Evaluation of the situational awareness effects for smart distribution networks under the novel design of indicator framework and hybrid weighting method

Abstract

As a key application of smart grid technologies, the smart distribution network (SDN) is expected to have a high diversity of equipment and complexity of operation patterns. Situational awareness (SA), which aims to provide a critical visibility of the SDN, will enable a significant assurance for stable SDN operations. However, the lack of systematic evaluation through the three stages of perception, comprehensive, and prediction may prevent the SA technique from effectively achieving the performance necessary to monitor and respond to events in SDN. To analyze the feasibility and effectiveness of the SA technique for the SDN, a comprehensive evaluation framework with specific performance indicators and systematic weighting methods is proposed in this paper. Besides, to implement the indicator framework while addressing the key issues of human expert scoring ambiguity and the lack of data in specific SDN areas, an improved interval-based analytic hierarchy process-based subjective weighting and a multi-objective programming method-based objective weighting are developed to evaluate the SDN SA performance. In addition, a case study in a real distribution network of Tianjin, China is conducted whose outcomes verify the practicality and effectiveness of the proposed SA technique for SDN operating security.

Keywords

distribution networks / operation and maintenance / expert systems

Introduction

Situational awareness (SA) has been widely used in various fields such as communication networks, power systems, intelligent transportation, evaluation practice, and emergency management in the new era of digitalization and informatization [¹–⁸]. Liu et al. [⁴] proposed the network security SA which can perceive the network threat from a global perspective. Garcia et al. [⁸] obtained the results which revealed seven key factors pertinent to evaluator competence in these domains by the SA technologies. Secured, reliable, and sustainable operation of modern smart distribution network (SDN) cannot be realized without a high level of SA. Meanwhile, the complexity and variety of SDN’s scales, elements, and structures call for a generic SA technique to realize a high operating security. The SA technique, which can improve the SDN visibility by its monitoring function in real-time and provide key information for active preventive control, will have a crucial impact on the SDN operating security as well as its operating benefits [⁹,¹⁰]. Evaluating and improving the SDN SA effectiveness, which is ready against potential hazardous events such as typhoons and storms [¹¹,¹²], can significantly enhance the SDN operation resilience as well as rapidness and correctness for the decision-making of the SDN system operators.

Inadequate SA effect has been identified as one of the causes of several recent large-scale electrical disturbances worldwide [¹³]. One of the main reasons for the inadequacy was a lack of accurate and reliable data acquisition units in required network locations. With the increasing application of new measurement units such as smart data concentrators, smart sensors, and smart meters, the difficulty for SDN data acquisition is gradually reduced, but the massive data corrected from the SDN may end up being unused or significantly increasing data processing burden [¹⁵]. The level of the SDN SA effect is pending validation.

As discussed in Refs. [¹⁶,¹⁷], the rapid commissioning of distributed energy resources (DER), electric vehicles, and energy storage units in SDN results in highly unpredictable power flow patterns and sophisticated stability issues, which in turn causes more difficulties for SA accuracy and effectiveness. The financial, environmental, and regulatory pressures are pushing the SDN toward its power capability limits, which has to be secured and improved by SA [¹⁸,¹⁹]. To tackle the challenges, Menke et al. [¹⁵] used a limited number of measurements to monitor the SDN with an artificial neural network (ANN)-based scheme. Xiao et al. [¹⁸] proposed a security region for the SDN SA based on the security distance. Su et al. [²⁰] proposed an online evaluation method for voltage security assessment, which could assist in operational decision-making. Zhang et al. [²¹] proposed a novel data-adaptive robust optimization method that enabled the economic dispatch of the distribution networks with a high penetration of renewable power generation. Gao et al. [²²] presented a methodology for evaluating the techno-economic feasibility of mixed AC and DC distribution networks with large-scale integration of photovoltaic energy sources. However, as the aforementioned works only focus on the SDN SA specific performance or the correlations between the SDN states and the SA evaluation indicators, the subjective experiences from system operators or experts were not embedded into objective data processing, which might lead to biased results from the SDN SA effect.

A specific SDN may pose challenges to systematically summarizing and comprehensively interpreting its SA effects. It is of great significance to both establish a generic evaluation indicator framework that can generically characterize these SA effects and design a reasonable SDN SA method based on the various links between these indicators and the SDN operational states. Therefore, this paper proposes a comprehensive evaluation indicator framework for SDN SA with multilevel and multidimensional indicators, and designs an implementation process using hybrid weighting systems. First, a new comprehensive SA architecture is proposed, which integrates the SDN data monitoring system, automation system, and state analysis systems for assessing the SDN SA effectiveness. Next, a SA evaluation indicator system is proposed to mathematically associate the SDN states with the SA effects for enhancing SDN operational visibility in both breadth and depth. Finally, to take advantage of both theoretical rules and practical experiences, a combined subjective and objective weighting method is proposed to execute the data processing of the SDN SA under the predefined evaluation in consideration of the knowledge of system operators or experts.

Situation awareness of smart distribution network

A candidate SDN SA system

In this paper, a candidate SA framework for SDN is employed as shown in Fig. 1, which consists of SA perception, comprehension, prediction as well as its communication networks over the physical SDN elements.

