Availability growth models and verification of power equipment

Abstract

The general availability growth models for large scale complicated repairable system such as electric generating units, power station auxiliaries, and transmission and distribution installations are presented. The calculation formulas for the maintenance coefficient, mathematical expressions for general availability growth models, ways for estimating, and fitting on checking the parameters of the model are introduced. Availability growth models for electric generating units, power station auxiliaries, and transmission and distribution installations are given together with verification examples for availability growth models of 320–1000 MW nuclear power units and 1000 MW thermal power units, 200–1000 MW power station auxiliaries, and 220–500 kV transmission and distribution installations. The verification results for operation availability data show that the maintenance coefficients for electric generating units, power station auxiliaries, transmission and distribution installations conform to the power function, and general availability growth models conform to rules of availability growth tendency of power equipment.

Graphical abstract

Keywords

repairable system / power equipment / electric generating unit / power station auxiliary / transmission and distribution installation / reliability / availability / availability growth model

1 Introduction

The “International Electro Technical Commission Standard” (IEC60050) has defined the “reliability growth” as “a process which improves the product reliability quantity with time.” In most cases, the reliability characteristic quantity of electrical-mechanical products is the mean time to failures M_TTF. The reliability growth models of M_TTF are the Duane model and the AMSAA (The US Army Material Systems Analysis Activity) model. The earliest developed and most frequently used reliability growth model was first proposed by Duane in 1964 [1]. The AMSAA model was developed by Crow in 1984 [2]. Healy provides an alternative to the Duane model that ignores early failures in 1987 [3].

Due to the difference of the availability characteristic quantities between electric generating units, power station auxiliaries, and transmission and distribution installations with M_TTF, there is no sufficient investigations of availability growth models of power equipment abroad. In 1999, a reliability growth model was proposed for the equivalent availability factor E_AF of domestic 300 MW thermal power generating units [4]. In 2009, the model was applied to pumped-storage power units [5], and in 2016, the model was applied to the reliability prediction of nuclear power units in service [6].

Based on the comprehensive analysis of mathematical expressions for the maintenance coefficients and availability characteristic quantities, the general availability growth model for large scale complicated repairable systems are investigated aiming at providing a theoretical guidance for the availability growth management of large scale complicated repairable systems such as electric generating units, power station auxiliaries, and transmission and distribution installations.

2 General availability growth model of large scale complex repairable system

2.1 Calculation formula for maintenance coefficient

When the active state of a large scale complicated repairable system such as electric generating units or power systems is treated as two states for “up” and “down” as shown in Fig. 1, the respective formulations for the availability A, the unavailability U, and the maintenance coefficient r_yx are

(1)A={{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i\left({\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i}+{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{y}_i}\right)}}^{-\mathrm{1}}={(\mathrm{1}+{\rho }_{yx})}^{-\mathrm{1}},

(2)U={\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{y}_i}{\left({\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i}+{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{y}_i}\right)}^{-\mathrm{1}}={\rho }_{yx}{(\mathrm{1}+{\rho }_{yx})}^{-\mathrm{1}},

(3){\rho }_{yx}={\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{y}_i}{\left({\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i}\right)}^{-\mathrm{1}}={\displaystyle \frac{\mathrm{1}-A}{A}}={\displaystyle \frac{U}{\mathrm{1}-U}},

where x₁, x₂, …, x_n are the uptimes for a large complex repairable system in the up state; y₁, y₂, …, y_n are the downtimes for a large complex repairable system in the down state; n is the number of down; and r_yx is the maintenance coefficient.

Under the condition that the availability of a large scale complicated repairable system such as electric generating units, power station auxiliaries, and transmission and distribution installations is being improved, the availability A and the unavailability U are not any fixed number any more but become some functions of time t, and the maintenance coefficient r_yx is yet functions of time t. The maintenance coefficient r_yx (t_i) in the t_ith year can be represented by

(4){\rho }_{yx}(t_i)=\frac{1-A(t_i)}{A(t_i)}=\frac{U(t_i)}{1-U(t_i)},

where A(t_i) is the statistical values of the availability in the t_ith year and U(t_i) is the statistical values of the unavailability in the t_ith year.

2.2 General availability growth models

According to a reliability growth model for 300 MW thermal power generating units [4], the model expressed in Eq. (5) is proposed for the general availability model for the maintenance coefficient of large scale complicated repairable systems.

