculation methods are based on a period of time for line loss calculation, that is, the load current, voltage and other factors are calculated in an average manner over a period of time [20]. It is generally assumed that the power flow state remains unchanged in period T as follows

$$\{\begin{array}{l}{P}_{L1}\equiv (W_{L1}^{(q_s)}-W_{L1}^{(q_1)})/T\\ P_G\equiv W_G^{(q_s)}-W_G^{(q_1)}/T\end{array}$$

(2)

However, due to the lack of metering data of bus 5, the average power of bus 5 in this period cannot be obtained. Therefore, deterministic power flow calculation cannot be used to obtain the line loss. However, in theory, the corresponding optimization model can be established according to the measurement information to calculate a fluctuation range of line loss within period T. The actual line loss is within this range, as shown below

$$\begin{array}{l}\underset{P_L,P_G}{\max /\min }{\displaystyle \underset{0}{\overset{T}{\int }}F(U_N(t),P_{L1}(t),P_{L2}(t),P_G(t))dt}\\ \begin{array}{cc}s.t.& (1)\end{array}\end{array}$$

(3)

The optimization model (3) could be extended to a general distribution network with n buses, as shown below

$$\begin{array}{l}\underset{P,Q}{\max /\min }{\displaystyle \underset{0}{\overset{T}{\int }}F(U_N,P,Q)dt}\\ \begin{array}{cc}s.t.& \{\begin{array}{l}{P}_i^{\min }\le P_i\le P_i^{\max },\forall i\in N,\forall t\in [0,T]\\ Q_i^{\min }\le Q_i\le Q_i^{\max },\forall i\in N,\forall t\in [0,T]\\ {\displaystyle \underset{0}{\overset{T}{\int }}{P}_{i}dt}=W{p}_i^{(q_s)}-W{p}_i^{(q_1)},\forall i\in A_P\\ {\displaystyle \underset{0}{\overset{T}{\int }}{Q}_{i}dt}=W{q}_i^{(q_s)}-W{q}_i^{(q_1)},\forall i\in A_Q\\ U_N(t)=U_N^{(t)},\forall t=q_1,q_{2,}\cdots ,q_s\end{array}\end{array}\end{array}$$

(4)

where, n represents the number of buses, and P, Q represent the injected active and reactive power of all buses, respectively. $W{p}_i^{(q)},W{q}_i^{(q)}$ represents the value of active watt-hour meter and reactive watt-hour meter of bus i at time q, respectively.$A_P$, $A_Q$ is the bus set with normal data acquisition of active watt hour meter and reactive watt hour meter, respectively. Equation (4) shows two optimization models which, respectively, calculate the upper and lower limits of the line loss interval. Since there are integrals in the constraint and objective function, it is difficult to solve Equation (4). This paper approximately solves the optimization model (4) by discretization and linearization, which will be given in Section 3.

3. Solution Method for Line Loss Interval Based on Discretization and Linearization

3.1. Influence of DG on Line Loss Calculation of Distribution Network

The traditional theoretical line loss calculation needs to calculate multiple time sections [20], such as 24-point calculation. In other words, the operator selects a time section every hour to calculate the theoretical line loss. Under the assumption that the time T is short and the system state does not change much, an estimated value of line loss can be obtained. However, the power system often shows irregular changes with time [21]. Many researches put the focus on the change rule of load, thinking that power system at a certain time is a form of several consecutive time segments. Nevertheless, it could be found that electric load curve that changes are often unpredictable [22], especially in the distribution network with DGs. Due to the large fluctuation of load in the distribution network and the random fluctuation of DG output, the power flow may change greatly in a short time. Figure 3a shows the real load curve of two families in the United States in one day (the measurement period is 5 min), and Figure 3b shows the power generation curve of wind power and photovoltaic in one day (the measurement period is 5 min). It can be seen from Figure 3 that the load fluctuation in the distribution network is violent as well as output of DG. The intermittence of output of DG is obvious, the output fluctuation range is large, and the fluctuation frequency is irregular. Therefore, the power flow of distribution network may also change greatly in a short time, and the traditional method such as 24-point method may cause a large error.

For example, it is assumed that transmission power through lines 2–4 in the 5-bus system of Figure 2 is as follows.

