METHODOLOGY FOR ECONOMIC OPTIMIZATION OF STATIC CONVERTER CIRCUITS

A. S. BATURIN

(Presented by the scientific seminar of the Department of Industrial Economics)

The technical development of static converter circuits is an important direction in the creation of advanced automation systems. However, each circuit solution can be implemented using various types (brands) of elements that have different prices and possess different operating characteristics (efficiency, failure rate, accuracy of output characteristics, etc.).

In addition, the operating conditions are also different, that is, the number of hours of amplifier operation per year, the ratio between time operating under load and at no load, the tariff for electrical energy, environmental conditions, and requirements for failure-free operation. One of the factors determining the production cost of a converter is also the volume of production and the conditions of the manufacturing plant. All the factors listed influence, on the one hand, the price of the static converter (capital investments associated with its use), and, on the other, the annual operating costs.

An assessment of cost-effectiveness for a given set of factors may be carried out by the formula for annual reduced costs:

\[ Z_g = C + E_n \cdot K, \tag{1} \]

where \(Z_g\) — annual reduced costs;
\(C\) — annual operating costs;
\(E_n\) — coefficient of comparative economic efficiency;
\(K\) — capital investments associated with the use of the static converter.

As the quantitative criterion of the cost-effectiveness of a static converter for given operating conditions, we take the minimum of annual reduced costs. Determining the cost-effectiveness of use requires enumerating a large number of variants, which makes it necessary to employ a computer.

In the present work, the development of a research methodology is considered for selecting an economically advantageous implementation variant of a static converter for various production and operating conditions.

Since in the formula for annual reduced costs \(E_n = 0.12\) is a standard value [1], the variable quantities are \(C\) and \(K\). The study should begin with capital costs, since without this it is impossible to determine the elements of operating costs.

Capital investments include: the price of the static converter, transportation expenses, and costs associated with installation

at the place of operation. The largest share in capital expenditures is the price. Therefore, first of all we shall determine the price of the converter.

The price of a static converter includes the full manufacturing cost and the enterprise’s profit. The full manufacturing cost can be determined most accurately by costing for specified manufacturing conditions.

In our case, when no technology is available, it is impossible to prepare a costing.

Given the information available (the circuit solution is known, as are the mathematical relationships for determining the parameters of the circuit elements, and the input and output parameters of the static converter), we can use aggregate methods for determining manufacturing cost. The method most acceptable for us will be that of determining cost by the specific weight of material expenditures in the full cost of analogous products. Mathematically this is expressed as follows:

\[ C_{\text{п}} = \frac{M_{\text{п}}}{У_{\text{м}}}\cdot 100, \tag{2} \]

where \(C_{\text{п}}\) is the full cost of the static converter;
\(M_{\text{п}}\) is the material expenditures of the converter;
\(У_{\text{м}}\) is the specific weight of material expenditures in the full cost of analogous products, in percent.

The material expenditures of a static converter comprise the cost of the circuit elements, including transport and procurement expenses.

To determine the cost of the elements, it is necessary, on the basis of the output and input parameters of the static converter, to find by formulas the parameters of the corresponding element, and then to select the element according to the corresponding price list and at the same time take the price of the element. Sequential determination of the elements and their prices makes it possible to determine material expenditures as

\[ M_{\text{п}}^{(j)} = \sum_{1}^{n} Ц_i, \tag{3} \]

where \(n\) is the number of circuit elements;
\(Ц_i\) is the list price of the \(i\)-th element.

This operation can be performed on an electronic computer (ECM). To carry it out, the parameters of the elements and their prices, which can be used for manufacturing the static converter, are entered into the machine memory. Along with this, mathematical relationships are entered into the machine memory, on the basis of which the parameters of the circuit elements can be determined. Using the input and output parameters, as well as the mathematical relationships, the ECM itself selects the elements according to the specified conditions. In this case, machine design of the static converter is carried out.

Since the functions of an element may be performed by different types (brands) of elements, the machine will sort through all possible sets of one and the same circuit and thereby determine different material expenditures \(\left(M_{\text{п}}^{j}\right)\).

It is known that the specific weight of material expenditures in analogous products \(У_{\text{м}}\) varies with the scale of production. Therefore we take \(У_{\text{м}}\) as varying from 30 to 50% and, accordingly, obtain a set of costs for different scales and conditions of production.