The SA perception is the data acquisition stage that obtains the required data for the SDN analysis and control. The core technologies may include the measurement optimization configuration technology, the phase measurement unit (PMU) configuration optimization and data application technology, and the advanced measurement system construction technology.

The SA comprehension is the state analysis stage that extracts knowledge from the collected data and analyzes the SDN states in terms of stable operations, economy, reliability, flexibility, network power capacity, load transfer, load access capacity and distributed generation capacity, etc. The core technologies may include power supply computation, distribution system flexibility analysis, survivability, and vulnerability analysis, power flow analysis, SDN status estimation, big data, and cloud computing technologies.

The SA prediction is the state forecast stage that predicts the potential variations in the SDN states, such as the changes in load, distributed generation, and electric vehicles. SA can evaluate and prompt the security risks of the system and warn system operators. The specific core technologies include load hierarchical prediction, power output prediction considering uncertainty, electric vehicle zoning prediction considering randomness, system safety risks analysis, and early warning techniques.

Challenges to SA evaluation for SDN

The goal of SDN SA effect evaluation is to have a thorough understanding of its effectiveness on various SDN operations by defining appropriate indicators. However, there are 5 main challenges for the SDN SA effect evaluation. To begin with, limited work has been done to quantify the effect and impact of the SDN SA system, as the SDN SA system lacks definitions of generic indicators framework which can uniformly characterize the SDN key operating states and requirements. Besides, with the rapid deployment of measurement units, the data acquired by SDN SA has also surged, which inevitably increases the pressure on redundant data processing for the SA evaluation. The large amount of data representing the good running state of the systems, and the lack of data representing the system failure or bad running state have resulted in the asymmetry of data acquired. To guarantee the SA evaluation efficiency, it is necessary to select the high-impact evaluation indicators and establish an effective evaluation model at a limited data-size. In addition, the system hierarchies and automation levels of SDNs in various regions can be significantly different. For effective management of the SDN, the requirement for SA effect models may not be generic and adaptable for all SDN cases. Moreover, compared with the traditional distribution networks, the SDN is characterized by the high penetration of DER. The DER increases the flexibility but also the uncertainty of SDN operations [¹⁹], which makes the SA effect evaluation more challenging to capture the SDN states due to the DER power volatility. Furthermore, there are practically certain gaps in the coverage levels of SDN data measurements compared to those of the transmission grid, in terms of lower number and lower precision of the measurement units [¹⁵], which can significantly compromise the SA evaluation performance.

An evaluation indicator framework for SDN SA

As observed from Fig. 1, the SDN SA may be required to deal with a large number of influencing factors. The SA evaluation indicators should be comprehensively and systematically selected and their corresponding algorithms have to be properly designed and interpreted. To avoid the evaluation indicators being incomplete, subjective, or biased, this paper proposes an evaluation indicator framework with three layers of indicator levels as demonstrated in Fig. 2, which generally covers the effects of SA perception, comprehension, and prediction.

Indicator layer A of SA perception effectiveness

The indicator layer A of the SA perception effect indicators framework is depicted in Fig. 3, whose key indicators are introduced below.

Observability A₁

An “observable” distribution network acts as the basis for SA comprehension and prediction. Practically, the data measurement units may not all cover the electrical nodes existing in one SDN. The observability of SDN indicates the level of the SDN state measurement and estimation. For an SDN with N_sp state phasors, the observability of node i can be expressed as

(1)A_i^{\text{'}}=b_i+{\displaystyle \sum _{j=\mathrm{1},j\ne i}^{N_{\mathrm{p}}}{a}_{j,i}}{b}_j+{\displaystyle \sum _{j=\mathrm{1},j\ne i}^{N_{\mathrm{p}}}{a}_{j,i}}{\mathrm{c}}_{j,i},

(2){\displaystyle \sum _{\mathrm{i}=\mathrm{1},i\ne j}^{N_{\mathrm{p}}}{a}_{j,i}}{\mathrm{c}}_{j,i}=z_j^{},

(3)a_{i,j}=a_{j,i}=\{\begin{array}{ll}\mathrm{1},& \begin{array}{lll}i=j& \mathrm{or}& \mathrm{node}i\mathrm{and\; node}j\mathrm{are\; connected}\end{array}\\ \mathrm{0}& \end{array}.

Accordingly, the observability A₁ of the SDN can be expressed as

(4)A_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{N_{\mathrm{p}}}}{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{p}}}{A}_i^{\text{'}}},

where i and j are the node indices in the SDN; A_i^{\text{'}} is the observability of electrical node i, N_p is the number of phasor nodes in the SDN; b_i is a logical binary for node i that indicates if the measurement units are installed or not (if the measurement units are installed, b_i = 1; otherwise, b_i = 0); and a_j,i is the (j, i)-th element of the network incidence matrix. Among the nodes connected to the zero-injection node, if there is only one node whose observability is 0, the state of the unobservable node can be calculated according to other observable nodes. z_j is a logical binary to indicate if it is a zero-injection node or not (when node j is a zero-injection node, z_j = 1; otherwise z_j = 0). c_j,i is a logical binary for node i to indicate if the state can be calculated or not (if the state can be calculated, c_j,i = 1; otherwise c_j.i = 0).