Due to the randomness of DG, the power fluctuates in the range of [$1-\sigma$,$1+\sigma$] within one hour. It is Assumed that the resistance of the branch connected to the DG is R, and the DG adopts PV control mode, that is, the voltage remains unchanged at 1 (p.u.). Ignoring the voltage drop of the branch, the electric quantity within one hour is $(1+{\sigma }^2)R$ measured by a watt hour meter. The traditional line loss calculation method assumes that the output of DG remains unchanged within one hour (that is, assuming that the branch power of one hour is shown by the dotted line in Figure 4). In this way, the calculated line loss is R, which is $1/(1+{\sigma }^2)$ times the actual line loss. The power flow fluctuation of the transmission network in a short time is small, which means $\sigma$ is small. Therefore, the difference between the calculated line loss and the actual line loss is small. However, in the distribution network with DG, this method may cause greater error since $\sigma$ is larger. In engineering, the shape coefficient of load curve is often used to correct the calculated line loss [9,10]. However, due to the irregularity of the output frequency of DG and the acquisition frequency of metering device, it is actually difficult to obtain an accurate shape coefficient.

3.2. Discrete Optimization Model Considering Power Constraint of the First Branch of Distribution Network

The optimization model (4) contains integral, which is difficult to solved. Therefore, the model needs to be simplified. First of all, the voltage fluctuation at the first bus is small, which can be approximately regarded as unchanged as follows.

$$U_N(t)\equiv U_N^{(q_1)}$$

(5)

Considering the fluctuation of system power flow in T period, T period is divided into h small periods. It is assumed that power flow in each small period is unchanged. Therefore, the optimization (4) can be simplified into the following form

$$\underset{P^{(1)},P^{(2)},\dots ,P^{(h)},Q^{(1)},Q^{(2)},\dots ,Q^{(h)}}{\max /\min }{\displaystyle \sum _{k=1}^{h}F(P^{(k)},Q^{(k)})·T/h}$$

(6a)

Subject to the following constraints

$$\{\begin{array}{c}{P}_i^{\min }\le P_i^{(k)}\le P_i^{\max },\forall k=1,\dots ,h,\forall i\in N\\ Q_i^{\min }\le Q_i^{(k)}\le Q_i^{\max },\forall k=1,\dots ,h,\forall i\in N\end{array}$$

(6b)

$$\{\begin{array}{l}{\displaystyle \sum _{k=1}^h{P}_i^{(k)}·T/h}=W{p}_i^{(q_s)}-W{p}_i^{(q_1)},\forall i\in A_P\\ {\displaystyle \sum _{k=1}^h{Q}_i^{(k)}·T/h}=W{q}_i^{(q_s)}-W{q}_i^{(q_1)},\forall i\in A_Q\end{array}$$

(6c)

where, $P^{(k)},Q^{(k)}$ represents the injected active and reactive power of all buses in the distribution network in the k-th small period.

Since the transformer connected to the upper power grid is usually equipped with high-frequency automatic metering device, the power information measured by the metering device is introduced into the above optimization model as a constraint to further compress the line loss interval. For each small period, the active and reactive power of the first section of the distribution network should be between the maximum and minimum power measured by the metering device, as shown below

$$\{\begin{array}{l}{P}_{\min }^{(k)}\le P_f^{(k)}\le P_{\max }^{(k)},\forall i\in N,\forall k=1,2,\dots ,h\\ Q_{\min }^{(k)}\le Q_f^{(k)}\le Q_{\max }^{(k)},\forall i\in N,\forall k=1,2,\dots ,h\end{array}$$

(6d)

where $P_f^{(k)}$,$Q_f^{(k)}$ represents the active and reactive power flowing through the first branch of the distribution network in k-th period. $P_{\min }^{(k)},P_{\max }^{(k)}$, respectively, represent the minimum and maximum values of active power measured by the metering device in k-th period. $Q_{\min }^{(k)},Q_{\max }^{(k)}$, respectively, represent the minimum and maximum values of reactive power measured by the metering device in k-th period.

3.3. Linear Optimization for Line Loss Interval

The difficulty in solving the optimization model (6) is that the objective function $F(P^{(k)},Q^{(k)})$ and the power transmitted by the first branch of the line $P_f^{(k)}$,$Q_f^{(k)}$ is non-linear function of $P^{(k)},Q^{(k)}$. Therefore, in this paper, the objective function (6a) and constraint (6d) are transformed into linear expressions and linear constraints by sensitivity analysis.

