Digital Current Control of a Voltage Source Converter With Active ...

WU AND LEHN: DIGITAL CURRENT CONTROL OF A VOLTAGE SOURCE CONVERTER 1367
where
Simple pole assignment may be used to determine the input
Fig. 5. Separation of a complex system into d- and q-axis components.
(24)
so that the closed-loop system has the desired state response.
The actual gains and may then be found using (20) and
(21), respectively.
C. Optimal Pole Assignment
In this study, linear quadratic (LQ) optimal control [10] is
used to determine the gains and , since optimal control
frees the designer from the mathematical abstraction of pole
assignment. Furthermore, LQ controllers offer good stability
margins and robustness. Analyses of stability margins for discrete-time
LQ regulators are given in [12] and [13].
The LQ regulator is designed by solving the algebraic Riccati
equation to determine a gain which minimizes the quadratic
cost function
(25)
and determine the quadratic cost weightings on the state
values and inputs, respectively. The th diagonal element of
determines the direct weighting on the th state, while off-diagonal
elements have more complicated effects and are set to zero
in this study.
Some empirical guidelines for their selection can be given.
• Heavy weighting should be placed on to ensure fast,
well-damped transient current response.
• The impedance of the capacitor is fairly high and little current
is shunted through ; thus and will behave similarly,
and an equal cost should also be placed on .
• should be allowed to move as dictated by the grid in
order to avoid converter saturation. Low or zero weighting
may be placed on this state.
• Weighting on should initially be close to or at zero.
While an increase in this weighting will limit the amount
of control effort used, it will also decrease damping and
stability. For the same reason, input weighting should
initially be set to one.
• A fairly high weighting should be placed on the integrator
state , in order to attain high bandwidth. However, this
increase comes at the expense of stability.
The transient response of the converter terminal voltage
should be verified during a 1 per-unit step in , to ensure that
it lies within the available converter terminal voltage, including
the voltage required to reject the bus voltage . may be
limited by increasing either the weighting on or the input
weighting .
D. Separation of and Axes
To implement and simulate the system, the controller and
model must be decomposed into real-valued d and -axis systems.
The result of a multiplication between vector and
matrix can be separated into and components using the
matrix equation
(26)
This process is shown graphically in Fig. 5. Equation (26) can
be applied to every gain in Fig. 3 to find real-valued, coupled
- and -axis controllers, and to the matrices of the closed-loop
state model for simulation and numerical analysis.
IV. SIMULATION RESULTS
To verify the controller’s performance, an LCL-filter was designed
following the guidelines presented in [14]. The sampling
and switching frequencies were 8 kHz and 4 kHz, respectively,
and the filter parameters were 1.0 mH 0.04 p.u.),
0.5 mH 0.02 p.u. , 14 F 20 p.u. ,
with a resonant frequency of 2.3 kHz. The 60 Hz, 1.4 kVA test
system had a rated voltage of 115 V (line–line rms).
Following the guidelines of Section III-C, the following LQ
weightings were chosen:
The per-unitization of and with base voltage 115 V
and current 7 A can be done by writing and in terms
of their per-unit values and
(27)
(28)