(5){\rho }_{yx}(t)=\eta t^{-m},

where r_yx(t) is the maintenance coefficient in the tth year, h is the scale parameter, and m is the availability growth factor.

The physical indication of the availability growth factor m is that m indicates the variation of the availability for large scale complicated repairable systems. When m is less than 0, the availability of large scale complicated repairable systems is considered to be decreasing. When m equals 0 and r_yx(t) is constant, the availability is considered to be constant. The availability is increasing when m is greater than 0. As m increases, the availability for large scale complicated repairable systems increases, too.

When t equals 1, r_yx(t) = η, and A = (1+ h)^–1 can be obtained. The scale parameter h indicates the maintenance coefficient of large scale reparable systems at the beginning of the availability following when t equals 1. The reciprocal of (1+ h) indicates the availability of large scale reparable systems at the beginning of the availability supervisory when t equals 1.

Equation (5) for the general availability growth model of large scale complicated repairable systems can be expressed as

(6)A(t)={(\mathrm{1}+\eta t^{-m})}^{-\mathrm{1}},

(7)U(t)=\eta t^{-m}{(\mathrm{1}+\eta t^{-m})}^{-\mathrm{1}}.

In general, Eq. (6) is adopted to indicate the growth model of the availability A of large scale complicated repairable systems while Eq. (7) is indicated the decrease model of the unavailability U of large scale complicated repairable systems. The decrease of the unavailability U indicates the growth of the availability A.

2.3 Parameter estimation and fitting check of general availability growth models

Since r_yx(t) in Eq. (5) is the power function, the nonlinear regression analysis is used in determining the undetermined parameters of the general availability growth model. Because the relation between t and r_yx(t) in Eq. (5) appears to be a linear function of t on a log-log plotting paper, parameters m and h can be determined by the least square method [7].

Checking whether Eq. (5) conforms to the rule that governs the availability growth model of the large scale complicated repairable systems is equivalent to examining the linearity of the statistical availability data of the system on a log-log paper. According to Ref. [7], the statistic checking magnitude F can be calculated by

(8)F={\displaystyle \frac{\widehat{b}\cdot S_{xy}\cdot (n-\mathrm{2})}{S_{yy}-\widehat{b}\cdot S_{xy}}},

where, \widehat{b}={\displaystyle \frac{S_{xy}}{S_{xx}}}, S_{xx}={\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i^{\mathrm{2}}-n{\overline{x}}^{\mathrm{2}}}, S_{xy}={\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i{y}_i-n\overline{x}}\overline{y}, S_{yy}={{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{y}_i^{\mathrm{2}}-n\cdot \overline{y}}}^{\mathrm{2}}, x_i = lnt_i, y_i = lnr_yx (t_i), \overline{x}={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{x}_i}, \overline{y}={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \underset{i=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{y}_i}.

Based on Eq. (8), the statistic checking magnitude F is obtained. With an assumed significance a, F_a (1, n–2) can be found by using an F-distribution table [7]. In case F>F_a (1, n–2), the general availability growth model (Eq. (5)) is acceptable, but it becomes unacceptable in case F≤F_a (1, n–2).

3 Availability growth models of power equipment

3.1 Availability growth models of electric generating units

The equivalent availability factor E_AF, the maintenance coefficient ρ_EA, the equivalent availability factor deducted planed outage hours from period hours E_AP, and the maintenance coefficient ρ_AP of electric generating units are respectively expressed as

(9)E_{\mathrm{AF}}={\displaystyle \frac{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}{t_{\mathrm{PH}}}}={\displaystyle \frac{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}{t_{\mathrm{AH}}+t_{\mathrm{UH}}}}={\displaystyle \frac{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}{(t_{\mathrm{AH}}-t_{\mathrm{EUNDH}})+(t_{\mathrm{UH}}+t_{\mathrm{EUNDH}})}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\rho }_{\mathrm{EA}}}},

(10){\rho }_{\mathrm{EA}}={\displaystyle \frac{t_{\mathrm{UH}}+t_{\mathrm{EUNDH}}}{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}}={\displaystyle \frac{\mathrm{1}-E_{\mathrm{AF}}}{E_{\mathrm{AF}}}},