Firstly, an operation point is determined as the central point, and the injected powers of each bus are determined by (7) and (8).

$$\{\begin{array}{l}\overline{P_i}=\left(W{p}_i^{(q_s)}-W{p}_i^{(q_1)}\right)/T,\forall i\in A_P\\ \overline{Q_i}=\left(W{q}_i^{(q_s)}-W{q}_i^{(q_1)}\right)/T,\forall i\in A_Q\end{array}$$

(7)

$$\{\begin{array}{l}\overline{P_i}=(P_i^{\min }+P_i^{\max })/2,\forall i\notin A_P\\ \overline{Q_i}=(Q_i^{\min }+Q_i^{\max })/2,\forall i\notin A_Q\end{array}$$

(8)

Line loss $P_{loss,0}$ and transmission power of the first branch $P_{f,0},Q_{f,0}$ can be obtained by calculating the power flow according to the above determined power injection. According to the sensitivity analysis, the linearized expression of the objective function can be obtained by (9).

$$F(P^{(k)},Q^{(k)})=P_{loss,0}+A_P^{loss}·(P_i^{(k)}-\overline{P_i})+A_Q^{loss}·(Q_i^{(k)}-\overline{Q_i})$$

(9)

The linearization expression of the transmission power of the first branch is as follows.

$$\{\begin{array}{c}{P}_f^{(k)}=P_{f,0}+A_P^{Pf}·(P_i^{(k)}-\overline{P_i})+A_Q^{Pf}·(Q_i^{(k)}-\overline{Q_i})\\ Q_f^{(k)}=Q_{f,0}+A_P^{Qf}·(P_i^{(k)}-\overline{P_i})+A_Q^{Qf}·(Q_i^{(k)}-\overline{Q_i})\end{array}$$

(10)

where, $A_P^{loss}$ and $A_Q^{loss}$, respectively, represent the sensitivity of line loss power to the injected active and reactive power of each bus. $A_P^{Pf}$ and $A_Q^{Pf}$, respectively, represents the sensitivity of the transmission active power of the first branch to the injected active and reactive power of each bus. $A_P^{Qf}$ and $A_Q^{Qf}$, respectively, represents the sensitivity of the transmission active power of the first branch to the injected active and reactive power of each bus. Combining Equations (9) and (10), the optimization model (6) can be transformed into the following linear optimization model.

$$\begin{array}{l}\underset{\begin{array}{l}{P}^{(1)},P^{(2)},\dots ,P^{(h)}\\ ,Q^{(1)},Q^{(2)},\dots ,Q^{(h)}\end{array}}{\max /\min }{P}_{loss,0}T+{\displaystyle \sum _{k=1}^h{A}_P^{loss}(P_i^{(k)}-\overline{P_i})·T/h}\\ +{\displaystyle \sum _{k=1}^h{A}_Q^{loss}(Q_i^{(k)}-\overline{P_i})·T/h}\\ s.t.\\ \{\begin{array}{l}{P}_i^{\min }\le P_i^{(k)}\le P_i^{\max },\forall k=1,\dots ,h,\forall i\in N\\ Q_i^{\min }\le Q_i^{(k)}\le Q_i^{\max },\forall k=1,\dots ,h,\forall i\in N\\ {\displaystyle \sum _{k=1}^h{P}_i^{(k)}·T/h}=W{p}_i^{(q_s)}-W{p}_i^{(q_1)},\forall i\in A_P\\ {\displaystyle \sum _{k=1}^h{Q}_i^{(k)}·T/h}=W{q}_i^{(q_s)}-W{q}_i^{(q_1)},\forall i\in A_Q\\ P_{\min }^{(k)}\le P_{f,0}+A_P^{Pf}·(P_i^{(k)}-\overline{P_i})+\\ A_Q^{Pf}·(Q_i^{(k)}-\overline{Q_i})\le P_{\max }^{(k)},\forall i\in N,\forall k=1,2,\dots ,h\\ Q_{\min }^{(k)}\le Q_{f,0}+A_P^{Qf}·(P_i^{(k)}-\overline{P_i})+\\ A_Q^{Qf}·(Q_i^{(k)}-\overline{Q_i})\le Q_{\max }^{(k)},\forall i\in N,\forall k=1,2,\dots ,h\end{array}\end{array}$$

(11)

The above linear optimization model can be solved by optimization solvers such as MOSEK and CPLEX.

