Nonlinear Compensation of Solar Array Simulators with Dual Power Regulation

During the tests of the spacecraft electrical systems there is a need for simulators of individual parts of the spacecrafts, in particular, solar array simulators (SAS). One of the topologies of medium and high power SAS simulators has dual control of consumed power and contains series or parallel connected linear and switching regulators. This topology allows to provide wide bandwidth and high efficiency, but the range of the resistance change of periodically switched SAS load is limited to the value of the stabilized attribute. Nonlinear compensator (NC) allows to reduce the average feedback voltage of the switching regulator in case of periodic load switching, which, in turn, allows to increase the average value of the stabilized attribute. The describing function method provides a mathematical description of the NC electrical circuit, which allows to select parameters of NC that eliminate the excitation of self-oscillation based on the SAS load switching frequency range and to study the switching regulator stability.


Introduction
Validation of electrical systems characteristics of models and experimental prototypes of the spacecraft's requires performing the special electrical tests. During these tests, as power supply sources is used devices that reproduce (with a specified error) the desired static and dynamic characteristics of the real primary power sources, in particular, solar arrays. Such devices are calling solar array simulators (SAS) [1][2][3][4][5] The quality of ground-based tests is determined by the properties of used simulators, namely:  accuracy of reproduction of static (current-voltage characteristic) and dynamic (internal impedance) characteristics of the actual solar array;  performance properties of simulator, such as the possibility of uninterrupted operation and the possibility of durable operation in maximum loading conditions. The most promising topology of medium and high power SAS at the moment is topology with dual control of consumed power which contains series or parallel connected linear and switching regulators [6].
A linear regulator (LR) in the SAS may be connected in series (Fig. 1a) or in parallel ( Fig.  1b) with the load. The LR provides the desired characteristics of a real solar cell (currentvoltage characteristics and impedance) and a switching regulator (SR) limits the power that dissipate the LR by stabilizing the LR voltage (series connection) or by stabilizing the LR current (parallel connection).  This topology combines the advantages of simulators containing only one regulator. Simulators based on individual linear regulator allow to achieve high performance rates and to ensure the accuracy of desired dynamic characteristics. Simulators based on individual switching regulator allow to achieve high values of efficiency and good weight/size ratio.
However, simulators with dual power control has problems with a broadband load switching. Since performance of the SR is considerably slower than performance of the LR, during the process of load switching from open-curcuit mode to short-curcuit mode there is a possibility that the LR will stop operate in the active region, which will lead to deterioration of the quality of electricity at a load.
In [7] was proposed the method of improving the quality of SAS operation by introducing a nonlinear compensator (NC) in the SR feedback loop. The NC belongs to the class of dynamic compensators, since the effect of its introducing becomes significant only during transients. During steady state (constant load resistance) stabilized LR attribute remains unchanged.
The purpose of the NC introduction in the feedback loop of SR -reducing the average SR feedback voltage in the case of a periodic load switching, which, in turn, will lead to an increase in value of the stabilized attribute:  voltage across the LR -in series type simulators;  current through the LR -in parallel type simulators.
Increasing the value of the stabilized attribute allows the LR to remain in active mode for a wider range of periodically switched resistance of SAS.
The NC may be made as a RC-circuit whose resistance R is shunted by diode VD (figure 2).
In this circuit VD RR , so the discharge time constant d T is much smaller than the capacitor charge time constant c T .Consequently, the average voltage across the capacitor decreases with a periodic load switching, which increases the average value of the LR stabilized attribute and increase switching load range, wherein the LR remains is in active mode. Thus, the introduction of the NC improves performance properties of SAS.
2. Mathematical description of solar array simulator with the nonlinear compensator One of the common methods for the study of nonlinear systems is the describing function method [8][9][10]. This method allows to study the stability of the nonlinear system described by differential equations of any order.
The nonlinear system represented as a feedback connected linear part of whole nonlinear system and a nonlinearity. Assume that the signal at the input of a nonlinear system () (2) The transfer function (2) corresponds to the absolute value Switching angle 1  (figure 4) can be found numerically from the equation: From the expressions (5)  (1 ( ) cos( )) 1 2  ( ) 1 ( ) sin( ) sin (2 ) (1 ( ) cos( )) 44 The expression (10) shows that the constant component *  T it necessary to check the stability of the SR control system for self-oscillations. Since the bandwidth of the SR is much smaller than the pulse modulation frequency, on the basis of Nyquist-Shannon sampling theorem we will consider the SR as a continuous system.
According to the describing function method [8] characteristic equation is written as negative reciprocal of a linear system transfer function () For example, we use the following form of the SR linear system transfer function consisting of a second-order filter and compensator: where 1 0.34s T  , 5 2 7.88 10 s T  ,   The figures show that the introduction of the NC is quite effective: in the same load resistance switching range a minimum voltage at the LR is greater than zero, so the LR operates in active mode, and the load voltage L v has the required form.

Conclusion
The paper studied the impact of parameters of the nonlinear compensator (NC) on the stability of medium and high power solar array simulator (SAS) control system by the describing function method. The parameters of the NC, which preclude the self-oscillations, were defined. The following results were obtained:  the introduction of the NC in the feedback loop of the switching regulator allows to extend the range of the periodically switched resistance of the SAS;  using the describing function method a mathematical description of the proposed NC circuit was developed, this description allows to select the parameters of NC based on load switching frequency range of the SAS;  the describing functions of the NC allow to explore the impact of the dynamic properties of the NC on the switching regulator stability. The simulation shows that the introduction of the NC is quite effective: in the same load resistance switching range a minimum voltage at the LR is greater than zero, so the LR operates in active mode, and the load voltage has the required form.