An Importance Analysis–Based Weight Evaluation Framework for Identifying Key Components of Multi-Configuration Off-Grid Wind Power Generation Systems under Stochastic Data Inputs

Abstract

Wind power systems have great potential due to its inexhaustible nature and benign environmental impacts. Especially in remote areas, where wind is plentiful, but it is difficult to get grid-connected power, an off-grid wind power system is an effective alternative for power supply. Reliable and safe operation of the generating system are essential for electricity production and supply. Importance analysis to identify key components of the system is a critical part of reliability assessment. This paper proposes an importance analysis–based weight evaluation framework for identifying key components of multi-configuration off-grid wind power generation systems under stochastic inputs. In the framework, the joint importance analysis based on Birnbaum importance and Criticality importance are introduced to analyze the system reliability and failure rate. Wind speed with stochastic characteristics, load demand with multiple scenarios, and energy transfer with different paths are also merged into the evaluation framework. The results reveal that the rectifier, battery, discharge load, and valve controller dominate the reliability of the off-grid wind power generation system. High priority should be placed on these components during the design phase and maintenance stage. The proposed approach is a positive step forward in promoting component importance analysis and providing more theoretical supports in system design, reliability analysis, and monitoring scheme formulation.

1. Introduction

Exploitation of renewable energy sources (RESs) has been prioritized around the world since the “energy crisis” of 1970s [1,2,3]. Among the RESs, wind power has become more attractive due to its short gestation period, mature level of technology, and negligible environmental pollution [4,5,6,7]. Consequently, many countries have set strategic plans to develop technologies for supporting sustainable wind power development, especially in areas with rich wind energy sources [8]. However, off-grid wind power systems have greater advantages in remote places such as mountainous areas, islands, coastal areas, and other places where wind energy is abundant and where it is temporarily unable to achieve grid-connected power supply [9,10,11]. The reliable and safe operation of wind power systems is one of the key factors for power transmission and distribution [12]. In the design phase of a wind power system, the importance calculation is needed to help the designer to identify the weak points of the system and provide support for the reliability improvement and optimization design. In the operation phase, the maintenance resources are reasonably assigned through the importance analysis to ensure that the maintenance cost is reduced and the system is in normal operation [13]. Therefore, importance analysis is a key factor in the reliability design and evaluation of wind power systems.

Importance analysis is a method to measure the importance of components in a system and the impact of a component on the overall system reliability [14,15]. Through the importance evaluation, the system operation mechanism is revealed from the perspective of reliability, and the various factors influencing the system stability are clearly defined, helping to improve the system design and formulate the monitoring programs [16,17]. With the improvement of reliability theory and engineering application, the importance analysis method has been rapidly developed and widely used in the fields of safety analysis, reliability analysis, and risk analysis of nuclear energy and complex equipment [18,19,20,21]. Zhang et al. [22] studied the reliability of the all-digital protection system, mainly considering the failure rate of the components and the overall reliability of the system. Based on the theory of trapezoidal fuzzy numbers, Liu et al. [23] proposed an importance evaluation method for power transformer fault modes. In view of the complex structure and strong correlation of components of power plants, the importance evaluation index was developed and analyzed by identifying the main influencing factors and constructs [24,25,26,27].

Although importance analysis has been widely applied, few of the studies have attempted to apply importance analysis to assist in the design of the wind power generation system, which frequently results in difficulties in figuring out which components or configurations of components are major contributors to system reliability or system failure. As the most commonly used definition, the Birnbaum importance (BI) of a component indicates the change in the system unavailability given that the component went down. Criticality importance (CI) is developed to prioritize reliability improvement activities and identify weak links in the system with high efficiency. A maintainable and well-designed wind power generation system can not only effectively judge the most critical components among them but also identify the most cost-effective component numbers. It is noteworthy that the weight parameters have significant effects on importance analysis of off-grid wind power generation system [28]. A number of studies have been conducted to systematically evaluate these effects, and it is demonstrated that weight is one of the widely used indicators for measuring the importance of each component in a system [29,30,31,32,33]. Nevertheless, few of them attempted to use importance analysis methods for identifying of weights without depending subject information. According to the existing weight evaluation methods, there could be an obvious trend to figure out those major components or configurations contributing to system reliability or failure. Nevertheless, there is a lack of weight evaluation framework that could sufficiently address the uncertainty input data such as wind speed and wind power supply and demands.

Therefore, this paper aims to propose an importance analysis–based weight evaluation framework for identifying key components of multi-configuration off-grid wind power generation systems under stochastic inputs. In importance analysis, Birnbaum importance (BI) and Criticality importance (CI) will be employed as importance indicators, and the wind power state weights are considered. Wind speed will be expressed as stochastic inputs to the evaluation framework due to its random characteristics. Load demands that are preset as multiple scenarios will also be taken into consideration within the evaluation framework. Outputs from the framework are expected to identify the key components with high importance levels. The identification results will provide necessary information for supporting system design, monitoring and maintenance.

2. Weight Evaluation Framework

The overall framework of the approach used in this study is illustrated in Figure 1. Two main parts are involved for the framework—the importance analysis index and the weight parameters during operational process. The main procedure of the proposed approach is presented as follows.

