On the Static Accuracy of an Electric Machine Equivalent of Autonomous Power Generators
Abstract Generated abstract
This paper examines the static accuracy of an electric-machine equivalent used to reproduce the volt-ampere characteristics of autonomous power generators, with particular attention to photovoltaic-converter sources. It derives relations between open-loop gain and reproduction error for a nonlinear feedback scheme, showing that the required gain in nominal operation can be reduced while maintaining about 1 percent accuracy, thereby easing stability and transient-performance constraints. The paper also analyzes errors caused by heating of the electric-machine amplifier compensating winding and proposes a thermistor-based temperature correction circuit, with calculation formulas for its elements, to improve stability of the equivalent characteristics during prolonged operation.
Full Text
On the Issue of the Static Accuracy of an Electric-Machine Equivalent of Autonomous Power Generators
M. V. Lukyanenko, A. N. Lovchikov, G. M. Grinberg
(Presented by the scientific and technical seminar of the Departments of Electrical Machines and Apparatus and General Electrical Engineering)
In the investigation and development of electric-power-supply systems (EPSS) for autonomous facilities, equivalents of primary power generators are widely used. Possible equivalent circuits and questions of their application are considered in detail in [1, 2]. The present work is devoted to questions of ensuring the static accuracy with which an electric-machine equivalent reproduces the volt-ampere characteristic (VAC) of a primary power generator. A simplified block diagram of the equivalent is presented in Fig. 1, where \(K_y\) is the gain of the dc voltage amplifier, \(W_{\mathrm{emu}}(p)\) is the transfer function of the electric-machine amplifier, \(Z_n\) is the load resistance, \(K_{\mathrm{os}}\) is the feedback gain, \(K_{\mathrm{fp}}\) is the nonlinear gain of the functional converter (FC), \(U_{\mathrm{sm}}\) is the bias voltage, and \(U_e\) is the reference voltage.
Diagram labels: \(U_{\mathrm{sm}}\), \(U_e\), \(K_y\), \(W_{\mathrm{emu}}(p)\), \(U'_{\mathrm{out}}\), \(I'_{\mathrm{out}}\), \(\dfrac{1}{Z_n}\), \(K_{\mathrm{os}}\), \(K_{\mathrm{fp}}\).
Fig. 1. Block diagram of the equivalent
The accuracy of reproducing the VAC is determined mainly by the voltage open-loop gain \(K_{\mathrm{rk}} = K_y \cdot K_{\mathrm{emu}}\). However, taking into account that the load resistance enters into the transfer function of the electric-machine amplifier (EMA), \(K_{\mathrm{emu}}\), as the operating mode changes, may vary by a factor of 10. This means that, having ensured in the rated mode an accuracy of reproduction of \(U_e\), for example, of 1% (i.e., \(K_{\mathrm{rk}} = 100\)), in the no-load mode we obtain \(K_{\mathrm{rk}} = 1000\). Thus, to a considerable extent we worsen the stability conditions in the no-load mode. On the other hand, by providing, with the corresponding correction, sufficient stability of the closed-loop system in the no-load mode, we substantially reduce the quality indi-
quality of the transient process in the nominal mode, to which high requirements are imposed. Taking into account the features of this circuit, namely the current feedback, which includes a nonlinear FP, it is possible to prove the possibility of simplifying the circuit.