Measurement redundancy A₂

Measurement redundancy is a guarantee of reliable power flow calculation and accurate state estimation. The measurement redundancy A₂ can be defined as

(5)A_{\mathrm{2}}\mathrm{=}{\displaystyle \frac{S_{\mathrm{indep}}}{S_{\mathrm{status}}}}\times \mathrm{100}\mathrm{\%,}

where S_indep and S_status represents the number of independent measurements and state variables, respectively.

Identifiability A₃

The performance of measurement and communication systems may be affected by certain factors of the system state variations, operating conditions, and human inventions, etc., which may incur bad data, affect the SA evaluation accuracy, and lead to operating decision errors. In practice, bad data can be detected by the method of combined detection of residual and mutation. To indicate the degree of bad data, the identifiability A₃ which represents the proportion of erroneous data are defined as

(6)A_{\mathrm{3}}={\displaystyle \frac{N_{\mathrm{error}}}{N_{\mathrm{sum}}}}\times \mathrm{100}\mathrm{\%,}

where N_error is the number of erroneous data and N_sum is the number of all the measured data.

Unobservable depth A₄

The SDN automation level may not be consistent for different cases, and the rural distribution networks often have more unobservable nodes than urban distribution networks, which cannot guarantee sufficient observability. To consider this impact, the unobservable depth A₄ is defined as the number of branches of the shortest path from the unobservable node i to other observable nodes, which is defined as

(7)A_{\mathrm{4}}\mathrm{=}\underset{\mathrm{1}\le i\le N_{\mathrm{p}}}{\max }\{{\eta }_i\},

where h_i is the unobservable depth of node i. If node i is observable, h_i is 0; otherwise, h_i equals the number of branches along the shortest path from the unobservable node i to other observable nodes.

Average communication delay A₅

For the operations of the SDN different components such as automatic voltage regulators (AVR) and substation auto-recloser switches, communication delay or time lag, which is the difference between the data transmission and data reception, is an un-negligible factor for the SDN SA. The average delay can be obtained by the specifications as well as test results for different types of communications. The SDN average communication delay A₅ is defined as

(8)A_{\mathrm{5}}=T_{\mathrm{r}}-T_t,

where T_t and T_r are the time point when the data are transmitted and received, respectively.

Indicator layer B of SA comprehension effectiveness

The proposed indicator layer B of SA comprehension effect has eight main indicators as exhibited in Fig. 4, whose key indicators are introduced below.

Low-voltage feeder power capacity B₁

This paper adopts the method of regional calculation to solve the feeder power capacity. The low-voltage feeder power capacity B_1-i in region i can be approximated as

(9)B_{\mathrm{1}-i}=\min \left\{S_{\mathrm{TD}-i},{\displaystyle \sum _{M\in {\Omega }_{\mathrm{LVL},i}}{S}_M}\right\},

where S_TD-i is the transformer power capacity in network area i, S_M is the sustainable transformer capacity of low voltage line M, and W_LVL,i is the numbering set of all outgoing lines in low-voltage feeder area i.

Regardless of the load transfer, the overall low-voltage feeder power supply capacity can be expressed as

(10)B_{\mathrm{1}}={\displaystyle \sum _{i\in {\Omega }_{\mathrm{LDN}}}{B}_{\mathrm{1}-i}},

where B₁ is the low-voltage distribution feeder power supply capacity and W_LDN is the set of all low-voltage distribution feeder area.

Medium-voltage feeder power capacity B₂

Compared with the low-voltage feeder typically with radial topology, the medium-voltage feeders have more complex network structures, and the influence of wiring methods such as tie switches should be considered. The medium-voltage feeder power capacity depends on the power supply capacity of the substations and load transfer capacity of the SDNs, which reflects the network frame of the existing structural problems and the actual distribution network power supply reliability. This paper calculates the power supply capacity of the medium-voltage feeder based on the N–1 principle.

Considering the situation where the loads can be transferred between multiple medium-voltage lines, the line power capability is defined as

(11)S_{{}^{{}^L}}^{{}^{\text{'}}}\mathrm{=}{\eta }_{\max ,L}{S}_L,

where S_L is the sustainable limit transmission capacity of the medium-voltage line L, S_L^{\text{'}} is the maximum transmission capacity of the line L, and h_max,L is its maximum load rate. Generally, the single radiating line is 0. Based on the N–1 principle, the overhead double contact is 0.667, the overhead single contact is 0.5, and the other cases are 0.75, including the overhead three contacts and above, the switch station with a busbar switch, and the cable double ring network.

Similar to the low-voltage feeder power capacity B_1–i, the power supply capacity of the medium-voltage feeder area i can be expressed as

(12)B_{\mathrm{2}-i}=\min \left\{S_{\mathrm{TD-}i},{\displaystyle \sum _{L\in {\Omega }_{\mathrm{VL}-i}}{S}_{{}^{{}^L}}^{{}^{\text{'}}}}\right\},

where B_2–i is the power supply capacity of the medium-voltage feeder area i, W_VL-i is the number set of all the outgoing lines of the medium-voltage feeder area i.