2.1. Wind Power System Structure

The composition (Ck) of a general off-grid wind power system is given in Figure 2, which includes wind turbine, braking device, permanent magnet synchronous generator, rectifier, controller, battery, valve controller, converter (include DC booster and inverter circuit), user load, and discharge load. The valve device is available for controlling valve opening or closing according to the system’s operations. The wind turbine operates when the wind speed raises above the designed cut-in speed and stops when the wind speed is below the designed cut-in speed. In such a case, the battery group would be used to meet the user demands. When wind turbine is operating, the generated AC power would be supplied to meet the requirement of user load, and any surplus electricity would be stored in the battery as a backup. With response to wind power generation, there is an inherent relationship among various sectors. Normally, the changes in the wind speed, storage capacity, and energy load would bring about variations in the system energy transfer or operating conditions. Therefore, it is particularly desired to develop comprehensive strategies for supporting wind power generation.

2.2. Design of Operation Schemes

The energy transfer associated with wind power generation system is presented in Figure 3. According to different wind speed range and electricity loads, four operation schemes can be preset as follows:

Case 1 (Ω₁): when the wind speed is lower than the cut-in speed, the wind turbine in the off-grid wind power system will fail to capture the wind energy. No electricity can be generated in this case. The battery should be started through a DC booster and inverted circuits to satisfy the load demand.

Case 2 (Ω₂): when the wind speed reaches the start-up speed required by the wind turbine (cut-in speed) yet is less than the maximum wind speed, the turbine will start to work and capture the wind energy. The captured energy will be transformed into mechanical energy and further transformed into AC electric power. This AC electric power will be transformed into the rated voltage 220 V to meet the load demand through the converter. In this case, the wind energy captured by the wind turbine will satisfy the demand of the users.

Case 3 (Ω₃): the power system can substantially capture the wind energy when an enough large wind speed is considered, which will be transferred through four lines (lines 1–4). Specific working procedures regarding the transfer lines are summarized as follows:

(1)

Ω₃₁: when the generated electricity power can satisfy the load demand, the excess power obtained from the wind turbine will be supplied for meeting the end-user demand through line 1, and for charging the battery through line 2.

(2)

Ω₃₂: if the load demand changes to be zero, the wind turbine will directly charge the battery through line 2 after rectifier circuit. But in the case, the battery will get inadequate electricity for charging.

(3)

Ω₃₃: if the load demand is completely met by line 1 and the battery is completely charged through line 2, the battery will stop replenishing by shutting down line 2. The excess wind power electricity will be sent to discharge load for reduction of energy through line 3.

(4)

Ω₃₄: when the wind speed is higher than maximum speed, line 4 (i.e., the mechanical brake device) will be activated. Turbine will be locked to avoid any further damage due to over speed.

Case 4 (Ω₄): when the wind speed is relatively low, but the load demand is large, the power will be satisfied directly from the wind turbine through line 5. If the load demand is still not met, the battery will supply the load through line 6. In such a case, the energy storage devices and generators will jointly supply power to the load.

Since the system cannot be continuously operated for 24 h, any of the four running cases are allowed to be operated in terms of the practical conditions considering wind speed and load demand.

Table 1. Running status of each component in each relation of transfer in the wind power control system.

Component	Ω1	Ω2	Ω31	Ω32	Ω33	Ω34	Ω4
C1	0	0&1	0&1	0&1	0&1	0	0&1
C2	0	0&1	0&1	0&1	0&1	0	0&1
C3	0	0&1	0&1	0&1	0&1	0	0&1
C4	0	0&1	0&1	0&1	0&1	0	0&1
C5	0	1	1	1	0	0	1
C6	0	0	0&1	0&1	0	0	0
C7	0	0	0&1	0&1	0	0	0
C8	1	0	1	1	0	0	1
C9	1	1	1	0	0	0	1
C10	0&1	0&1	0&1	0	0	0	0&1
C11	0&1	0&1	0&1	0	0	0	0&1
C12	0&1	0&1	0&1	0	0	0	0&1
C13	0	0	0	0	0	1	0
C14	0	0	0	0	1	0	0

Note: where 0 represents the component does not work, 1 represents the component works, and 0&1 means the component does not work or works.

shows the running status of system components under different operation schemes.

2.3. Importance Analysis for Weight Evaluation

BI and CI are both popular measures to evaluate the reliability of system components by ranking the importance level of each component. In BI, the reliability and failure rate of the component are independent, which means that the two components may have a similar index value, but the levels of reliability can differ substantially. CI is an extension of BI with the unreliability or failure rate are included. In this study, the joint importance analysis based on BI and CI are introduced to analyze the reliability and failure rate of the wind power system. The definitions of BI ($I_k^B$) and CI ($I_k^{CR}$) can be expressed as follows:

$$I_k^B(t)=\partial R(t)/\partial R_k(t)=E[{\delta }_k(X)]$$

(1)

$$I_k^{CR}(t)=\frac{\partial R(t)}{\partial R_k(t)}\cdot \frac{R_k(t)}{R(t)}$$

(2)

where R(t) represents reliability function of the system at time t; R_k(t) represents reliability of component Ck at time t, where Ck means component k in a system; X represents operational state of the system, where X = {x₁, x₂,…x_n}. Component Ck works when x_k = 1 and does not when x_k = 0; ${\delta }_k(X)=r\left(1_k,X\right)-r(0_k,X)$ for 1≤ k ≤ n. The detailed calculation procedures of the two indicators are followed as reference [34].

