On the Assessment of the Reliability of Charged Particle Accelerators
Abstract Generated abstract
The paper considers how to assess the reliability of charged particle accelerators as complex technical systems for which conventional indices such as time to failure or failure rate do not adequately characterize practical usability. It reviews standard operational coefficients, including availability, utilization, downtime, and operating cost, and argues that these give insufficient economic evaluation and do not account for modernization during service. The authors propose a generalized operational characteristic based on output product over a calendar period, incorporating preventive maintenance duration, timing of maintenance, modernization cost, failure cost, and operating time. They suggest optimizing this functional, particularly with respect to preventive maintenance parameters, and note that mathematical programming methods, including stochastic programming with computer implementation, can be used for such reliability and operation planning problems.
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Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Volume 156 1969
On the Assessment of the Reliability of Charged Particle Accelerators
V. A. Kochegurov, N. I. Sablin
Reliable operation of accelerators is an important condition for their effective use. An accelerator may be assigned to the class of complex systems. For complex systems, such reliability criteria as time to failure, probability of failure-free operation, failure rate, etc., are “poorly adapted” and clearly do not express the usability of the system.
Therefore, at the present time the reliability of a system [1] should be understood as the stability of effectiveness, taking into account the reliability of the parts that make up the system. Reliability is one of the properties that make it possible to operate technical systems successfully. By operation [2] we shall understand the entire set of works that make it possible to use the system for its intended purpose:
- Preventive maintenance work.
- Restoration work.
- Training of operators.
- Development of methods and equipment for monitoring.
- Development of methods and equipment for forecasting, etc.
The operational characteristics currently used should include [2]:
- Availability coefficient
\[ K_{\mathrm{g}}=\frac{T}{T+T_{\mathrm{v}}}, \tag{1} \]
where \(T\) is the mean operating time,
\(T_{\mathrm{v}}\) is the mean restoration time.
- Utilization coefficient
\[ K_{\mathrm{u}}=\frac{t_{s\mathrm{i}}}{t_{s\mathrm{i}}+t_{s\mathrm{pr}}}, \tag{2} \]
where \(t_{s\mathrm{i}}\) is the total time of serviceable operation,
\(t_{s\mathrm{pr}}\) is the total time of forced downtimes.
In the present case, forced downtimes are understood as downtimes caused by failures and technical servicing.
- Downtime coefficient
\[ K_{\mathrm{pr}}=\frac{t_{s\mathrm{pr}}}{t_{s\mathrm{i}}+t_{s\mathrm{pr}}}. \tag{3} \]
- An economic assessment of operational properties is made with the aid of the operating-cost coefficient
\[ K_x=\frac{C_1}{C_0}, \tag{4} \]
where \(C_1\) is the cost of operation over the course of a year,
\(C_0\) is the cost of the apparatus.
In all three of the first cases, the coefficients are assessed for one and the same calendar period.
The principal shortcomings of the above operational characteristics are:
- A weak economic assessment of operational properties.
- The absence of criteria for assessing the modernization of systems, since in the process of operation the overwhelming majority of systems are improved. By modernization we shall understand a set of measures aimed at improving and increasing the reliability of systems.
For a stronger economic assessment and for taking account of the effect of modernization, a generalized operational characteristic (GOC) is proposed:
\[ Y(T)=F(t_{\mathrm{pr}}, \Theta, Ц_1, Ц_2, T), \tag{5} \]
where \(T\) is a specified calendar period,
\(t_{\mathrm{pr}}\) is the duration of preventive maintenance,
\(\Theta\) is the time interval from the beginning of operation to the beginning of preventive maintenance,
\(Ц_1\) is the price of modernization,
\(Ц_2\) is the price of failure,
\(Y(T)\) is the output product.
The output product of a system should be understood as the integral value of the system’s output parameter over the specified calendar period.
For example, for accelerators the output product may be taken as
\[ Y(T)=\sum_{i=1}^{n}\int_{t_{i-1}}^{t_i} y\,dt, \tag{6} \]
where \(y\) is an intensity varying according to some law;
\(t_i\) is the time of fault-free operation of the accelerator;
\(n\) is the number of \(t_i\)’s over the specified calendar period.
The price of modernization is the funds by means of which a set of measures aimed at modernizing the system is carried out:
\[ Ц_1=\sum_{i=1}^{n} Ц_{1i}, \tag{7} \]
where \(Ц_{1i}\) are the expenditures associated with the modernization of individual devices of the system (a differential parameter).
The price of failure is the total losses associated with the occurrence of a failure:
\[ Ц_2=\sum_{i=1}^{n} Ц_{2i}, \tag{8} \]
The price of failure consists mainly of four components:
\(Ц_{21}\) is the cost of the experiment,
\(Ц_{22}\) is the wages of the operating personnel and experimenters,
\(Ц_{23}\) is the loss of output product,
$C_{24}$ — repair of the system (cost of materials, wages, etc.).
Consequently,
\[ C_2 = C_{21} + C_{22} + C_{23} + C_{24}. \tag{9} \]
In practice, however, the cost of a failure is usually averaged and taken as constant, which is not an entirely correct solution.
The expression for the OEC is a functional, whose analytical expression depends on the specific type of system. Naturally, one should require an optimum of the OEC. The optimum of the OEC gives us the optimal value of system use
\[ Y(T) = F(t_{\mathrm{pr}}, \Theta, C_1, \tag{10} \]
\[ C_2, T) = \max. \]
With the aid of heuristic reasoning it can be shown that $Y(T)$ depends on its components in accordance with the dependences shown in Fig. 1.
As is seen from Fig. 1, the curves $Y(T)=f_1(t_{\mathrm{pr}})$ and $Y(T)=f_2(\Theta)$ have extrema at points $A$ and $B$, the curve $Y(T)=f_4(C_2)$ decreases monotonically, and the curve $Y(T)=f_3(C_1)$ increases monotonically but has an inflection point at point $C$. The inflection can be explained by the running-in of the modernized devices together with the system.
Fig. 1. Dependence of $Y(T)$ on
a) $Y(T)=f_1(t_{\mathrm{pr}})$,
b) $Y(T)=f_2(\Theta)$,
c) $Y(T)=f_3(C_1)$,
d) $Y(T)=f_4(C_2)$
It follows from the above that it is necessary to optimize the OEC only in a three-dimensional space, with subsequent allowance for $C_1$ and $C_2$, i.e.,
\[ Y(T) = F\,[Y_1(T), C_1, C_2, T], \tag{11} \]
where
\[ Y_1(T) = F_1(t_{\mathrm{pr}}, \Theta, T) = \max. \tag{12} \]
The problem of choosing optimal $t_{\mathrm{pr}}$ and $\Theta$ was considered in works [3, 4, 5, etc.]. It should be noted that such problems can be solved by methods of mathematical programming (for example, stochastic programming), whose main advantage is the extensive use of digital computers.
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