(11)E_{\mathrm{A}\mathrm{P}}={\displaystyle \frac{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}{t_{\mathrm{PH}}-t_{\mathrm{P}\mathrm{O}\mathrm{H}}}}={\displaystyle \frac{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}{t_{\mathrm{AH}}+t_{\mathrm{U}\mathrm{O}\mathrm{H}}}}={\displaystyle \frac{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}{(t_{\mathrm{AH}}-t_{\mathrm{EUNDH}})+(t_{\mathrm{U}\mathrm{O}\mathrm{H}}+t_{\mathrm{EUNDH}})}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\rho }_{\mathrm{A}\mathrm{P}}}},

(12){\rho }_{\mathrm{A}\mathrm{P}}={\displaystyle \frac{t_{\mathrm{U}\mathrm{O}\mathrm{H}}+t_{\mathrm{EUNDH}}}{t_{\mathrm{AH}}-t_{\mathrm{EUNDH}}}}={\displaystyle \frac{\mathrm{1}-E_{\mathrm{A}\mathrm{P}}}{E_{\mathrm{A}\mathrm{P}}}=\frac{1-P_{\mathrm{O}\mathrm{F}}-E_{\mathrm{A}\mathrm{P}}}{E_{\mathrm{A}\mathrm{P}}}},

where t_AH is the available hour, t_EUNDH is the equivalent unit derating hour, t_P is the period hour, t_POH is the planned outage hour, t_UH is the unavailable hours, and t_UOH is the unplanned outage hour.

By reference to Eq. (5), the availability growth models for maintenance coefficients of electric generating units are respectively expressed as

(13){\rho }_{\mathrm{EA}}(t)={\eta }_{\mathrm{1}}{t}^{-m_{\mathrm{1}}},

(14){\rho }_{\mathrm{AP}}(t)={\eta }_{\mathrm{2}}{t}^{-m_{\mathrm{2}}},

where h₁ and h₂ are the scale parameters, and m₁ and m₂ are the availability growth factors.

Referring to Eq. (6), the availability growth models of electric generating units with the availability characteristic quantity for E_AF and E_AP are respectively expressed as

(15)E_{\mathrm{AF}}(t)={(\mathrm{1}+{\eta }_{\mathrm{1}}{t}^{-m_{\mathrm{1}}})}^{-\mathrm{1}},

(16)E_{\mathrm{AP}}(t)={(\mathrm{1}+{\eta }_{\mathrm{2}}{t}^{-m_{\mathrm{2}}})}^{-\mathrm{1}}.

3.2 Availability growth models of power station auxiliaries

The use availability A_U and the maintenance coefficient ρ_US of power station auxiliaries are respectively expressed as

(17)A_U={\displaystyle \frac{t_{\mathrm{SH}}}{t_{\mathrm{SH}}+t_{\mathrm{UOH}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\rho }_{\mathrm{US}}}},

(18){\rho }_{\mathrm{US}}={\displaystyle \frac{t_{\mathrm{UOH}}}{t_{\mathrm{SH}}}}={\displaystyle \frac{\mathrm{1}-A_{\mathrm{U}}}{A_{\mathrm{U}}}},

where t_SH is the service hour.

By reference to Eq. (5), the availability growth model for the maintenance coefficient of power station auxiliaries is expressed as

(19){\rho }_{\mathrm{US}}(t)={\eta }_{\mathrm{3}}{t}^{-m_{\mathrm{3}}},

where h₃ is the scale parameter, and m₃ is the availability growth factor.

Referring to Eq. (6), the availability growth model of power station auxiliaries with the availability characteristic quantity for A_U is expressed as

(20)A_U(t)={(\mathrm{1}+{\eta }_{\mathrm{3}}{t}^{-m_{\mathrm{3}}})}^{-\mathrm{1}}.

3.3 Availability growth models of transmission and distribution installations

The transmission and distribution installations include transformers, reactor, circuit breakers, current transformers, voltage transformers, isolating switches, arresters, composite apparatus, and so on. According to DL/T837, the major availability characteristic quantity of transmission and distribution installations are the availability factor A_F. The availability factor A_F and the maintenance coefficient r_AF of transmission and distribution installations are respectively expressed as

(21)A_F={\displaystyle \frac{t_{\mathrm{AH}}}{t_{\mathrm{AH}}+t_{\mathrm{UH}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\rho }_{\mathrm{AF}}}},

(22){\rho }_{\mathrm{AF}}={\displaystyle \frac{t_{\mathrm{UH}}}{t_{\mathrm{AH}}}}={\displaystyle \frac{\mathrm{1}-A_F}{A_F}}.