When the two methods are developed, the running status of the wind power system should be considered. Therefore, the weights of operation statuses under different wind speed and load demand scenarios are merged when computing the BI and CI. The states can be represented by symbols Ω_ij (seen Section 2.2), and the corresponding weight is defined as W_ij, where 1 ≤ i ≤ n, 1 ≤ j ≤ m. The weight for each operation status W_i is calculated by Equation (3), and the sum of weights in each power supply scenario is 1. The improved methods for importance analysis of the multi-configuration system can be expressed as Equations (5) and (6).

$${\displaystyle \sum _{j=1}^m{W}_{ij}}=W_i$$

(3)

$${\displaystyle \sum _{i=1}^n{W}_i}=1$$

(4)

$$I_{\Omega k}^B(t)={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m\left[W_{ij}\cdot \partial R(t)/\partial R_k(t)\right]}}\cdot {\Omega }_{ij}$$

(5)

$$I_{\Omega k}^C(t)={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m\left[W_{ij}\cdot \partial R(t)R_k(t)/\partial R_k(t)R(t)\right]}}\cdot {\Omega }_{ij}$$

(6)

2.4. Parametric Analysis for Weight Evaluations

2.4.1. Wind Speed Estimation

The instability of natural wind creates heavy fluctuation in the output of a wind power plant. If the wind force is beyond the design requirements, it would be a threat to the security of the electrical power generating system. Therefore, it is necessary to study the distribution characteristics of the wind speed for a wind power station. As an important factor of wind power generation, wind speed has a relationship with the system running status. In addition, wind speed is one of the major factors for not only evaluating the running performance of a multi-configuration system but also identifying the key components of the system.

The average wind speed is the average value of wind speed along the horizontal direction at certain time and at a certain point. The formula is expressed as

$$\overline{v}=\frac{1}{t_2-t_1}{\displaystyle {\int }_{t_1}^{t_2}v(t)}dt$$

(7)

As a consequence of fluctuant property of wind speed, it can be represented as a stochastic variable with probabilistic density function being provided. The wind speed has been demonstrated to follow Weibull distribution and expressed as Equation (8).

$$f(v)=\frac{k}{c}{\left[\frac{v}{c}\right]}^{k-1}\exp \left\{-{\left[\frac{v}{c}\right]}^k\right\},$$

(8)

where v is the wind speed (m/s), f(v) is the Weibull distribution function, and k and c are shape coefficient and scale coefficient, respectively.

As operation schemes are set based on the relation between wind speed and electricity loads, the weight parameter W_ij of the multi-configuration wind power system is determined only by the wind speed. The running modes of the system change with the variation of the wind speed, so the weights of different operating schemes are determined by Equation (9) to Equation (13).

$$W_1=P(0\le v<v_{ci})={\int }_0^{v_{ci}}f(v)dv$$

(9)

$$W_2+W_4^{\prime }=P(v_{ci}\le v<v_r)={\int }_{v_{ci}}^{v_r}f(v)dv$$

(10)

$$W_{31}+W_{32}+W_{33}+W_4^{″}=P(v_r\le v<v_{co})={\int }_{v_r}^{v_{co}}f(v)dv$$

(11)

$$W_{34}=P(v_{co}\le v<+\infty )={\int }_{v_{co}}^{+\infty }f(v)dv$$

(12)

$$W_4=W_4{}^{\prime }+W_4{}^{″}$$

(13)

where V_ci is the cut-in wind speed, V_co is the cut-out wind speed, V_r is the rated wind speed, W₄’ represents the condition that wind speed is small, and W₄’’ represents the condition that wind speed is high.

2.4.2. Wind Power Estimation

There is a nonlinear relationship between wind power output and wind speed, and that relationship can be described by the corresponding operational parameters of the wind turbines. The relationship between wind speed and wind power output is given in Figure 4, and the power output is expressed as Equation (14) [35].

$$P_{\mathrm{out}}=\{\begin{array}{c}0\\ P_r(A+Bv+C{v}^2)\\ P_r\\ 0\end{array}\begin{array}{ccc}& & \begin{array}{cc}& \end{array}\end{array}\begin{array}{c}0\le v\le v_{ci}\\ v_{ci}\le v\le v_r\\ v_r\le v\le v_{co}\\ v\ge v_{co}\end{array}$$

(14)

Among them, the function relations between parameters A, B, C, and V_ci, V_r are given by Equations (15)–(17) [36].

$$A=\frac{1}{(v_{ci}-v_r{)}^2}\left[v_{ci}(v_{ci}+v_r)-4(v_{ci}\times v_r){(\frac{v_{ci}+v_r}{2v_r})}^3\right]$$

(15)

$$B=\frac{1}{{(v_{ci}-v_r)}^2}\left[4(v_{ci}+v_r){(\frac{v_{ci}+v_r}{2v_r})}^3-(3v_{ci}+v_r)\right]$$

(16)

$$C=\frac{1}{{(v_{ci}-v_r)}^2}\left[2-4{(\frac{v_{ci}+v_r}{2v_r})}^3\right]$$

(17)

Further, consider a time span of 24 h and ignore the impact of events occurring with small probability on the system. The weights can be classified to the following situations in view of power balance, assuming P_N and P_out represent the electricity demand and electrical output, respectively.

