DC-Bus Voltage Control

A PI controller for the DC-bus voltage is implemented in the motulator.grid.control.DCBusVoltageController class, whose base class is motulator.common.control.PIController. In the following, the tuning of the DC-bus voltage controller is considered in the continuous-time domain.

Controller Structure

In order to have a linear closed-loop system, it is convenient to use the DC-bus energy as the controlled variable, [Hur et al., 2001]

(1)\[\hatWdc = \frac{1}{2}\hatCdc \udc^2 \qquad \Wdcref = \frac{1}{2}\hatCdc (\udcref)^2\]

where \(\hatCdc\) is the DC-bus capacitance estimate, \(\udc\) is the measured DC-bus voltage, and \(\udcref\) is the DC-bus voltage reference. Using these variables, the output power reference \(p_\mathrm{c}^\mathrm{ref}\) for the converter is obtained from a PI controller,

(2)\[p_\mathrm{c}^\mathrm{ref} = -\kp (\Wdcref - \hatWdc) - \int \ki (\Wdcref - \hatWdc) \D t\]

where \(\kp\) is the proportional gain and \(\ki\) is the integral gain. The negative signs appear since the positive converter output power decreases the DC-bus capacitor energy, see (3).

Closed-Loop System Analysis

For simplicity, the capacitance estimate being accurate, \(\hatCdc = \Cdc\). Furthermore, the converter is assumed to be lossless and its output power control ideal. The DC-bus energy balance is

(3)\[\frac{\D \Wdc}{\D t} = p_\mathrm{dc} - p_\mathrm{c}\]

where \(p_\mathrm{dc}\) is the external power fed to the DC bus, \(p_\mathrm{c}\) is the converter output power, and \(\Wdc = (\Cdc/2) \udc^2\) is the energy stored in the DC-bus capacitor. The power \(p_\mathrm{dc}\) is considered as an unknown disturbance.

In the Laplace domain, the closed-loop system resulting from (2) and (3) is given by

(4)\[\Wdc(s) = \frac{\kp s + \ki}{s^2 + \kp s + \ki} \Wdcref(s) + \frac{s}{s^2 + \kp s + \ki} p_\mathrm{dc}(s)\]

where it can be seen that \(\Wdc = \Wdcref\) holds in the steady state. Furthermore, it can be shown that also \(\udc = \udcref\) holds in the steady state (independently of the errors in the capacitance estimate \(\hat{C}\), since both reference and measured values in (1) are scaled by the same estimate).

Gain Selection

Based on (4), the gain selection

(5)\[\kp = 2\alphadc \qquad \ki = \alphadc^2\]

results in the double real pole at \(s = -\alphadc\). The closed-loop bandwidth is approximately \(\alphadc\).