Extensive simulations have been carried out to test the proposed solution. The tur- bine characteristics reported in 6.3.3 have been considered, moreover, the following non- idealities and bounds have been added to account for a realistic pitch actuator: a first order dynamics between pitch command and actual pitch position with a time constant
τ = 20ms, a slew rate limitation at±10◦/s and a limited pitch angle range, β ∈ [0, 60]◦.
Finally, in order to prove the proposed solution effectiveness for actual wind energy conver- sion systems, a discrete-time implementation of the controller has been carried out; taking
into account the common performances of turbine controllers, a sampling time Ts = 4ms
6.5. Simulation results 0 100 200 300 400 0 15 30 45
Actuated pitch angle 0 100 200 300 400 0 2 4 6 8 10
Reference angular speed
0 100 200 300 400 0
1.5 3.5
5x 10
4 Generator torque and bound (red)
0 100 200 300 400 −0.2
−0.10 0.1 0.2
Angular speed error
0 100 200 300 400
0 15 30 45
Mean wind speed
V [m / s] ω ∗ [r a d / s] ˜ω [r a d / s] TG , TG s a t [N m ] β [d e g ] time [s]
(a) Controller performance un- der abrupt wind speed varia- tions. 25 50 75 100 125 150 0 1 2 3
Actuated pitch angle 25 50 75 100 125 150 0
2 4 6
8 Reference angular speed
25 50 75 100 125 150 0
1.5 3.5
5x 10
4Generator torque and bound (red) 25 50 75 100 125 150 −0.05
−0.0250 0.0250.05
Angular speed error 25 50 75 100 125 150 0
10 20 30
Mean wind speed
V [m / s] ω ∗ [r a d / s] ˜ω [r a d / s] TG , TG s a t [N m ] β [d e g ] time [s]
(b) Controller behavior under wind step producing generator torque saturation. 0 1000 2000 3000 4000 0 5 10 15
Actuated pitch angle 0 1000 2000 3000 4000 0
4 8
12Reference and optimal angular speed (red)
0 1000 2000 3000 4000 0
1.5 3 5x 10
4Generator torque and bound (red)
0 1000 2000 3000 4000 0 1 2 3x 10 5 Generator Powerr 0 1000 2000 3000 4000 0 5 10 15 20 Wind speed V [m / s] ω ∗, ωo pt [r a d / s] PG [W ] TG , TG s a t [N m ] β [d e g ] time [s] (c) MPPT algorithm combined with discrete time controller per- formance.
In Fig. 6.4(a) up and down high wind steps have been reproduced, in order to test the stability domain of the proposed solution, it’s further to notice that pejorative conditions with respect to those considered for stability analysis in 6.3.2 has been considered; a slowly varying reference speed covering the entire nominal range has been adopted, and a wind
step from 10 to 30 m/s, causing an initial error ˜χ(tstep) > ˜χmax has been produced. The
reference tracking is ensured even when an abrupt wind speed increase occurs and the power limit is reached, causing the generator torque to drop, hence it can be reasonably assumed that the stability properties of the proposed solution go beyond the basin of at- traction theoretically estimated in 6.3.2.
In Fig. 6.4(b) the benefits produced by taking into account system thermal dynamics, mentioned in 6.2.2, are clearly shown. Starting from a wind speed not requiring any torque-power limitation, a wind step, such that the torque saturation limit is exceeded, is produced. It can be noted how the generator torque reaches higher values than the
maximum RMS value TGRM Smax for few seconds, then, when the torque dynamic bound
decreases due to receding horizon thermal constraints, the proposed controller starts pitch angle variation with a quite smooth trajectory, according to time-varying torque bound. Finally, Fig. 6.4(c) reports the results obtained integrating the MPPT algorithm sketched in 6.4, with the discrete-time controller. In order to test the reliability of the MPPT algorithm, realistic turbulent wind speed conditions have been reproduced, by adding a stochastic component, generated according to the widely accepted Von Karman spectrum
representation ([38]), to the wind speed mean value. It can be noted how the optimal an-
gular speed ωopt= V λRopt is tracked quite accurately by the perturb and observe algorithm
when the wind speed is lower then the rated value, then, when power saturation, caused by the wind step at time 2000s, occurs, the angular speed is steered to the constant value ω = 7.6 rad/s to reduce generator power losses, while the torque-pitch coordinated action limits the captured power. During the last part of the simulation the wind speed drops below the rated limit and the MPPT algorithm is restarted to track the new optimal power value.