(1) when 0 ≤ v < v_ci, then

$$W_1={\int }_0^{v_{ci}}f(v)dv$$

(18)

The weight value is not affected by the condition of equilibrium of supply and demand of electricity when the wind speed is in this range.

(2) when v_ci ≤ v < v_r, then

$$P_N>P_{out}=P_r(A+Bv+C{v}^2),\begin{array}{cc}& W_2\end{array}=0\begin{array}{cc}& W_4^{\prime }\end{array}={\int }_{v_{ci}}^{v_r}f(v)dv$$

(19)

$$P_N\le P_{out}=P_r(A+Bv+C{v}^2),\begin{array}{cc}& W_2\end{array}={\int }_{v_{ci}}^{v_r}f(v)dv\begin{array}{cc}& W_4^{\prime }\end{array}=0$$

(20)

(3) When v_r ≤ v <v_co, then

$$P_N<P_{out}=P_r,\begin{array}{cc}& W_{31}\end{array}+W_{33}={\int }_{v_r}^{v_{co}}f(v)dv\begin{array}{cc}& W_4^{″}=W_{32}\end{array}=0$$

(21)

$$P_N=P_{out}=P_r,\begin{array}{cc}& W_{31}\end{array}=W_{32}=W_{33}=W_4^{″}=0\begin{array}{cc}& W_2\end{array}={\int }_{v_r}^{v_{co}}f(v)dv$$

(22)

$$P_N>P_{out}=P_r,\begin{array}{cc}& W_{31}\end{array}=W_{32}=W_{33}=0\begin{array}{cc}& W_4^{″}={\int }_{v_r}^{v_{co}}f(v)dv\end{array}$$

(23)

$$P_N=0,\begin{array}{cc}& W_{32}\end{array}+W_{33}={\int }_{v_r}^{v_{co}}f(v)dv\begin{array}{cc}& W_4^{″}=W_{31}=0\end{array}$$

(24)

(4) when v_co ≤ v < +∝, then

$$W_{34}={\int }_{v_{co}}^{+\infty }f(v)dv$$

(25)

The weight value is not affected by the supply equilibrium condition and the power demand when the wind speed is in this range.

2.4.3. Probabilistic Function Distribution of Data

Table 2. Evaluation method of weight parameters during operational processes.

	${\mathit{P}}_{\mathit{N}}<{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}$	${\mathit{P}}_{\mathit{N}}={\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}$	${\mathit{P}}_{\mathit{N}}>{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}$	${\mathit{P}}_{\mathit{N}}=0$
$W_1$	${\int }_0^{v_{ci}}f(v)dv$	${\int }_0^{v_{ci}}f(v)dv$	${\int }_0^{v_{ci}}f(v)dv$	0
$W_2$	${\int }_{v_{ci}}^{v_r}f(v)dv$	${\int }_{v_{ci}}^{v_{co}}f(v)dv$	0	0
$W_{31}$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(1+k_1\right)$}\right.{\int }_{v_r}^{v_{co}}f(v)dv$	0	0	0
$W_{32}$	0	0	0	${\int }_{v_{ci}}^{v_{co}}f(v)dv$
$W_{33}$	$\raisebox{1ex}{$k_1$}\!\left/ \!\raisebox{-1ex}{$\left(1+k_1\right)$}\right.{\int }_{v_r}^{v_{co}}f(v)dv$	0	0	$k_2{\int }_{v_{ci}}^{v_{co}}f(v)dv$
$W_{34}$	${\int }_{v_{co}}^{+\infty }f(v)dv$	${\int }_{v_{co}}^{+\infty }f(v)dv$	${\int }_{v_{co}}^{+\infty }f(v)dv$	${\int }_{v_{co}}^{+\infty }f(v)dv$
$W_4^{\prime }$	0	0	${\int }_{v_{ci}}^{v_r}f(v)dv$	0
$W_4^{″}$	0	0	${\int }_{v_r}^{v_{co}}f(v)dv$	0
$W_4$	0	0	${\int }_{v_{ci}}^{v_{co}}f(v)dv$	0

lists the evaluation method of weights during operational processes. k₁ and k₂ are the linear correlation coefficients and their values ranges from 0 to 1. The coefficients can do stochastic-optimization processing for system analysis in a specified value range. Through integration of two factors (i.e., wind speed and electricity load), the estimated values indicating system operation status and operation of the weighting parameters in the considered time span are listed in Table 2.

P_N = 0 is a special case that does not need wind power generation system (i.e., no electricity load). This case has no research significance. In the long run (continuous operation of the system), however, this case must be taken into account.

Here, four conditions are considered, namely, P_N < P_out, P_N = P_out, P_N > P_out, and P_N = 0 and are denoted by the proportion σ_m, where m = 1, 2, 3, 4. Finally, the weight parameter of wind power system can be obtained by Equations (26).

$$W_{(\Omega )\mathrm{ij}}={\displaystyle \sum _{m=1}^4{W}_{\mathrm{ij}}}{\sigma }_m$$

(26)

The operation of this system is not continuous for a long time. In a period of continuous operation time, there are a set of alternative operations, whose running status need to be determined by two impact factors: wind speed and electricity load. Considering the influence of two factors, Ω₃₃ is subordinate to Ω₃₁ and Ω₃₂. The basic symbols are also presented in Appendix A.

