ON OPTIMAL CONTROL OF SYSTEMS WITH MULTIPHASE PULSE-WIDTH MODULATION

V. A. BEINAROVICH, A. I. SAPOZHNIKOV

(Presented by the scientific-technical seminar of the Research Institute of Automation and Electromechanics at TPI)

When several identical loads controlled by pulse-width modulation (PWM) according to a law identical for all loads are supplied from a common DC source of limited power, the question arises of choosing a rational circuit for the power section of the pulse modulator and determining an optimal control law that ensures minimum energy losses. Such a problem arises, for example, in the development of a pulsed system for rheostatless control of a two-motor traction drive of mine storage-battery electric locomotives.

At present, in the construction of the indicated systems there are two tendencies:

1) the use of single-phase power PWM systems, in which there is one power switch feeding simultaneously all loads connected in parallel;

2) the use of multiphase PWM systems, in which there are several power switches, each of which feeds one load, while the circuits consisting of a power switch and a load are connected in parallel to the source, usually have identical parameters, and are switched sequentially with a time shift.

Let us carry out a comparative analysis of losses in the storage-battery power source in a circuit with several identical loads fed through individual power switches, with the aim of determining the optimal control law that ensures minimum losses in the storage battery under the following assumptions:

1) the loads are represented by identical active-inductive circuits with constant parameters \(R, L\) and a constant back emf \(E\);

2) the power switches and shunting diodes possess ideal properties—the switching occurs instantaneously, the resistance in the on state is zero, and in the off state is infinity;

3) the storage battery is represented by a source of constant emf \(E_u\) and with unchanged internal resistance \(R_u\);

4) the current consumed from the source by each load has a rectangular form;

5) losses from currents associated with artificial commutation processes are neglected.

Taking the above into account, the equivalent circuit of the system has the form shown in Fig. 1.

With multiphase pulse-width modulation with period \(T\),

each of the power switches can be turned on with a time shift by a fraction of the period \(\lambda T\), and the relative shift of the turn-on instant of one switch with respect to another, \(\lambda\), may take values from 0 to 1. The pulse duration \(\gamma T\) is determined by the pulse duty factor \(\gamma\), which in the regulation process also varies from 0 to 1.

Fig. 1.

In the absence of a phase shift \(\lambda=0\) (or \(\lambda=1\)), with \(\gamma=1\) and any number \(n\), the mode of single-phase PWM is obtained, when all power switches operate synchronously. In this case the current drawn from the source is intermittent and, over the pulse interval, is equal to the sum of the currents of all loads. The power of the energy losses in the power supply in this case, for any value of \(n\), will have the maximum value

\[ \Delta P_0=(nI)^2 R_u \gamma, \tag{1} \]

where

\(n\) is the number of loads;
\(I\) is the current of one load.

With a phase shift \(\lambda \ne 0\), with \(\gamma \ne 1\) and the number of loads \(n>1\), a mode of multiphase PWM is obtained, when the power switches operate with a phase shift in a definite sequence. Depending on the ratios of the quantities \(\lambda\), \(\gamma\), and the number of loads (or power switches) \(n\), it is possible to obtain various modes of current consumption from the power supply over the switching-period interval: intermittent current with a constant or stepwise varying pulse amplitude, continuous current with a stepwise character of variation of the magnitude in time, or constant current.

Let us consider the most common practical case of supplying two loads, \(n=2\) (for example, two-motor drives). With a phase shift \(0<\lambda<1\) and variation \(0\le \gamma \le 1\), such systems yield two-phase PWM, and the following modes of current consumption from the source are possible.

The mode of intermittent current with constant amplitude over the pulse intervals, equal to the current of one load, arises under the condition that simultaneous supply of the two loads is absent,

\[ \begin{cases} \gamma \le \lambda,\\ \gamma \le 1-\lambda. \end{cases} \tag{2} \]

The boundary of this mode corresponds to tangency of the current pulses of the two loads and is determined by the condition

\[ \begin{cases} \gamma=\lambda_1,\\ \gamma=\lambda_2. \end{cases} \tag{3} \]

The power of the losses in the power supply in the mode of intermittent current with constant pulse amplitude is determined by the following expression:

\[ \Delta P_{2}=2I^{2}R_{\mathrm{u}}\gamma . \tag{4} \]

The continuous-current mode arises in the case of overlap and continuous contact of the current pulses consumed from the source (Fig. 2), which corresponds to the condition

\[ \begin{cases} \gamma \geq \lambda,\\ \gamma \geq 1-\lambda . \end{cases} \tag{5} \]

Fig. 2.