Depending on the current-voltage characteristic of the primary generator, various curves can be set up on the FP. The paper considers an AEP in which a battery of photovoltaic converters is used as the primary generator. The equation of the current-voltage characteristic of a photovoltaic converter has the following form:
\[ I = I_{\text{sc}}\left\{1 - e^{\frac{[\ln(1-j)](U-U_{\text{oc}})}{U_{\text{oc}}(i-1)}}\right\}, \]
where
\[ j=\frac{I_{\text{nom}}}{I_{\text{sc}}}; \qquad i=\frac{U_{\text{nom}}}{U_{\text{oc}}}. \]
Then, in the ideal case, the output current and voltage are related by the relation
\[ I_{\text{out}} = I_{\text{sc}}\left(1-k_1 e^{k_2 U_{\text{out}}}\right), \]
where
\[ k_1=\frac{1}{e^{\frac{\ln(1-j)}{i-1}}}; \qquad k_2=\frac{\ln(1-j)}{U_{\text{oc}}(i-1)}. \]
In a real system
\[ U'_{\text{out}} = U'_{\text{in}}\cdot m, \]
\[ I'_{\text{out}} = I_{\text{sc}}\left(1-k_1 e^{k_2\frac{U'_{\text{out}}}{m}}\right), \]
where
\[ m=\frac{K_{\text{rk}}}{1+K_{\text{rk}}}; \quad U_{\text{in}} \text{ is the voltage corresponding to } U_{\text{e}}, \xi U'_{\text{out}}, U'_{\text{in}}, I'_{\text{out}} \]
are the parameter values in the real system.
We find the error
\[ \Delta I = I_{\text{sc}}\cdot k_1\left(e^{k_2\frac{U'_{\text{out}}}{m}} - e^{k_2 U_{\text{out}}}\right), \]
\[ \frac{\Delta I}{I} = \frac{ k_1\left(e^{k_2\frac{U'_{\text{out}}}{m}} - e^{k_2 U_{\text{out}}}\right) }{ 1-k_1 e^{k_2 U_{\text{in}}} }. \]
Let us denote the system error by
\[ \frac{\Delta I}{I}=\frac{\Delta U}{U}=\delta, \]
where
\[ \Delta U = U_{\text{out}} - U'_{\text{out}}; \]
\[ \Delta I = I_{\text{out}} - I'_{\text{out}}. \]
Then
\[ \delta - k_1 \delta e^{k_2 U_{\text{out}}} = k_1 \left(e^{k_2 \frac{U'_{\text{out}}}{m}} - e^{k_2 U'_{\text{out}}}\right). \]
We obtain
\[ \delta = k_1 \cdot e^{k_2 U_{\text{out}}} \left[e^{k_2\left(\frac{U'_{\text{out}}}{m} - U_{\text{out}}\right)} - (1-\delta)\right]. \tag{1} \]
Transform the expression in the exponent in the brackets of equation (1):
\[ \frac{U'_{\text{out}}}{m} - U_{\text{out}} + \frac{U_{\text{out}}}{m} - \frac{U_{\text{out}}}{m} = U_{\text{out}}\left(\frac{1-\delta}{m} - 1\right). \]
Consequently, expression (1) can be written as
\[ \frac{\delta}{k_1 \cdot e^{k_2 U_{\text{out}}}} + 1 - \delta = e^{k_2 U_{\text{out}}\left(\frac{1-\delta}{m} - 1\right)} . \tag{2} \]
Taking the logarithm of expression (2),
\[ \ln \left[ \delta \left(\frac{1}{k_1 \cdot e^{k_2 U_{\text{out}}}} - 1\right) + 1 \right] = k_2 U_{\text{out}}\left(\frac{1-\delta}{m} - 1\right). \tag{3} \]
From expression (3) we find
\[ m = \frac{1-\delta} { \frac{ \ln \left[ \delta \left(\frac{1}{k_1 \cdot e^{k_2 U_{\text{out}}}} - 1\right) + 1 \right] }{ k_2 U_{\text{out}} } + 1 }. \tag{4} \]
On the basis of expression (4) we find the required value of the gain coefficient of the open loop so that the error \(\delta\) does not exceed the specified value:
\[ k_{\text{ol}} = \frac{k_2 U_{\text{out}}(1-\delta)} { \ln \left[ \delta \left(\frac{1}{k_1 \cdot e^{k_2 U_{\text{out}}}} - 1\right) + 1 \right] + \delta k_2 U_{\text{out}} }. \tag{5} \]
As a result of analyzing expression (5), it was found that, for reproducing the V–A characteristic of this system with an accuracy of no less than 1% in the nominal mode, it is possible to reduce the value of \(K_{\text{ol}}\) by approximately 50%.