3. Case Study and Results

3.1. Data Inputs

In this study, an off-grid wind power generation system is established by taking household electricity as an example. The daily average electricity consumption of a family is about 0.63 KWh, and the annual average is about 227 KWh (

Table 3. Electric power consumption of household appliances.

Load Name	Power (W)	Daily Electric Hours (h)	Annual Electric Hours (h)	Daily Power Consumption (KWh)	Annual Power Consumption (KWh)	Note
Television	50	3.5	1280	0.175	63.9	Based on the average power
Electric light	30	4.5	1650	0.135	49.3
Air-blower	25	1.5	550	0.04	14.6
Electric blanket	30	-	340	-	10.2	4.5 months in winter
Recorder	15	4.0	1460	0.06	22.0
Water supply pump	60	1.5	550	0.09	32.9
Other	100	-	-	0.095	34.6
Total				0.595+	227

). The wind turbines are required to supply more than 65% of annual electrical power consumption for this family consumer. Besides, data of monthly average wind speed and frequency distribution are collected, as shown in

Table 4. Average monthly wind velocity.

Wind Velocity	Month
	Jan.	Feb.	Mar.	Apr.	May	Jun.	Jul.	Aug.	Sep.	Oct.	Nov.	Dec.
v (m/s)	6.7	7.3	7.4	7.1	7.4	6.9	6.6	5.7	6.7	7.3	7.9	8.9
$\overline{v}$ (m/s)	7.33

and Figure 5.

3.2. Selection of Off-Grid Wind Power Turbines

In an independently operated off-grid wind power generation system, the efficiencies of the power transfer, utilization and transformation must be considered. The minimum quantity of the output can be calculated by Equation (27).

$$E_W=E_0{K}_D/\eta$$

(27)

where E_W is the minimum annual generating capacity of the wind turbine generator output (KWh); E₀ is the power consumed by the load annually (KWh); K_D is the proportion of the wind power generation amount required by the users to the total power supply; and η represents the comprehensive efficiencies of wind energy transfer, effective utilization and transformation, and the value ranges from 0.6 to 0.9. In this case, the comprehensive efficiency is about 0.8, i.e., η = 0.8. The wind power system needs to provide at least 65% of the total annual electricity consumption for the home user (i.e., K_D = 0.65). According to Equation (27), the annual energy production of the wind turbine installed in the system should be no less than 184.4 KWh.

$$E_{\mathrm{w}}=227\times 0.65÷0.8=184.4$$

(28)

It is a valid assumption to consider the utilization coefficient of equipment as K_q in the initial evaluation of wind power generation capacity. For the off-grid wind power generation system, K_q value varies from 0.22 to 0.32 with the local wind conditions. Here, K_q is considered as 0.29. The primary capacity of the wind turbine is derived as follows:

$$P_n=E_W/(8760\times K_q)=184.4÷8760÷0.29=72.6$$

(29)

The turbine parameters provided by the wind power manufacturers are listed in

Table 5. Part of parameters for small-scale wind turbines.

Product Model	Rotor Diameter (m)	Centre Height of the Wind Wheel (m)	Start-Up Wind Speed (m/s)	Rated Wind Speed (m/s)	Downtime Wind Speed (m/s)	Rated Power (W)	Rated Voltage (V)	Leaf Number
FD2-100	2.0	5	3	6	18	100	28	2
FD24-150	2.0	6	3	7	40	150	28	2
FD2.1-200	2.1	7	3	8	25	200	28	3
FD2.5-300	2.5	7	3	8	25	300	42	3
FD3-500	3.0	7	3	8	25	500	42	3
FD4-1K	4.0	9	3	8	25	1000	56	3
FD5.4-2K	5.4	9	4	8	25	2000	110	3
FD6.6-3K	6.6	10	4	8	20	3000	110	3
FD7-5K	7.0	12	4	9	40	5000	220	2
FD7-10K	7.0	12	4	11.5	60	10,000	220	2

. Based on the wind range and rated power, FD2-100 type of off-grid wind turbine model is selected in this study. Wind turbine power rating is 100 W, annual energy output is about 260 kWh, its rotor diameter is 2 m, start-up wind speed is 3 m/s, the rated wind speed is 6 m/s, downtime wind speed is 18 m/s, and the running speed is from 3 m/s to 18 m/s. The annual power generated from the turbine is estimated based on the local wind resources. The power generation capacity is about 12.7% higher than the required. The turbine capacity is basically feasible.

3.3. Weight Analysis

Some information can be obtained from the statistical material, i.e., $\overline{v}$ being 7.33 m/s, statistical time T being one year, and the statistical coverage of wind speed being from 0 m/s to 25 m/s (v_max). In this study, v_ci is 3 m/s and v_r is 6 m/s. It should be specially noted that the cut-out wind speed is 18 m/s (v_co). According to the weight calculation method in Table 2 and the above data, the weight parameters in different operating states can be obtained as Equations (30)–(34).

$$W_1=0.1308639469$$

(30)

$$W_2+W_4^{\prime }=0.2911810777$$

(31)

$$W_{31}+W_{32}+W_{33}+W_4^{″}=0.5693688002$$

(32)

$$W_{34}=0.0085861752$$

(33)

$$W_4=W_4^{\prime }+W_4^{″}$$

(34)

Considering the two main factors with respect to load demand and wind speed, the system’s states and the run weighting parameters can be obtained in

Table 6. Evaluation results of weight parameters at different operational conditions.