4. Case Study

The proposed algorithm is verified on the IEEE 33-bus system as shown in Figure 5. The algorithm is programmed on MATLAB 2016a platform and runs on a PC with i5-7300 CPU, 2.50 GHz and 8 GB RAM. The linear optimization program is written by yalmip toolbox and solved by MOSEK solver.

4.1. IEEE 33-Bus Distribution System

The EEE 33-bus distribution system is shown in Figure 5.

For time-varying systems, the load of each bus is variable at different times. It is assumed that the loads of each bus include residential, commercial and industrial loads, and distributed power generation includes distributed photovoltaic and wind power. The distribution of various loads at each bus is different, but the change curve of the same type of load or DG is the same. The typical load curve of residents and the output curve of wind power and photovoltaic are shown in Figure 3. The commercial and industrial load curves are shown in Figure 6.

Let i denote the bus number, s denotes the load or distributed power type, j denotes the time point, $M_{i,s}$ denotes the proportion of class s load or distributed power in bus i, $C_{s,j}$ denotes the component of class s load at time j, and $L_{Ni}$ denotes load of bus i. Then the load of bus i at time j is $L_{i,j}={\displaystyle \sum _{s=1}^5{L}_{Ni}{M}_{i,s}{C}_{s,j}}$. The proportion of various loads and DG in each bus is shown in

Table A1. Proportion of various loads and DG in each bus.

Bus Index	Rated Load (kw)	Residential 1	Residential 2	Industrial	Commercial	Photovoltaic	Wind
1	0	0	0	0	0	0	0
2	100	0.3	0.2	0.2	0.3	0	0
3	90	0.6	0.3	0	0.1	0	0
4	120	0.35	0.45	0.2	0.1	−0.1	0
5	60	0.9	0	0.1	0	0	0
6	60	0.1	0.6	0.2	0.1	0	0
7	200	0.1	0.1	0.8	0	0	0
8	200	0.9	0.2	0.2	0.5	−0.8	0
9	60	0.2	0.8	0	0	0	0
10	60	0.8	0	0.1	0.1	0	0
11	45	0.4	0.6	0.1	0.1	−0.2	0
12	60	1	0	0	0	0	0
13	60	0	1	0	0	0	0
14	120	0.1	0.1	0.8	0	0	0
15	60	0.3	0.6	0	0.1	0	0
16	60	0.8	0.3	0.1	0.3	−0.5	0
17	60	0.5	0.8	0.2	0.3	−0.8	0
18	90	0.9	0.3	0.1	0	−0.3	0
19	90	0.95	0.1	0.3	0.1	−0.45	0
20	90	0.3	0.25	0.6	0.15	−0.3	0
21	90	0.6	0.3	0.05	0.05	0	0
22	90	0.8	0.1	0	0.1	0	0
23	90	0.35	0.85	0.1	0.3	−0.6	0
24	420	0.1	0.3	0.9	0.5	0	−0.8
25	420	0.2	0.4	1	0.1	0	−0.7
26	60	0.1	0.8	0.1	0.4	−0.4	0
27	60	0.7	0.3	0.3	0.1	−0.4	0
28	60	0.45	0.45	0.3	0	−0.2	0
29	120	0.75	0.2	0.15	0.3	−0.4	0
30	200	0.1	0.15	0.3	0.8	0	−0.35
31	150	0.3	0.2	0.2	0.7	0	−0.4
32	210	0.2	0.1	0.85	0.1	−0.25	0
33	60	0.8	0.1	0.1	0.1	−0.1	0

in the Appendix A.

In this paper, it is assumed that the metering cycle of all metering devices in the distribution network is one hour, and the metering cycle of metering devices at the transformer where the distribution network is connected to the upper power grid is 5 min. Set the number of time periods h to 6.

4.2. Single Bus, Single Point Data Is Lost or Abnormal

Firstly, the proposed method is tested on a 33-bus distribution system. It is assumed that the singe datum of bus 8 is abnormal or missing, and the data of other time buses are normal. The proposed method is used to calculate the fluctuation range of line loss. The result is compared with the estimate line loss by state estimation method [12,13], as shown in the following figure.