Thus, in designing an electric-machine equivalent, it is necessary to take into account the special feature of this scheme, which greatly simplifies correction of the system and improves the quality indices of the transient process.
High requirements are imposed on the equivalent of autonomous power generators with respect to the stability of the characteristics [1]. In
During prolonged operation of the equivalent in the rated mode, heating of the EMU windings occurs, which changes the parameters of the system transfer function. The greatest influence on the stability of the equivalent characteristics is exerted by heating of the compensating winding, since, in order to form the reference voltage, a voltage proportional to the load current is used; this voltage is taken from the EMU compensating winding, amplified, and fed to the input of the function converter. Instability of the voltage at the input of the function converter introduces a significant error into the operation of the system, which is inadmissible under the operating conditions of the equivalent.
It was established experimentally that, after half an hour of operation of the electric-machine equivalent in a mode close to rated, the resistance of the compensating winding increases by approximately 8%. When the resistance of the compensating winding changes due to heating, the voltage at the input of the function converter changes by the amount
\[ \Delta U=\Delta R_{k0}\cdot I_n\cdot K_{oc}, \]
where
\[
\Delta R_{k0}
\]
—is the change in the resistance of the compensating winding upon heating,
\[
I_n
\]
—is the load current,
\[
K_{oc}
\]
—is the current-feedback amplification factor.
To eliminate the influence of heating of the EMU compensating winding on the stability of the equivalent characteristics, a special correction has been introduced into the system. The essence of this correction is that the voltage taken from the compensating winding is fed to a divider, one arm of which is a thermistor with a negative temperature coefficient of resistance. The thermistor is placed inside the EMU in the region of the compensating winding. The circuit for connecting the thermistor is shown in Fig. 2. According to the circuit, the increase in voltage due to heating of the compensating winding is compensated by a decrease in the total resistance \(R_c=R_t+R_2\), i.e., by decreasing the share of the voltage \(U_k\) taken from \(R_c\). The voltage at point \(A\) is equal to
Fig. 2. Circuit of temperature stabilization of the equivalent
\[ U_A=\frac{U_k R_c}{R_3+R_c}. \]
As a result of heating of the EMU compensating winding, the voltage at point \(A\) is determined by the following expression:
\[ U_{AT}=\frac{(U_k+\Delta U_k)(R_c-\Delta R_c)}{R_3+R_c-\Delta R_c}. \]
The voltage increment as a result of heating is equal to
\[ \Delta U_A=\frac{\Delta U_k(R_c-\Delta R_c)}{R_3+R_c-\Delta R_c} -\frac{U_k\Delta R_c R_3}{(R_3+R_c-\Delta R_c)(R_3+R_c)}. \]
It is necessary that \(\Delta U_A=0\); then
\[ \Delta U_{\mathrm{k}}(R_{\mathrm{c}}-\Delta R_{\mathrm{c}})R_{\mathrm{c}} = U_{\mathrm{k}}\Delta R_{\mathrm{c}}R_3 - \Delta U_{\mathrm{k}}(R_{\mathrm{c}}-\Delta R_{\mathrm{c}})R_3 . \tag{6} \]
From expression (6) we find the value of the resistance \(R_3\),
\[ R_3= \frac{\Delta R_{\mathrm{k}}(R_{\mathrm{c}}^2-\Delta R_{\mathrm{c}})} {R_{\mathrm{k}}\Delta R_{\mathrm{c}}-\Delta R_{\mathrm{k}}R_{\mathrm{c}}+\Delta R_{\mathrm{k}}\Delta R_{\mathrm{c}}}. \]
With a change in temperature, the resistance of the compensating winding changes linearly [3].
\[ R_{\mathrm{kt}}=R_{\mathrm{k}20}\left[1+\alpha(T-T_0)\right], \]
where
\(R_{\mathrm{kt}}\) is the resistance of the compensating winding at temperature \(T\),
\(R_{\mathrm{k}20}\) is the resistance of the compensating winding at temperature \(T_0=20^\circ\mathrm{C}\),
\(\alpha\) is the temperature coefficient of resistance of the compensating winding.