	PN < Pout	PN = Pout	PN > Pout	PN = 0
$W_1$	0.13086	0.13086	0.13086	0
$W_2$	0.29118	0.86055	0	0
$W_{31}$	0.17098	0	0	0
$W_{32}$	0	0	0	0.86055
$W_{33}$	0.39839	0	0	0.13086
$W_{34}$	0.00859	0.00859	0.00859	0.00859
$W_4^{\prime }$	0	0	0.29118	0
$W_4^{″}$	0	0	0.56937	0
$W_4$	0	0	0.86055	0

. Parameters K₁ and K₂ are determined to be 2.33 and 0.15, respectively. The proportion of P_N < P_out, P_N = P_out, P_N > P_out, P_N = 0 is 30%, 40%, 20%, and 10%, respectively.

3.4. Reliability Analysis of Off-Grid Wind Turbines

The research regarding component reliability mainly involves the wind turbine, permanent magnet generator, rectifier, controller, battery energy storage device, the valve controller and the converter. The reliability or failure rate of some of these components can be found in the relevant literature, nameplate, and technical parameters. Details are shown in

Table 7. Components’ reliability of the within prescribed time limit for security.

Component	Name	Reliability (%)	MTBF (h)	MTTFF (h)
C1	Wind turbine	95	3500	3000
C2	Generator	95	4000	3000
C3	Wind turbine	95	3500	3000
C4	Generator	95	4000	3000
C5	Rectifier	95	3000	2000
C6	Controller	90	2000	1500
C7	Controller	90	2000	1500
C8	Battery	98	5000	3000
C9	Valve controller	95	3500	3000
C10	Converter	90	2000	1500
C11	Converter	90	2000	1500
C12	Converter	90	2000	1500
C13	Braking device	90	3000	2000
C14	Discharge load	85	1500	1500

, wherein Mean Time to First Failure (MTTFF) and Mean Time Between Failures (MTBF) of components are calculated as Equations (35) and (36).

$$MTTFF=\frac{1}{r}\left[{\displaystyle \sum _{i=1}^r{t}_i+(n-r)t_c}\right]$$

(35)

$$MTBF=\frac{n{t}_c}{{\displaystyle \sum r^{\prime }}}$$

(36)

where r is the number of components that have occurred first failure in all test components, $r^{\prime }$ is the total failure number of test components, t_i is the cumulative working hours when the tested component i occurred first failure (h), n is the number of the tested components, and t_c is the prescript total test time (h).

In this study, time intervals are considered as one month, three months, six months, and a year. Maintenance projects include a) minor repairs (one month, three months), namely inspection, cleaning, adjustment, and lubrication grease, and b) middle repair (six months) and overhaul (a year). In this study, the time series of six nodes are set up—360 h, 720 h, 1080 h, 1440 h, 1800 h, 2160 h.

Table 8. Variation of components’ reliability during different operational time.

Rk(t)	Time Hours
	360	720	1080	1440	1800	2160
R1(t)	0.902256	0.814066	0.734495	0.662703	0.597928	0.539484
R2(t)	0.913931	0.835270	0.763379	0.697676	0.637628	0.582748
R3(t)	0.902256	0.814066	0.734495	0.662703	0.597928	0.539484
R4(t)	0.913931	0.835270	0.763379	0.697676	0.637628	0.582748
R5(t)	0.886920	0.786628	0.697676	0.618783	0.548812	0.486752
R6(t)	0.835270	0.697676	0.582748	0.486752	0.406570	0.339596
R7(t)	0.835270	0.697676	0.582748	0.486752	0.406570	0.339596
R8(t)	0.930531	0.865888	0.805735	0.749762	0.697676	0.649209
R9(t)	0.902256	0.814066	0.734495	0.662703	0.597928	0.539484
R10(t)	0.835270	0.697676	0.582748	0.486752	0.406570	0.339596
R11(t)	0.835270	0.697676	0.582748	0.486752	0.406570	0.339596
R12(t)	0.835270	0.697676	0.582748	0.486752	0.406570	0.339596
R13(t)	0.886920	0.786628	0.697676	0.618783	0.548812	0.486752
R14(t)	0.786628	0.618783	0.486752	0.382893	0.301194	0.236928

presents the variation of components’ reliability with time.

3.5. Importance Analysis of the Components

In order to evaluate the importance of components, an improved importance model of BI and CI are developed, and the weight parameters of the different operating states of the system are considered. The evaluation of these parameters is based on the probability distribution of wind speed and the equilibrium of supply and demand. According to the probability distribution of wind speed, the running weight coefficient of the system is calculated and further carried out based on the balance relation between supply and demand. Then, the obtained results are classified, and importance level of the components is discussed. Finally, a summary of the importance of components in the wind power generation system and the evaluation results of importance based on BI and CI are presented.