Accordingly, the medium-voltage feeder power capacity B₂ can be expressed as

(13)B_{\mathrm{2}}={\displaystyle \sum _{i\in {\Omega }_{\mathrm{DN}}}{B}_{\mathrm{2}-i}},

where B₂ is the medium-voltage feeder power capacity and W_DN is the set of all area numbers of all medium-voltage distribution networks.

Voltage deviation B₃

The voltage deviation B₃ is the average of the voltage deviation of each SDN node from nominal values

(14)B_{\mathrm{3}}\mathrm{=}{\displaystyle \frac{\mathrm{1}}{N_{\mathrm{p}}}}\times {\displaystyle \sum _{i\mathrm{=}\mathrm{1}}^{N_{\mathrm{p}}}\left|{\displaystyle \frac{U_{i-\mathrm{n}}-U_{i-\mathrm{rt}}}{U_{i-\mathrm{n}}}}\right|\times \mathrm{100}\%},

where U_i-n and U_i-rt are the nominal and real-time voltage of node i.

Loss of connectivity B₄

When the fault occurs, the formulation of the load transfer strategy needs to be based on the current network connectivity situation. Loss of connectivity B₄ evaluates the intactness of SDN network topology in real-time, quantifies the level of connectivity between the SDN substations and the loads, and directly affects the load transfer capacity of the SDN.

(15)B_{\mathrm{4}}\mathrm{=}\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{N_{\mathrm{LD}}}}{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{LD}}}\frac{N_{\mathrm{f-}i}}{N_{\mathrm{s-}i}}},

where N_LD is the number of loads in the distribution network, N_f-i is the number of substations connected to load i after supply failures, and N_s-i is the number of substations initially connected to load i.

Distribution network voltage out-of-limit risk B₅

When the distribution network is under maintenance or in failure, it is hoped that through the reconfiguration of the distribution network structure, the affected load will be transferred without the system exceeding the limit. The distribution network voltage out-of-limit risk B₅ is defined to consider the node voltage out-of-limit risk.

(16)B_{\mathrm{5}}\mathrm{=}{\displaystyle \frac{\mathrm{1}}{T_{\mathrm{w}}{N}_{\mathrm{p}}}}{\displaystyle \sum _{i=\mathrm{0}}^{N_{\mathrm{p}}}{\displaystyle {\int }_{\mathrm{0}}^{T_{\mathrm{w}}}{H}_{\mathrm{node}}^{\lim }(t\mathrm{)d}t}},

(17)

where T_w represents the statistical time range in the specific time window, H_{\mathrm{node}}^{\lim }(t) is the risk of the node voltage exceeding the limit at time t, V_i-min is the lower voltage limit of node i, V_i(t) represents the voltage of node i at time t, and V_i-max is the upper voltage limit of node i.

Distribution network power flow out-of-limit risk B₆

The distribution network power flow out-of-limit risk B₆ is defined to consider the branch power flow out-of-limit risk.

(18)B_{\mathrm{6}}\mathrm{=}{\displaystyle \frac{\mathrm{1}}{T_{\mathrm{w}}{N}_l}}{\displaystyle \sum _{i=\mathrm{0}}^{N_l}{\displaystyle {\int }_{\mathrm{0}}^{T_{\mathrm{w}}}{H}_{\mathrm{line}}^{\lim }(t\mathrm{)d}t}},

(19)H_{\mathrm{line}}^{\lim }(t)=\{\begin{array}{ll}\mathrm{0},& S_i(t)\le S_{i\mathrm{-max}}\\ {\displaystyle \frac{S_i(t)-S_{i\mathrm{-max}}}{S_{i\mathrm{-max}}}},& S_i(t)>S_{i\mathrm{-max}}\end{array},

where N_l is the number of branches in SDN, H_{\mathrm{line}}^{\lim }(t) is the risk of the branch power flow exceeding the limit at time t, S_i(t) is the power flow of branch i at time t, and S_i-max is the line flow capacity of branch i.

Mean fault of repair time B₇

The average fault repair time reflects the time that it takes for the distribution network to detect faults, respond accordingly, and restore normal operation.

(20)B_{\mathrm{7}}=T_{\mathrm{h}}-T_{\mathrm{f}},

where T_h indicates the time when the fault of distribution network occurs and T_f indicates the time when the fault of distribution network has been repaired and normal operation can be resumed.

Dispatcher maloperation rate B₈

The dispatcher maloperation rate reflects the cumulative number of electrical maloperations of all dispatchers within certain statistical time frame.

(21)B_{\mathrm{8}}=N_{\mathrm{w}}/T_{\mathrm{w}},

where N_w represents the cumulative times of wrong operations by distribution network dispatchers within the statistical time.

Indicator layer C of SA prediction effectiveness

The indicator layer C of SA prediction is illustrated in Fig. 5, which consist of 5 main indicators.

Load fluctuation degree C₁

Grid-connected distributed generation can be used as a supplement to distribution network, and accurate distributed power output prediction can play a role in peak load shifting. This paper combines load hierarchical prediction and electric vehicle distributed charging prediction to formulate a reasonable grid-connected distributed power plan, which can further reduce the large fluctuations in the distribution network and help the stable operation of the distribution network.

