# Category Photovoltaic Sources

## Complete State Feedback (Pole Placement Technique)

As it is known, the poles position is related to the transient response of the system, then, by the knowledge of the time-domain characteristic response, it is possible to determine the system poles which have caused that response.

The complete state feedback technique is based on the sensing of all the state variables of the system that are multiplied for a suitable gain; it allows the poles to be assigned. This last characteristic is of particular importance for system emu­lation. As a matter of fact, a correct system emulation requires the matching of transient response of the emulator with that of the system to be emulated. In the case of PV emulation, this matching can be achieved by imposing the closed-loop poles to the DC/DC converter used for the source emulation.

A continuous-tim...

## PID Compensation

The PID compensation is obtained by adding three signals: the first one is the input signal multiplied for a gain Kp, the second is the integral of the input signal multiplied for a gain Kt, and the last is the derivative of the input signal multiplied for a gain Kd.

A simple block diagram of a PID implemented in Simulink® is given in Fig. 8.9.

The corresponding transfer function has a pole at the origin and two zeros.

The corresponding transfer function is:

To obtain a suitable crossover frequency, two high frequency poles can be added. In this case, Eq. (8.13) becomes:

Vc = K (s + z1)(s + z2)

Sin s(s + P1)(s + p2)

Figure 8.10 shows the bode diagram of both a PID compensator with Kd = 3...

## PI Compensation

The PI compensation is obtained by adding the input signal, multiplied for a gain Kp, to an other signal obtained by the integration of the input signal multiplied for a gain Kj. A simple block diagram of a PI implemented in Simulink® imple­mentation is given in Fig. 8.6.

The corresponding transfer function has a pole at the origin and a zero whose value is given by z = K/Kp.

It can be noted that Eq. (8.11) exhibits a constant gain for high frequencies. On the other hand, in the case of a DC/DC converter, it is mandatory that, near to the switching frequency, the open-loop gain is sufficiently low to reduce the ripple at the switching frequency. In this case, and additional low pass filter is added to the PI as shown in Fig. 8.7...

## Compensation Networks

The aim of a compensation network is to assure the stability of the feedback system by a suitable gain and phase margin. Moreover, a correct design allows the closed-loop system poles to give the desired transient time-domain response.

A great part of compensating networks are based on a linear combination of the error signal of its integral and of its derivative. Among them, the proportional- integrative (PI) network and the proportional-integrative-derivative (PID) network are commonly used.

The design of a compensation network comes from a trade-off among several constraints. The three main system characteristics are:

1. Steady-state performance: they are usually expressed in terms of the maximum error for a given input (step or ramp) and are improved by the presence of integrators.

2...

## Feedback Network Transfer Function

The feedback networks can be formed by a voltage transducer or simply by a voltage divider (see Sect. 7.2.1); their aim is to manage a reduced signal compared to the output value. In the case of a voltage divider, the transfer function is given by:

(8.9)

8.2.1 Pulse Width Modulator Transfer Function

The pulse width modulation (PWM) provides the duty cycle D obtained, for example, by comparing a sawtooth signal with the control voltage Vc. This operating mode is called voltage mode control.

The duty cycle ranges from zero to unitary value. The sawtooth frequency corresponds to the DC/DC converter switching frequency. When the control voltage equals the maximum value of the sawtooth, the duty cycle approaches 1. The modulator gain is given by:

< 1 V > Vsw

Gpwm = % V~ 0 > Vc > VSw...

## Stability Analysis

The loss of stability for a system, defined by Eq. (8.1), can be described by the following conditions:

/ lbA| = 1 (84,

phase(b) + phase(A) = 180° ( : )

In this case, the denominator of Eq. (8.1) has a zero that lies in the imaginary axis. From a physical point of view, if the conditions expressed by Eqs. (8.4) occur, the input signal of the feedback system goes through the loop formed by the block described by transfer function A, the feedback network b, then it is rotated of 180° by the minus sign and it is superimposed to the original signal with the same amplitude and phase. A new signal with an amplitude twice of the original is obtained and so on, until a nonlinearity occurs...

## Closed-Loop Gain

Figure 8.1 shows a block diagram of a closed-loop system. The original system has a transfer function A; a feedback network with transfer function b is added.

The input of the closed-loop system is the desired output value Vref; the obtained output Vo is processed by the feedback network, then the difference with the reference value, that represents the error signal, is the input of the system to be controlled.

M. C. Di Piazza and G. Vitale, Photovoltaic Sources, Green Energy and Technology, DOI: 10.1007/978-1-4471-4378-9_8, © Springer-Verlag London 2013

Fig. 8.1 Block diagram representation of a closed – loop system

The closed-loop transfer function is given by:

 Vo A (8.1) Vef 1 + bA The open loop transfer function is defined as: L = bA (8.2)

Some considera...