To pass from cost to price and then to capital investments, we shall use coefficients, that is,

\[ K=C_{\text{п}}\cdot K_1\cdot K_2, \tag{4} \]

where \(K\) is the capital investment associated with the use of a static converter;

\(K_1\) is a coefficient taking into account the enterprise profit; we take \(K_1=1.10\) [2];

\(K_2\) is a coefficient of additional costs for transportation to the consumer and installation; we take \(K_2=1.05\).

The generalized formula for determining capital costs will be

\[ K=\frac{100\cdot K_1\cdot K_2\displaystyle\sum_{1}^{n} \Ц_i}{У_{\text{м}}}. \tag{5} \]

The notation is the same as in formulas (1), (2), (3), (4), given earlier.

On the basis of formula (5), a set of values of capital costs \((K)\) will be obtained, which are used for further investigations.

The annual operating expenses \((C)\) for a static converter include the following costs:

a) depreciation for renovation, since the product under consideration is not subject to capital repair;

b) expenses for current repairs;

c) the cost of the electrical energy consumed.

Wage costs for maintenance personnel are not included, since the circuit operates in automatic mode.

Depreciation for renovation for static converters, according to the current norms, is 10%. In this case the depreciation period is 10 years. However, it is known that the full service life of a static converter is 30,000 hours. Consequently, a depreciation rate of 10% corresponds to 3,000 or fewer hours of operation per year. With a larger number of operating hours, namely 4,000, 6,000, 8,760 hours per year, the service life of the converter is shortened and therefore the depreciation rate for renovation must be increased; that is, at 4,000 hours of operation,

\[ \frac{4000}{30000}\cdot 100=13.3\%, \]

at 5,000 h—17%, at 6,000 h—20%, and at 8,760 h (continuous operating mode)—30%.

The annual depreciation amount is determined by the formula

\[ A_{\text{г}}=\frac{H_{\text{a}}\cdot K}{100}, \tag{6} \]

where \(A_{\text{г}}\) is the annual amount of depreciation for renovation;

\(H_{\text{a}}\) is the depreciation rate, in percent.

For our case, with an annual number of operating hours up to 3,000 h—10%; 4,000 h—13.3%; 5,000 h—17%; 6,000 h—20%; 8,760 h—30%.

Expenses for current repairs may be calculated, proceeding from the failure rates of the individual elements of the circuit, by the following formula:

\[ P=T\cdot K_3\cdot \sum_{1}^{m} H_i \lambda_i \Ц_i, \tag{7} \]

where \(P\) is the annual repair expenses;

\(T\) is the number of hours of converter operation during the year; we take:

\(T = 1000,\ 2000,\ 3000,\ 4000,\ 5000,\ 6000,\ 8760\) h.

\(m\) — number of different types of elements in the converter circuit;
\(\mathrm{H}_i\) — number of elements of the \(i\)-th type (make);
\(\lambda_i\) — failure rate of an element of the \(i\)-th type (make);
\(\mathrm{Ц}_i\) — price of the \(i\)-th element according to the current price list;
\(K_3\) — aggregate coefficient taking into account: transport and procurement costs associated with acquiring circuit elements (1.05), wages, and overhead expenses. We take the value of \(K_3\) within the range from 2 to 3, which corresponds to the specific weight of material costs of 30–50%.

The cost of the energy consumed can be determined by the formula

\[ Э = T_э \cdot \left( T_p \frac{P_в}{\eta} + T_x \cdot P_x \right), \tag{8} \]

where \(Э\) — annual costs associated with energy consumption;
\(T_p\) — operating time of the converter under full-load conditions;
\(P_в\) — output power of the converter;
\(\eta\) — efficiency of the converter; according to technical research data it may be taken as 0.90;
\(T_э\) — two-rate electricity tariff, which for different power systems ranges from 0.0063 to 0.090 rub./kWh, and the average value is 0.013 rub./kWh. In studies this indicator may be varied;
\(T_x\) — no-load operating time, h/year;
\(P_x\) — power consumed in no-load operation.