The boundary of this mode is also determined by expression (3), since it takes into account the condition of contact of the current pulses, i.e., the condition for the transition of the current from intermittent to continuous.

The power loss in the power source in the continuous-current mode is determined by the expression

\[ \Delta P_{2}=2I^{2}R_{\mathrm{u}}(3\gamma-1). \tag{6} \]

For all other ratios of the quantities \(\gamma\) and \(\lambda\) that do not satisfy conditions (2) and (5), the current consumed from the source will be intermittent, with a stepwise change in the magnitude of the current over the pulse intervals.

The power loss in the power source in the intermittent-current mode with a stepwise varying pulse shape is determined, depending on the ratios, by the following expressions:

\[ \text{for}\qquad \lambda \leq \gamma \leq 1-\lambda \]

\[ \Delta P_{3}=2I^{2}R_{\mathrm{u}}(2\gamma-\lambda) \tag{7} \]

\[ \text{for}\qquad \lambda \geq \gamma \geq 1-\lambda \]

\[ \Delta P_{4}=2I^{2}R_{\mathrm{u}}(2\gamma+\lambda-1). \tag{8} \]

For convenience and clarity in determining the mode of current consumption from the power source as a function of the values of \(\gamma\) and \(\lambda\), for a system with two loads, Fig. 3 presents a diagram of the modes, obtained from the boundary conditions (2) and (5). The boundaries of the modes are represented in the plane of the parameters \(\gamma\) and \(\lambda\) by two straight lines intersecting at the point \(\gamma=0.5\), \(\lambda=0.5\). The entire plane of possible values of the parameters \(\gamma\) and \(\lambda\) is divided into four regions corresponding to the following modes of current consumption from the source:

Fig. 3

region \(A\)—the intermittent-current mode with constant pulse amplitude;

region \(B\)—the intermittent-current mode with a stepwise varying pulse shape;

\(C\)—the continuous-current mode with a stepwise variation of its magnitude.

The expressions obtained make it possible to determine quantitatively the dependence of the power losses in the source on the magnitude of the phase shift \(\lambda\) for different values of \(\gamma\). In Fig. 4 a diagram presents the dependence of the ratio of the power losses for two-phase PWM to those for single-phase PWM on the magnitude \(\lambda\) for various constant values of \(\gamma\), from which it is seen that the optimal value of the phase shift for two loads is equal to \(0.5\).

Fig. 4

Carrying out an analogous analysis for pulse systems with a number of loads greater than two, one can determine the optimal values of the phase ...

shift for each particular number of loads \(n\), and expressions for the loss power. Further, from the particular expressions obtained, by induction one can obtain general conditions for the minimum losses in the power supply. The loss power, for any constant values of \(\gamma\) and \(n\), gradually decreases as \(\lambda\) increases from 0 to

\[ \frac{1}{n}, \]

reaches a minimum at \(\lambda = \frac{1}{n}\), and then increases again as \(\lambda\) further increases from \(\frac{1}{n}\) to 1. As a result, it turns out that the optimal phase shift, which ensures a minimum of losses in the source, is equal to the reciprocal of the number of loads,

\[ \lambda_{\mathrm{opt}} = \frac{1}{n}; \tag{9} \]

and the loss power then has the minimum value

\[ \Delta P_{\mathrm{opt}} = I^2 R_{\mathrm{u}}\,[\,n\gamma(1 + 2k) - k(k + 1)\,], \tag{10} \]

where \(k\) is the integer part of the ratio of \(\gamma\) to \(\lambda\), equal to \(0,1,2,\ldots,(n-1)\).

As the number of loads is increased, the loss power decreases. The theoretically attainable minimum value of the loss power in the source, when the condition of the optimal equation (9) is satisfied and \(n \to \infty\), is determined by the expression

\[ \Delta P_{\mathrm{opt}} = I^2 R_{\mathrm{u}}(n\gamma)^2 . \tag{11} \]

Figure 5 presents the dependences of the loss power, in relative units, on the magnitude \(\gamma\) and the number of loads \(n\). The curve of the minimum attainable loss power for \(n \to \infty\) is shown by a dotted line.

Fig. 5.

From consideration of Fig. 5 it follows that the most expedient systems with pulse-width modulation may be considered those containing two to four switches. A further increase in the number of PWM phases will significantly complicate the system but will not yield appreciable energy savings through reduction of losses in the power supply.