Specifically, the load fluctuation degree C₁ represents the power change rate on a unit time scale of the distributed power source, and indirectly reflects the operation effect of the core technology of SA prediction, which is defined as

(22)C_{\mathrm{1}}={\displaystyle \sum _{\mathrm{t}=\mathrm{0}}^{T_{\mathrm{w}}}\left|{\displaystyle \frac{P_{\mathrm{DG}}(t+\mathrm{1})\mathrm{+}{P}_{\mathrm{EV}}(t+\mathrm{1})+P_{\mathrm{LD2}}(t+\mathrm{1})-P_{\mathrm{DG}}(t)-P_{\mathrm{EV}}(t)-P_{\mathrm{LD2}}(t)}{P_{\mathrm{DG}}(t)+P_{\mathrm{EV}}(t)+P_{\mathrm{LD2}}(t)}}\right|},

where P_DG(t) represents the output power of the distributed generation connected to the distribution network at time t, P_DG(t+ 1) represents the output power of the distributed generation connected to the distribution network at time t + 1, P_EV(t) represents the charging power of electric vehicles connected to the distribution network at time t, P_EV(t+ 1) represents the charging power of electric vehicles connected to the distribution network at time t + 1, P_LD2(t) represents the output power of the remaining loads connected to the distribution network at time t, and P_LD2(t+ 1) represents the output power of the remaining loads connected to the distribution network at time t + 1.

Load prediction average relative error C₂

Hierarchical load prediction at each node is the key to SA prediction. The load prediction average relative error C₂ reflects the average relative error of load prediction by the SA.

(23)C_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{N_{\mathrm{LD}}}}{\displaystyle \sum _{i\mathrm{=}\mathrm{1}}^{N_{\mathrm{LD}}}\left|{\displaystyle \frac{P_{\mathrm{for-}i}-P_{\mathrm{real-}i}}{P_{\mathrm{real-}i}}}\right|},

where P_for-i represents the predicted load of node i and P_real-i represents the actual load of node i.

System equilibrium C₃

The integration of DERs can change the power flow of distribution networks. Extremely uneven distribution of the load in the distribution network can significantly affect the user’s power quality and increase the line loss of the system. To achieve a balanced load distribution, the reconfiguration of the distribution network can change the power supply path and transfer the load on the heavy load line to the lighter load line. The SDN combines the SA prediction results to redistribute the system and balance the power flow distribution. The system equilibrium C₃ indicates the equilibrium degree of the power flow distribution in the distribution system, and indirectly reflects the effectiveness of the load transfer capacity at a no-fault state.

(24)C_{\mathrm{3}}={\displaystyle \sum _{i=\mathrm{1}}^{N_l}{\left({\displaystyle \frac{S_i(t)}{S_{i\mathrm{-}\max }}}-{\displaystyle \frac{\mathrm{1}}{N_l}}{\displaystyle \sum _{k=\mathrm{1}}^{N_l}\frac{S_k(t)}{S_{k\mathrm{-}\max }}}\right)}^{\mathrm{2}}}.

System load control degree C₄

System load control degree C₄ represents the proportion of interruptible loads in the distribution network to all loads in the distribution network and describes the friendliness of loads. It is the basis of system dispatching, load transfer, and optimal operation, affecting the decision-making effect based on SA prediction.

(25)C_{\mathrm{4}}\mathrm{=}{\displaystyle \frac{L_{\mathrm{A}}}{L_{\mathrm{A}}+L_{\mathrm{B}}}}\times \mathrm{100}\%,

where L_A represents the power of the interruptible load in the distribution network, and L_B represents the power of the uninterruptible load in the distribution network.

Power prediction accuracy C₅

The power prediction accuracy C₅ reflects the difference between the actual power consumption of the distribution network terminal and the power prediction value obtained based on the experience or the model in period T_w.

(26)C_{\mathrm{5}}={\displaystyle \frac{\mathrm{1}}{N_{\mathrm{LD}}}}{\displaystyle \sum _{i\mathrm{=}\mathrm{1}}^{N_{\mathrm{LD}}}(\mathrm{1}-{\displaystyle \frac{\left|P_{\mathrm{predicted-}i}-P_{\mathrm{real}-i}\right|}{P_{\mathrm{real}-i}}})\times \mathrm{100}\%},

where P_predicted-i represents the predicted power of node i.

Hybrid system weighting method for SA indicators

While the indicator framework as designed in Section 3 presents a comprehensive view of the SDN SA impact factors, it requires a systematic weighting system to be implemented in the SDN SA evaluation process. At present, there are mainly three types of weighting methods including subjective weighting methods, objective weighting methods, and hybrid subjective and objective weighting methods. This paper presents a hybrid system weighting method, which proposes an improved interval analytic hierarchy process (IIAHP) to calculate the indicator’s subjective weight and a multi-objective programming method (MOPM) to calculate the indicators’ objective weight, respectively. The combined subjective and objective indicator weights can then be obtained through linear weighting by using the hybrid system weighting method.

IIAHP

The subjective weighting methods include the analytic hierarchy process, the expert survey method and so on [⁹]. In this paper the IIAHP is used, considering the expert scoring ambiguity.