In a number of cases \(T_x\) has an insignificant specific weight, and then the formula takes the form

\[ Э = T_p \cdot \frac{P_в}{\eta} \cdot T_э . \tag{9} \]

In general, annual operating expenses \((C)\) may be determined by the formula

\[ C = \frac{\mathrm{H}_з \cdot K}{100} + T \cdot K_э \cdot \sum_{i}^{m} \mathrm{H}_i \cdot \lambda_i \mathrm{Ц}_i + T_э \left( T_p \cdot \frac{P_в}{\eta} + T_x \cdot P_x \right). \tag{10} \]

The initial data for conducting the study are:

1) schematic diagrams of static converters;
2) the list of elements included in each circuit;
3) the types (makes) of elements that can perform functions in the circuits;
4) data on the main technical parameters and prices for the manufactured types (makes) of elements that may be used in static-converter circuits;
5) input and output parameters of static-converter circuits;
6) mathematical relationships for determining the parameters of the elements and the procedure for calculating them;
7) variable values for conducting the study, which were mentioned earlier;
8) formula of the quantitative criterion.

\[ Z_{\mathrm{g}} = \left( E_{\mathrm{n}} + \frac{H_{\mathrm{a}}}{100} \right) \frac{100 \cdot K_1 K_2 \sum_{1}^{n} C_i}{U_{\mathrm{m}}} + \]

\[ + T \cdot K_3 \sum_{1}^{m} H_i \lambda_i C_i + T_{\mathrm{e}} \left( T_{\mathrm{r}} \frac{P_{\mathrm{v}}}{\eta} + T_{\mathrm{x}} P_{\mathrm{x}} \right), \tag{11} \]

where

\[ K = \frac{100 K_1 K_2}{U_{\mathrm{m}}} \cdot \sum_{1}^{n} C_i \tag{5}. \]

The procedure for carrying out the investigation on a computer is predetermined by the formula of the quantitative criterion (11).

First of all, all constant quantities are entered into the machine memory: \(E_{\mathrm{n}}\), tables of parameters and correlation dependences of element prices on parameters, \(\lambda_i\), the coefficients \(K_1\), \(K_2\), \(K_3\), and the variable values \(U_{\mathrm{m}}\), \(T\), \(H_{\mathrm{a}}\), \(T_{\mathrm{r}}\), \(T_{\mathrm{x}}\), \(T_{\mathrm{e}}\).

Then the input and output parameters of the static converter \((U_{\mathrm{p}}, I_{\mathrm{n}}, Z_{\mathrm{n}})\), as well as the mathematical formulas for selecting elements, are entered into the machine memory.

The solution of the problem begins with calculating the parameters of the elements from the formulas and selecting them from the machine memory according to a definite logical criterion. As a result, we find \(C_i\), or their set, if there are different types (brands) of elements capable of performing the specified functions in the circuit. This stage is the machine design of the static converter. Enumeration of all elements creates different circuit variants and makes it possible to determine all possible values \((K)\) of the capital costs associated with the use of the static converter. At this stage it is possible to obtain data on capital costs for different sets of variable parameters: types (brands) of elements, \(U_{\mathrm{m}} = 30, 35, 40, 45, 50\). Along with this, one can additionally vary the values of \(U_{\mathrm{p}}\) and \(I_{\mathrm{n}}\) at one and the same power of the static converter and thereby find the optimal \(U_{\mathrm{p}}\) and \(I_{\mathrm{n}}\), at which the minimum capital costs of the converter are achieved (5).

As a whole, the annual-cost formula (11) is implemented. We obtain a set of values of \(Z_{\mathrm{g}}\) for different variants. From this set one should select the minimum values for definite variants and operating conditions. Along with this, one should have data for comparing them with the maximum values of the annual reduced costs in the given set.

The solution of the problem in full can be represented as follows. We take one varying parameter, vary its values, and find the annual reduced costs (11) at constant values of the remaining variable parameters. The machine selects the minimum from the obtained series and its basic data. In addition, it is necessary to compare the minimum found with the maximum value in order to establish the degree of economy of the obtained solution. As a result, we have the first optimal solution.

Then the second variable parameter is changed at constant values of the others. This is repeated until all the variable parameters have been enumerated. For each enumeration we shall have an economic solution, and the entire region will have been investigated.

References

1. Standard methodology for determining the economic efficiency of capital investments. Moscow, 1969.

2. A. A. Koshuta. Price and economic efficiency of machines. Moscow, “Ekonomika,” 1969.

3. E. K. Smirnitskii. Economics and the machine. Moscow, “Ekonomika,” 1970.