By reference to Eq. (5), the availability growth model for the maintenance coefficient of transmission and distribution installations is expressed as

(23){\rho }_{\mathrm{AF}}(t)={\eta }_{\mathrm{4}}{t}^{-m_{\mathrm{4}}},

where h₄ is the scale parameter, and m₄ is the availability growth factor.

Referring to Eq. (6), the availability growth model of transmission and distribution installations with the availability characteristic quantity for A_F is expressed as

(24)A_F(t)={(\mathrm{1}+{\eta }_{\mathrm{4}}{t}^{-m_{\mathrm{4}}})}^{-\mathrm{1}}.

4 Verification for availability growth models of power equipment

4.1 Verification for availability growth models of electric generating units

By use of the availability statistical data from China Electric Reliability Management Center, the analytical results for parameters of the availability growth models of some electric generating units are obtained. The analytical results for parameters of the availability growth models for the equivalent availability factors E_AF of all the 320–1000 MW nuclear power units in China between 2008 and 2017 are given in Table 1. It is observed that F is larger than F_a, which implies that the regulation of the availability growth for the 320–1000 MW nuclear power units is in accordance with the general availability growth models for large scale complicated repairable system in Eq. (5) and the regulation of availability growth models for electric generating units in Eqs. (13) and (15). From Table 1, it is observed that since m₁ is greater than 0, the equivalent availability factors E_AF of 320–1000 MW nuclear power units does have a tendency of continuous growth. Given that t_i = 1 in 2008, t_i = 8 in 2015, and t_i = 10 in 2017, with the parameters m₁ and h₁ of the availability growth model in 2008 and 2015, the predicted values and the relate error E_r of the equivalent availability factors E_AF in 2016 and 2017 can be determined and given in Table 2. The statistical values and the predicted values of the equivalent availability factors E_AF is shown in Fig. 2.

The analytical results for parameters of the availability growth models for the equivalent availability factor deducted planed outage hours from period hours E_AP of two 1000 MW thermal power generating units in Zouxian Thermal Power Station are listed in Table 3. With the parameters m₂ and h₂ of the availability growth model before 2014, the predicted values and the relate error E_r of the equivalent availability factors in 2015 and 2016 can be determined and tabulated in Table 4. The statistical values and the predicted values of E_AP are respectively demonstrated in Figs. 3 and 4. It is observed that F is larger than F_a, which implies that the regulation of the availability growth for two 1000 MW thermal power generating units in Zouxian Thermal Power Station are in accordance with the general availability growth models for large scale complicated repairable system in Eq. (5) and the regulation of availability growth models for electric generating units in Eqs. (14) and (16). From Table 3, it is observed that since m₂ is greater than 0, the equivalent availability factor deducted planed outage hours from period hours E_AP of two 1000 MW thermal power generating units in Zouxian Thermal Power Station does have a tendency of continuous growth.

4.2 Verification for availability growth models of power station auxiliaries

By use of the availability statistical data for 200–1000 MW power station auxiliaries in 2008 and 2017, the analytical results for parameters of the availability growth models for the use availability A_U of coal mills, feed water pump sets, forced-draft fans, and high-pressure heaters of 200–1000 MW power station auxiliaries in China between 2008 and 2017 are presented in Table 5. With the parameters m₃ and h₃ of the availability growth model in 2008 and 2015, the predicted values and the relate error E_r of the use availability A_U in 2016 and 2017 can be determined and given in Table 6. The statistical values and the predicted values of A_U are respectively exhibited in Figs. 5 and 8. From Table 5, it is observed that since m₃ is greater than 0, the availability of the power station auxiliaries given in Table 5 does have a tendency of continuous growth. It is observed that F is larger than F_a, which implies that the regulation of the availability growth for the power station auxiliaries are in accordance with the general availability growth models for large scale complicated repairable system in Eq. (5) and the regulation of availability growth models for power station auxiliaries in Eqs. (19) and (20).