3.5.1. Balance of Energy Supply - Demand

When P_N = 0, the BI and CI levels of 14 components at different operational durations are shown in Figure 6. The results from the two evaluation approaches present the same importance rankings of the system components. Apparently, the importance level of components C5 and C8 are obviously larger than other components, so they are marked as key characteristics [A]. This highlights the fact that special attention needs to be paid to these components during the system maintenance. Components C1, C2, C3, C4, C6, and C7 have similar importance values. They are normal objects during the system maintenance and are marked as significant characteristics [B]. Due to the peculiarity of the system function, components C13 and C14 are not required to work frequently in the normal situation. Hence, they are ranked the lowest importance level as general characteristic [C]. As there is no electronical load demand, the energy generated by the wind only charges the battery until full. In this case, the components C9, C10, C11, and C12 do not work, so the BI and CI values are zero. It means that they have little effect on the system. From the perspective of runtime, the importance of the components decreases continually with longer running time. However, the rankings on the importance of the components remain unchanged. In addition, after taking the operational status of the multi-group in the system into account, there is no significant linear relationship between the importance of the components and their reliabilities or failure rates. That is, the importance of the components cannot be ascertained from their reliabilities or failure rates. This is a significant difference from the single-group operational status of the system.

When P_N > P_out, the BI and CI values for 14 components at different operational time are illustrated in Figure 7. As for the importance rankings of the system components, the results of the two evaluation models depict little difference from a qualitative perspective. Components C5, C8, and C9 are more important than others and have great impacts on the top event when the entire system failed (the definition of a top event is found in reference [34]). The impacts of components C5, C8, and C9 are particularly prominent, and they are marked as key characteristics (A). The importance levels of components C1, C2, C3, and C4 has been evaluated to be the same but greater than that of components C10, C11, and C12, which are the same product and have the same characteristics and importance in the system. These seven components belong to the general maintenance objects and are marked as significant characteristic (B). As components C13 and C14 are not required to work frequently in the normal situation, they are ranked the lowest importance level as general characteristic (C), and longer interval of maintenance can be set. As the components C6, C7 and C14 does not work on this occasion, the importance values are zero. With the system running for a longer period, the results of the two evaluation models show an obvious difference. This is due to the fact that Birnbaum importance became more important than Criticality importance, and also greater than harmfulness importance. Despite the difference between the BI and CI values at longer running time, the rankings on the importance of the components remain unchanged.

Figure 8 displays the BI and CI values for components at the optimal solutions as the supply and demand is balanced (P_N = P_out). Importance levels of the components evaluated by the two methods are the same. The results of components C6, C7, and C14 are zero because of their absence during system operation. From the figures, the importance levels of components of C5 and C9 are prominent, especially component C9, leading to a greater influence on the top event of the system. Thus, the characteristics of components C5 and C9 can be assumed as (A). The BI and CI values of components C10, C11 and C12 are the same due to their similar features, and that of components C1, C2, C3 and C4 also tend to be consistent but their values are larger than that of components C10, C11 and C12. In terms of the characteristics, the above seven components can be assumed as (B). Importance of component C13 ranks in the last position, mainly because it does not work frequently under normal circumstances attributed to its special nature for the entire system. The characteristics of component 13 can be assumed as (C) or not labeled while processing. Generally, from the time variation trend of the importance of the components, it can be concluded that each system component has maximum time during which there is a significant impact on the overall operation of the system. After this time, the importance of the component is getting smaller, and the maximum time node of each component is different, which is related to the average failure time of the components.

Figure 9 and Figure 10 indicate the importance values and ranking bars when the power demand is less than the amount of electricity (P_N < P_out), all components of the system involving in the operation of the system. Two kinds of important levels are consistent. Among them, the importance levels of components C5, C8, C9, and C14 are high, wherein the effect of the component C9 is the greatest, closely followed by C5, C8 and C14. Thus, the characteristics of these components can be assumed as (A). The importance results of components C1, C2, C3, and C4 are almost the same but their importance results are greater than components C10, C11, and C12. The characteristics of the seven components can be stated as (B). Nevertheless, regarding importance result, component C13 is last because of its special nature. The similar trends can be seen for components C6 and C7. So, the characteristics of components C6, C7, and C13 can be viewed as (C).

Importance analysis of component under all circumstances of supply and demand of electric quantity have shown that components C5, C8, C9, and C14 dominate the reliability of the off-grid wind power generation system. These components should be more carefully monitored and maintained. Due to their special location structure, the rectifier, battery, discharge load, and valve controller have significant importance. However, importance has no direct relationship with the average time of failure or failure rate. In spite of its special location structure compared to other devices, the component C13 (mechanical braking device) generally has the minimum importance. This component is present to protect the system and does not directly take part in the process of electricity generation. Components C6 and C7 are both controllers. When the user’s load demand for electricity is 0 or the power supply exceeds demand, the controllers can only have a significant importance level in stabilizing the system. Under other circumstances, the importance of controllers is always 0. The controllers have little influence on the electricity generation and the output of power supply: they only need to be in regular inspection and maintenance.