Knowing the resistance at temperature \(T_0\), we determine the value of \(R_{\mathrm{kt}}\) for the specified range of variation in the temperature of the compensating winding.
The temperature characteristic of the thermistor has the form [4]:
\[ R=R_0\cdot e^{\left(\frac{B}{T}-\frac{B}{T_0}\right)}, \tag{7} \]
where
\(B\) is a constant coefficient depending on the type of thermistor,
\(R_0\) is the resistance of the thermistor at temperature \(T_0\).
\(R\) is the resistance of the thermistor at temperature \(T\). From expression (7) it is evident that the temperature characteristic of the thermistor is nonlinear. Taking into account that the resistance of the compensating winding changes according to a linear law, the use of an uncorrected thermistor will lead to inadmissible deviations of the current–voltage characteristic of the equivalent. Correction of the temperature characteristic of the thermistor consists in deforming this characteristic by means of linear resistances (Fig. 2). The temperature characteristic of such a two-terminal network is best approximated to a linear one if, at the end points of the specified temperature range, it intersects the required linear characteristic, and at the midpoint it has a point of inflection. The latter condition is satisfied when the equality
\[ R_1+R_2=\frac{B-2T_{\mathrm{avg}}}{B+2T_{\mathrm{avg}}}\cdot R_{\mathrm{avg}}. \tag{8} \]
is fulfilled.
To calculate the two-terminal network we shall use condition (8) and the condition that the increment of the compensator be equal to the specified value
\[ R_{q_1}-R_{q_2}=\Delta R_q, \tag{9} \]
where
\[ R_{q_1}=\frac{R_{T_1}+R_2}{R_{T_1}+R_2+R_1}\cdot R_1 \]
is the resistance of the two-terminal network at temperature \(T_1\),
\[
R_{q_2}=\frac{R_{T_2}+R_2}{R_{T_2}+R_2+R_1}\cdot R_1
\]
— the resistance of the two-terminal network at temperature \(T_2\).
Solving equation (9) with respect to \(R_1\), we obtain
\[ R_1=\sqrt{\frac{\Delta R_q\left[R_{P_0}\cdot R_T+(R_{T_0}-R_T)(R_1+R_2)+(R_1+R_2)^2\right]}{R_{T_0}-R_T}}. \]
Thus, the proposed circuit makes it possible to significantly increase the static accuracy of an electric-machine equivalent of autonomous power generators.
REFERENCES
-
V. F. Trefilov, M. V. Lukyanenko, E. A. Mizrakh, V. S. Vlasov. Study of the possibility of creating a physical model of a generator with special characteristics. Research report. All-Union Scientific and Technical Information Center. State registration No. 69042563, inventory No. B034364, 1969.
-
M. V. Lukyanenko, V. F. Trefilov, E. A. Mizrakh, N. B. Kobelev. Method of physical modeling of the volt–ampere characteristics of autonomous power sources. Proceedings of the Second Regional Conference Dedicated to Radio Day. Krasnoyarsk, Institute of Physics, USSR Academy of Sciences, 1969.
-
E. S. Bleiz, Yu. N. Semenov, B. K. Chemodanov, N. M. Yakimenko. Dynamics of Electric-Machine Tracking Systems. Moscow, “Energiya,” 1967.
-
Yu. V. Zaitsev, A. N. Marchenko. Semiconductor Resistors in Radio Circuits. Moscow, “Energiya,” 1971.