4.3 Verification for availability growth models of transmission and distribution installations

By use of the availability statistical data for 220–500 kV transmission and distribution installations in 2008 and 2017, the analytical results for parameters of the availability growth models for the availability factor A_F of transformers, reactors, circuit breakers, current transformers, voltage transformers, isolating switches, arresters, composite apparatus of 220–500 kV transmission, and distribution installations in China between 2008 and 2017 are given in Table 7. With the parameters m₄ and h₄ of the availability growth model in 2008 and 2015, the predicted values and the relate error E_r of the availability factor A_F in 2016 and 2017 can be determined and given in Table 8. The statistical values and the predicted values of A_F are respectively displayed in Figs. 9 and 16. From Table 7, it is seen that since m₄ is greater than 0, the availability of the transmission and distribution installations given in Table 7 does have a tendency of continuous growth. It is observed that F is larger than F_a, which implies that the regulation of the availability growth for the transmission and distribution installations are in accordance with the general availability growth models for large scale complicated repairable system in Eq. (5) and the regulation of availability growth models for power station auxiliaries in Eqs. (23) and (24).

5 Analysis and discussion

The analytical examples for the availability growth of nuclear power units, thermal power units, power station auxiliaries, and transmission and distribution installations verify that the maintenance coefficients for large scale complicated repairable systems such as electric generating units and power equipment meet the regulation of the power function. It is obtained that the power function is suitable to describe the regulation of maintenance coefficients for large scale complicated repairable systems such as electric generating units and power equipment. For large scale complicated repairable systems whose availability characteristic quantity is expressed by percentage, the maintenance coefficients meet the regulation of the power function.

The verifying results given in Tables 1, 3, 5, and 7 of practical availability statistics confirm that the E_AF of 320–1000 MW nuclear power units, the E_AP of 1000 MW thermal and nuclear power units, the A_U of 200–1000 MW coal mills, feed water pump sets, forced-draft fans and high-pressure heaters, the A_F of 220–500 kV transformers, reactors, circuit breakers, current transformers, voltage transformers, isolating switches, arresters, and composite apparatus conform to the availability growth models given in this paper. The availability growth models for E_AF, E_AP, A_U d A_F can been expressed by the function (1+ ηt^–m)^–1. The general availability growth models given in this paper can be used for fitting the operation availability history data of large scale complicated repairable systems such as electric generating units and power equipment.

For the power function r_yx(t) = ηt^–m, r_yx(t) is a positive monotonic function when t>0. The availability function A(t) = (1+ ηt^–m)^–1 or the unavailability function U(t) = ηt^–m(1+ t^–m)^–1 is between 0 and 100% which satisfy A(t) + U(t) = 1. For practical large scale complicated repairable systems such as electric generating units and power equipment, the availability A(t) and unavailability U(t) are both between 0 and 100%. For the large scale complicated repairable system whose availability characteristic quantities are expressed by percentage, if the power function obtained from availability statistics data in the previous years is utilized, the predicted availability characteristic quantities would not exceeds 100% or below 0. This is in accordance with practical engineering situation which can be adopted in the engineering practice of availability growth management and availability prediction.

The availability growth analytical examples of nuclear power units, thermal power units, power station auxiliaries, and transmission and distribution installations verify that the availability changing regulation of electric generating units and power systems conforms to the general availability growth models proposed in this paper. This project has proposed the general availability growth model which reveals the availability changing regulation and characteristics of the large scale complicated repairable systems such as electric generating units and power equipment.

The existing Duan model and AMSAA model are common reliability models which can be used to represent the growth rule of the reliability feature quantity MTTF of various mechanical products with different failure mechanisms. The growth model of general availability A proposed in this paper can also be used to represent the availability growth rule of large and complex repairable systems with different failure mechanisms, but using percentages to indicate the availability characteristics.

6 Conclusions

The proposed general availability growth models for large scale complicated repairable systems such as electric generating units and power equipment conform to the availability changing regulation for nuclear power units, thermal power units, power station auxiliaries, and transmission and distribution installations, whose availability characteristic quantities are expressed by percentage.

The quantitative analysis of availability growth of electric generating units and power equipment is realized using the availability growth models proposed in this paper. The models can be utilized to quantitatively follow and evaluate the availability improvement of electric generating units and power equipment which can provide evidence for availability improvement management.

The availability growth models of electric generating units and power equipment can be applied to the availability improvement analysis and prediction of other large scale complicated repairable systems such as metallurgy, chemical equipment, and other repairable electromechanical products. Besides they can also provide guidance for the availability assessment, availability improvement management and availability prediction of the large scale complicated repairable systems.

References

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[3] Healy J. A simple procedure for reliability growth modeling. In: Proceedings of the Annual Reliability and Maintainability Symposium, New York, USA, 1987

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