From Figure 7, it can be seen that the actual line loss is within the line loss interval calculated by the method proposed in this paper. There is an error between the estimated line loss obtained by state estimation method and actual line loss. However, it is difficult to determine whether the estimated line loss is larger or smaller than the actual line loss as well as the error. (For example, the estimated value is smaller under missing data at 5 o’clock, but bigger under missing data at 11 o’clock). Compared with state estimation method, the line loss interval obtained by proposed method could give a reliable interval which actual line loss locates.

4.3. Single Bus, All Day Data Is Lost or Abnormal

In this section, the proposed algorithm is verified on the case of abnormal or missing measurement data of all day at a single bus. Two cases are set. In case 1, the measurement data of bus 12 are missing. In case 2, measurement data of bus 26 are missing. Since it is difficult to supplement the data with the method of state estimation due to the loss of data of a day, the calculation results are only compared with the actual values. The line loss interval at 24 h calculated by the proposed algorithm is shown in Figure 8.

As can be seen from Figure 8, the 24-h line loss interval calculated by the algorithm proposed in this paper is close to the corresponding real value of line loss per hour. When the measurement data of bus 12 are missing, the actual line loss is closer to the lower limit of the calculated line loss interval, while when the measurement data of bus 26 are missing, the actual line loss is closer to the upper limit of the calculated line loss interval. However, the real line loss at all 24 h falls within the line loss interval.

Figure 9 shows the power flow distribution of each branch corresponding to the maximum and minimum line loss when the metering data of bud 26 is missing from 23:00 to 24:00. As can be seen from Figure 9, in most branches, the average power transmission of each branch corresponding to the maximum and minimum line loss is similar. This is due to the fact that the power measurement value of the subsequent buses of these branches is known. The power transmitted by the branches is approximately equal to the sum of the power of the subsequent buses (the line loss is relatively small). It can be seen from Figure 8 that there is a large gap in the power flow distribution corresponding to the branches numbered 1, 2, 3, 4, 5, and 25. The branches numbered 1, 2, 3, 4, 5, and 25 are branches 1–2, 2–3, 3–4, 5–6, and 6–26, respectively. These branches are “upstream” of bus 26. Since the metering data of bus 26 are missing, the power transmitted by these branches can vary in a large range.

In addition, it can be seen from Figure 8 that branch 23 (corresponding to branch 23–24) has power reverse transmission. This is due to the fact that bus 24 is connected with large capacity wind power. From 23:00 to 24:00, the wind power generation is more than the local load and cannot be consumed locally. Therefore, power is supplied to other buses through the power grid.

4.4. Multiple Buses, All Day Data Is Lost or Abnormal

The proposed algorithm is verified on the case of abnormal or missing measurement data of multiple buses. Two cases are set. In case 1, the measurement data of bus 8 and 30 are missing. In case 2, the measurement data of bus 8, 12, 26, 30 and 33 are missing. The line loss interval value at 24 h calculated by the proposed algorithm is shown in Figure 10.

Comparing the line loss intervals of Figure 8 and Figure 10, it can be seen that the line loss interval of Figure 10b is greater than that of Figure 10a, and the line loss interval of Figure 10a is greater than that of Figure 8b. This is in line with the actual situation, since the more missing bus measurement data, the greater the uncertainty of distribution network power flow, so the greater the range of possible fluctuation of line loss.

5. Conclusions

Aiming at the problem of line loss calculation in distribution network with DG, an interval line loss calculation method based on power flow calculation and linear optimization is proposed in this paper. The proposed method establishes upper and lower power constraints for buses with abnormal or missing data, constructing a nonlinear optimization model. The optimization model is discretized and linearized according to sensitivity of line loss to the injected power and sensitivity of transmission power of first branch to the injected power, thus obtaining a line loss interval by linear optimization. The simulation results in IEEE 33-bus distribution network system show that the proposed method can give a reliable line loss interval under missing data, which the actual line loss locates.

Due to the changeable distribution power flow and the difficulty of data collection, even if the traditional theoretical line loss calculation method gives an estimated line loss value, it is often not accurate enough and cannot give an error range. Considering the variable characteristics of distribution network power flow and abnormal data, the proposed method could give a reliable line loss fluctuation interval to provide a basis for the line loss analysis and management of distribution network system with DG.