Interval judgment matrix A derivation

Base on the proposed evaluation indicator framework for the SDN SA, N experts are selected to compare and score the different indicators at each layer. The judgment matrix A is established as

(27)A\mathrm{=}\left[\begin{array}{llll}[a_{\mathrm{11}}^{\text{'}}]& [a_{\mathrm{12}}^{\text{'}}]& \dots & [a_{\mathrm{1}n}^{\text{'}}]\\ [a_{\mathrm{21}}^{\text{'}}]& [a_{\mathrm{22}}^{\text{'}}]& \dots & [a_{\mathrm{2}n}^{\text{'}}]\\ ⋮& ⋮& & ⋮\\ [a_{n\mathrm{1}}^{\text{'}}]& [a_{n\mathrm{2}}^{\text{'}}]& \dots & [a_{nn}^{\text{'}}]\end{array}\right],

(28)

where n denotes the indicator number in the selected indicator layer; i = 1, 2, …, n and j = 1, 2, …, n; [a_{ij}^{\text{'}}] is an interval value consisting of its midpoint value a_ij and the width m; a_ij determines the average importance difference between the indicator i and indicator j; and m represents the uncertainty of the indicator importance, which is presented by experts.

Interval proportional scale derivation

When the indicator i is treated more importantly than the indicator j, [a_{ij}^{\text{'}}]≥1, i ≠ j. When the indicator j is considered to be more important than the indicator i, 1/[a_{ij}^{\text{'}}]≥1, i ≠ j. [a_{ij}^{\text{'}}] can be determined by

(29)[a_{ij}^{\text{'}}]\mathrm{=}[a_{ij}^{\text{'}}-\mu ,a_{ij}^{\text{'}}+\mu ].

Interval scaling to real values

The weighting factor q is introduced to convert the values in uncertain intervals into the value range of [0, 1]

(30)a_{ij}^{\text{'}\text{'}}=\theta (a_{ij}^{\text{'}}-\mu )+(\mathrm{1}-\theta )(a_{ij}^{\text{'}}+\mu ).

Improved interval judgment matrix A_I derivation

The element of the interval judgment matrix A_I which considers Eq. (27)–(30) is then written as

(31)[a_{ij}^{\text{'}}\mathrm{]=}\{\begin{array}{lll}{a}_{ij}^{\text{'}\text{'}},& & i\ne j\\ {\displaystyle \frac{\mathrm{1}}{a_{ij}^{\text{'}\text{'}}}},& & j\ne i\\ \mathrm{1},& & i=j\end{array}.

Eigenvector derivation

The eigenvector x and the largest eigenvalue l_max of the judgment matrix A_I are calculated as

(32)A_{\mathrm{I}}\xi ={\lambda }_{\max }\xi .

Consistency check

To check the rationality and consistency of judgment matrix A_I, the consistency check is introduced and expressed as

(33)\{\begin{array}{l}\mathrm{CR}={\displaystyle \frac{\mathrm{CI}}{\mathrm{RI}}}\\ \mathrm{CI}=({\lambda }_{\max }-\mathrm{1})/(n-\mathrm{1})\end{array},

where the freedom index RI takes values according to Table 1 [⁹], if consistency ratio CR of A_I is less than 0.1, A_I passes the consistency check and can be accepted; and consistency indicator CI is an intermediate variable.

Interval weight of each index

After completing the above steps successively by N experts, N eigenvectors {\omega }_i^{\text{'}} from various experts can be obtained. The final subjective weight eigenvector a is calculated as

(34)a={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^N{\omega }_i^{\text{'}}},

where {\omega }_i^{\text{'}} is the eigenvector presented by the i-th expert.

MOPM

The objective methods include the entropy method, the principal component analysis method, and so on. In this paper, the proposed MOPM aims to simultaneously evaluate the extremum of multiple objective functions under certain constraints.

Indicators relative membership degree matrix R

According to the actual indicator value of SA effectiveness in described in Section 3, q proposed schemes are formed, and each scheme needs to consider p indicators. The relative member degree matrix is defined by

(35)R\mathrm{=}\left[\begin{array}{llll}{r}_{\mathrm{11}}& r_{\mathrm{12}}& \dots & r_{\mathrm{1}q}\\ r_{\mathrm{21}}& r_{\mathrm{22}}& \dots & r_{\mathrm{2}q}\\ ⋮& ⋮& & ⋮\\ r_{p\mathrm{1}}& r_{p\mathrm{2}}& \dots & r_{pq}\end{array}\right].

The indicator k in the scheme j can be expressed by using the target matrix (r_jk)_p×q. When the evaluation result is positively related to the indicator value, the corresponding element of R can be determined by

(36)r_{jk}=\{\begin{array}{ll}(x_{jk}-x_{k\min })/(x_{k\max }-x_{k\min }),& x_{k\max }\ne x_{k\min }\\ \mathrm{1},& x_{k\max }=x_{k\min }\end{array}.

where x_jk represents the value of indicator k in the scheme j, x_kmax represents the maximum value of indicator k, and x_kmin represents the minimum value of the indicator k.

When the evaluation result is negatively related to the indicator value, the r_jk is formulated by

(37)r_{jk}=\{\begin{array}{ll}(x_{k\max }-x_{jk})/(x_{k\max }-x_{k\min }),& x_{k\max }\ne x_{k\min }\\ \mathrm{1},& x_{k\max }=x_{k\min }\end{array}.