3.5.2. Quota Ratio of P_N and P_out

In the study of weights of running states for the off-grid wind power generation system, quotient parameter k₂ is approximate 0.15, parameter k₁ is approximate 2.33. The ratio of P_N < P_out, P_N = P_out, P_N > P_out, P_N = 0 equates to 3 : 4 : 2 : 1. Figure 11 reveals that components C9, C5, and C8 have the greatest importance. This means that valve controller, rectifier, and battery are the crucial components influencing the safe and effective operation of the system. They need to be carefully monitored and maintained. Due to its function of assistant operation belonging to a non-normal working range, the braking device (component C13) has the least influence in the system importance assessment. In spite of its higher failure rate than that of other components, the results of importance assessment for component C13 are not relatively important. Therefore, the maintenance interval of the braking device can be prolonged to reduce waste of manpower and material resources, and more attention can be paid to the crucial components. The controllers (components C6 and C7) have little influence on the importance assessment and operation of system, although they are relative important components and have high failure rates and a key position of structure in the system. Because they fall under the multi-configuration operation system, they are not in normalization of operation. Components C1, C2, C3, C4, C10, C11, C12, and C14 are in the medium situation of the importance assessment and can be marked as significant characteristic (B). They are the common objects of maintenance and optimization design.

4. Discussion

Importance analysis is carried out due to few similar researches particularly on small wind power system. BI and CI are popular measures for prioritization of reliability improvement activities, and identification of weak links in the system. However, previous studies focus more on the system reliability analysis, including the electricity transmission system, bulk electric system, power transformer system, multiple energy systems, etc. [27,32,37,38]. This study attempts to systematically analyze a small wind power system from the perspective of component reliability, with operating status and quota ratio being jointly considered for reliability evaluation. The reliability evaluation results of components under different operating conditions and scenarios integration show that the most critical parts of the wind power system are the rectifier, battery, discharge load, and valve controller. This result is obtained based on the premise that, although the wind speed changes, the power output should meet the load demand.

Researches have been conducted on the reliability studies, failure modes and effects analysis of offshore wind turbines [39]. Results shown that higher reliability of indirect drive turbines is revealed. Electrical subassembly and gearbox are confirmed to be the main problems of wind turbines, among which the inverter and electronics have higher failure rates than the gearbox. These findings have certain similarities with the results of this study. However, importance levels of electrical subassemblies are subtly studied and further ranked in this study. The implication in this study is a positive step forward in transferring system analysis to component analysis, providing more theoretical bases in support of system design, process optimization, probability safety assessment, reliability analysis, etc.

5. Conclusions

This paper proposes an importance analysis framework for the evaluation of running weight parameters for a multi-configuration (or multi-component) off-grid wind power system. BI and CI are used as the importance indicators. Wind speed with stochastic characteristics, load demand with multiple scenarios, and energy transfer with different paths are merged into the evaluation framework. To better understand the components’ importance, the probability distribution of wind speed and the equilibrium of supply and demand are comprehensively considered. According to the probability distribution of wind speed, the running weight coefficient of the system are then calculated. A household-type wind power generation system is applied to further verify the effectiveness and stability of the proposed model. Results from the case study reveal that (a) the components rectifier (C5), battery (C8), discharge load (C14), and valve controller (C9) rank the highest in importance level in the off-grid wind power generation system in all circumstances of demand and supply, principally because of their special location structure, but importance has no direct relationship with the average time of failure or failure rate; (b) the component C13 (braking device) has the minimum importance level; (c) components C6 and C7 (controllers) have a significant importance level in stabilizing the system when the users’ load demand for electricity is 0 or the power supply exceeds demand, but under other circumstances, the importance of controllers is always zero.

The weights of a multi-configuration system are divided into four classes according to the power balance relationship between supply and demand. Every class has its own parameters distribution, which is different from each other. However, the final parameters evaluated in this paper should be determined by the four classes. Thus, the ratio of power supply and demand, which is determined for specific place or conditions, is also important. For system designing or management, it plays an important role through dealing with uncertainty process. It is observed that the importance of components in a wind power system and the balance between power supply and demand influence each other. The ratio between power supply and demand has a great impact on the components’ importance level. Conversely, the design of power supply and demand can be achieved indirectly through changing the failure rate and reliability of system components, which has a high reference value for the incessant improvement and development of wind power system.

It should be mentioned that the wind speed data in this paper are directly referenced to the effective wind speed, such as cut-in wind speed and cut-out wind speed. Besides, during the process of system component importance analysis, the discussed time intervals refer to a continuous period of time for system operation. Actually, the wind turbine usually works in manual operation, relisting in an artificial shutdown times, such as maintenance downtime. However, this study ignores such artificial shutdown times, so the corresponding solutions are limited to the requirements of the continuous operation of the system. Moreover, in this paper, the proposed method is only determined by two main factors (i.e., wind speed and load demand) without other considerations such as planned production, natural disaster, and system shutdown. Only one year of the data regarding wind speed are collected in this study, which may not substantially support the importance analysis results. This may be addressed by using more available data with increasing observation years. Furthermore, this study centers mostly on a small-scale wind power generation system, but it frequently encounters difficulties, especially when facing a more complex energy system. It is thus desirable to discuss a more complex energy network system with multi configuration based on importance analysis. With regard to weights determination, it can be achieved by using the analytic hierarchy process. Through the above treatment, the reliability of a wind power generation system could be further improved.