When the value of indicator k is a fixed value x* k, the evaluation result is the highest, and r_jk is formulated by

(38)r_{jk}=\{\begin{array}{ll}\mathrm{1}-\left|x_{jk}-x_k^{*}\right|/{△}_k,& x_{jk}\ne x_k^{*}\\ \mathrm{1},& x_{jk}=x_k^{*}\end{array},

(39){△}_k\mathrm{=}\underset{\mathrm{1}\le j\le q}{\max }\{\lceil x_{jk}-x_k^{*}\rceil \}.

where D_k represents the deviation degree of x_jk.

When the indicator value is within a certain range [d_k, d_k^{\text{'}}], the evaluation result is the highest, expressed as

(40)

(41){\sigma }_k\mathrm{=max}\{d_k-d_{k\min },d_{k\max }-d_k^{\text{'}}\}.

where d_k, d_k^{\text{'}}, d_kmin, d_kmax, and s_k represent the best lower limit, the best upper limit, the minimum lower limit, the maximum upper limit, and the maximum deviation of indicator k, respectively.

As the relative membership degree of the indicator increases, the effectiveness of the affiliated scheme increases. When the relative membership degree of all the indicators is 1, a relatively optimal scheme is obtained, which is defined as the base point scheme G₀.

(42)G_{\mathrm{0}}={(\mathrm{1},\mathrm{}\mathrm{1},\mathrm{}\cdot \cdot \cdot ,\mathrm{}\mathrm{1},\mathrm{}\mathrm{1})}_{\mathrm{1}\times p}^T.

Optimal solution search

Let the weight vector W corresponding to p indicators be

(43)W={({\omega }_{\mathrm{1}},\mathrm{}{\omega }_{\mathrm{2}},\mathrm{}\cdot \cdot \cdot ,\mathrm{}{\omega }_{p-\mathrm{1}},\mathrm{}{\omega }_p)}^T.

As the scheme k is getting closer to the scheme G₀, the degree of deviation decreases, and the more likely it is to choose this scheme. When the scheme k is adopted, the degree of deviation from the optimal scheme g_j(\omega ) can be measured using Eq. (44).

(44)g_j(\omega )={\displaystyle \sum _{k=\mathrm{1}}^p{\omega }_k(\mathrm{1}-r_{jk})}\mathrm{,}j=\mathrm{1},\mathrm{2},\cdots ,q,

where w_k is the weight of the indicator k.

A multi-objective programming model [²¹]

To obtain the minimum value of g_j(w) for all indicators, a multi-objective planning model is proposed as

(45)\min {\displaystyle \sum _{j=\mathrm{1}}^q{g}_j(\omega )}\mathrm{s}.\mathrm{t}.\{\begin{array}{l}{\displaystyle \sum _{k=\mathrm{1}}^p{\omega }_k{}^{\mathrm{2}}}=\mathrm{1}\\ {\omega }_k\ge \mathrm{0},\mathrm{}k=\mathrm{1},\mathrm{2},\cdots ,p\end{array}.

Weight calculation

To solve the MOPM equation, the Lagrange function G(\omega ,\lambda ) is constructed as

(46)G(\omega ,\lambda )={\displaystyle \sum _{j=\mathrm{1}}^q{\displaystyle \sum _{k=\mathrm{1}}^p{\omega }_k(\mathrm{1}-r_{jk})}}+\lambda ({\displaystyle \sum _{k=\mathrm{1}}^p{\omega }_k{}^{\mathrm{2}}}-\mathrm{1}),

(47)\{\begin{array}{l}{\displaystyle \frac{\partial G}{\partial {\omega }_k}}={\displaystyle \sum _{j=\mathrm{1}}^q(\mathrm{1}-r_{jk})}+\mathrm{2}\lambda {\omega }_k=\mathrm{0}\\ {\displaystyle \frac{\partial G}{\partial \lambda }}={\displaystyle \sum _{k=\mathrm{1}}^p{\omega }_k{}^{\mathrm{2}}}-\mathrm{1}=\mathrm{0}\end{array}.

Equation (47) is solved to obtain the weight of each indicator. The normalized weight of the i-th indicator b_i is formulated by

(48)b_i={\displaystyle \sum _{j=\mathrm{1}}^q(\mathrm{1}-r_{jk})}/{\displaystyle \sum _{j=\mathrm{1}}^p{\displaystyle \sum _{k=\mathrm{1}}^q(\mathrm{1}-r_{jk})}}.

Subjective and objective hybrid method

To combine the subjective and objective weight, the linear weighting method is adopted in this paper [²³].

(49){\omega }_i=\alpha a_i+(\mathrm{1}-\alpha )b_i,

where a is the subjective preference coefficient and a∈[0, 1], and a_i and b_i are the subjective and objective weights of the i-th indicator, respectively.

The weight of the indicator i is multiplied by the indicator i, and the sum of the 18 indicators is used to obtain the comprehensive evaluation score of SA effects of SDN.

The flowchart of the hybrid system weighting method with IIAHP and MOPM is displayed in Fig. 6.

Case studies

Data set

To verify the proposed evaluation framework for the SDN SA, this paper selects a section of distribution networks in the city of Tianjin, China as the case study whose topological structure is illustrated in Fig. 7 and the key SA effect indicators are presented in Tables S1 and S2 (see Electronic Supplementary Material). The data set which is adopted by this paper is the collected data of nine continuous periods from January to October 2018.

It selects 20-year control engineers for the Tianjin distribution network operation as the experts. According to the expert scoring results, the criterion layer evaluation matrices, the indicator layer A, B, and C of SA perception, comprehension and prediction of the AHP and IIAHP are presented in Tables S3 to S7 respectively.

Comparison of different evaluation methods

The weights of the AHP and IIAHP were calculated from the input data in Tables S3 to S6. The weights of the multi-objective programming method were calculated from the indicator data of the distribution network in Tables S1 and S2 from January to October 2018. The weights of the proposed hybrid method were calculated in Section 4.

Comparison of IIAHP and AHP

As the weighting factor q changes from 0.1 to 1 with a step size of 0.1, the indicator weights in Fig. 8 show the comparison between IIAHP and AHP.

Since both methods require expert advice, the trends are consistent. This proves the feasibility of the two methods.

Based on the AHP method, the IIAHP method comprehensively considers the uncertainty of the expert’s comparison and evaluation of the indicators. The interval wideband value of the indicators ranges from 0 to 1, and the subdivision differences between the indicators can be described in detail.

With the weighting factor q changing in the normal value range, the weighting value of the indicator can reverse. It can be observed from Fig. 8 that when the weighting factor q is 0.1, the indicator weighting is reversed compared to the other weighting factors.

When the weighting factor q is 0.5, the runtime and process complexity of the IIAHP and AHP are the same.

Comparison of different weighting factors q and subjective preference coefficients a

To determine the optimal weighting factor q and subjective preference coefficient a, this paper further studies the impact of different q and a values on the weighting results. Figure 9 presents the weights for the proposed hybrid method with respect to weighting factor q and the subjective preference coefficient a. The subjective preference coefficient a ranges from 0.1 to 1 with the step size of 0.1.

Different subjective preference coefficient a derives different weights for the comprehensive evaluation model. It can be observed from Fig. 9 that when the subjective preference coefficient a is 0.5, the weight for the proposed hybrid method is between the IIAHP and multi-objective programming method. It should be a relatively balanced value. The proposed hybrid method is a balance between the IIAHP and the multi-objective programming method by using different subjective preference coefficient a. When the subjective preference coefficient a is 1, the hybrid method is reduced to the IIAHP, and when the subjective preference coefficient a is 0, the hybrid method is reduced to the multi-objective programming method.

Comparison of the proposed hybrid method, the IIAHP, and the multi-objective programming method

The indicators weights in Table 2 show the comparison of the proposed hybrid method, the IIAHP, and the multi-objective programming method.

Due to the difference between subjectivity and objectivity, the IIAHP method and multi-objective programming method have some certain variations in weight distribution.

The proposed hybrid weighting strategy balances the differences between the other two methods.

The results obtained from the IIAHP rely strongly on the subjective experience of experts while the results obtained from the multi-objective programming method rely strongly on the collected data. Compared with the above two methods, the hybrid method combines the actual data of the device and the expert’s score while considering the ambiguity of the expert and the device. Therefore, the hybrid method can effectively deal with the bias of experts, which is more suitable for objective evaluation.

Comparison of the SA impacts on SDN

The actual operation data of urban and suburban distribution networks were collected, and the aforementioned 18 indicators were calculated from A₁ to C₅ in Tables S1 and S2. After the min-max normalization and conversion of negative correlation indicators into positive correlation indicators, Fig. 10 shows the comparison of the average value of the 18 indicators between urban and suburban distribution network areas in Tianjin, China.

It can be concluded from the results in Fig. 10 that the urban distribution networks demonstrate significant advantages in observability, measurement redundancy, identifiability, unobservable depth, operating risk, and system equilibrium. The overall effectiveness of SA urban distribution networks is more positive than that of suburban networks, which are in line with the reality. Besides, urban and suburban distribution networks in Tianjin are generally reliable in terms of the average fault repair time and the power supply capacity of medium and high voltage distribution networks. However, both of them require improvements in mean fault repair time and power prediction accuracy. In addition, the management department in Tianjin should ameliorate load stratification prediction in SA to improve the power prediction accuracy and the efficiency of maintenance personnel.

The comprehensive evaluation results for the SA effects of Tianjin SDN in Table 3 show that the urban distribution network scores higher than the suburban distribution network in the different indicator weighting methods of subjective, objective, and comprehensive aspects, respectively. The comprehensive score balances the differences between subjective and objective scores. The above analysis proves the effectiveness of the proposed comprehensive evaluation model whose detailed calculation process is presented in Table S7.

Conclusions

This paper proposed a systematic framework to evaluate the effectiveness of SA in SDN. The framework defines innovative evaluation weight indicators to establish a comprehensive evaluation of a systematic, accurate and scientific evaluation of SA with consideration of various factors. A case study with real data from the city of Tianjin, China, proves that the framework can provide an effective support for the operation planning, scheduling, and maintenance of SDN. With the continuous emergence of new smart grid technologies, SA will continue to expand in the future with more abundant data for evaluation. Machine learning algorithms and other data-driven techniques can be further integrated into the evaluation model in the future to improve the evaluation accuracy and efficiency of distribution networks.

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