Part III
Adaptive Nonlinear Estimation of
Power Electronic and
Electromechanical Systems
Parameters
Polar Coordinates Observer for
Robust line grid parameters
estimation under unbalanced
conditions
In this chapter adaptive observers are designed to cope with the problem of es- timating amplitude, phase and frequency of the main component of three-phase line voltage, under unbalanced conditions. Different solutions, corresponding to particular reference frame selections are discussed, the convergence properties are formally proved, and a careful sensitivity analysis w.r.t harmonic distortion and the so-called negative sequence voltage components, generated by voltage un- balancing, is carried out. In this respect is is showed how a nonlinear adaptive solution obtained exploiting a synchronous coordinates set can improve robustness to unbalancing with respect to traditional solutions
7.1
Problem statement
Accurate three-phase line voltage information is required for high performance control of power electronic applications, in particularly the correct reconstruction of phase-angle is of utmost importance for control purpose. An example of such applications has already been discussed in ch. 2 for what concerns Shunt Active Filter;indeed, both the unconstrained current controller and the proposed anti-windup scheme have been designed exploiting a suitable coordinates transformation, which bring the system into a synchronous reference frame aligned with the line voltage vector.
In order to perform such change of coordinates, accurate informations on the line phase- angle and frequency need to be available (see 2.7). The same considerations apply for other power conditioning equipment such as statcom VAR compensator or Uninterrupt- able Power Supplies (UPS). Obviously phase-angle is not available for measurement and
7.1. Problem statement
need to be reconstructed by elaborating the phase voltage signals. The estimation method should be able to accommodate frequency fluctuations, not so uncommon in industrial en- vironments, and could be expected to be even more significant in the next generation more complex and smart grid networks. Furthermore the estimate has to be robust to source voltage disturbances. Beside harmonics distortion, a typical grid condition to deal
with in power applications is voltage unbalance ([121]); this situation occurs when several
single-phase loads are connected to a distribution system, the fluctuating power required by each of these loads can produce unbalance in power system, moreover, if a voltage sag takes place in one or two phases of a three-phase power system, it produces a temporary
unbalancement [122].
Representing the three phase system with the method of the symmetrical components, it is possible to show that voltage unbalance generates voltage terms rotating with oppo- site phase respect to the mains voltage, for this reason they are usually called negative
sequence or counter rotating components [123]. Even though the European regulation
limits the amount of supply voltage sags ([124]), and several countries introduced specific
power quality regulation ([125], [126]), the sensitivity to negative sequence disturbances
has to be considered in order to accomplish the estimation accuracy needed in most of the applications.
Various solutions are commonly employed for phase-angle estimation; Phase Locked Loop (PLL) based solutions are broadly adopted, however, from a control theory standpoint,
also several estimation algorithms have been proposed. In [127] a least squares estimation
algorithm is presented, while in [128] a sensorless estimation algorithm for a specific PWM
rectifier is proposed.
Here nonlinear adaptive observers of the line voltage main component are considered; as mentioned, beside adaptation with respect to modifications of line frequency (slowly- varying variations of few percents around the nominal value are admissible), the key estimation objectives is to ensure high selectiveness with respect to line voltage harmonics and, at the same time, robustness to negative sequence at line frequency. Two different
solutions, first proposed in [43], are presented. The first one is straightforwardly derived
by LTI model of the line voltage in the so-called “stationary reference-frame”, adding a suitable adaptation law for the line frequency. By means of mathematical analysis and simulations, it is shown that, adopting pseudo-linear techniques, it is hard to achieve at the same time robustness to negative sequence and good selective behavior with respect to. harmonics. The second solutions exploits nonlinear line voltage model expressed in a generic “synchronous reference frame” which is enforced to be aligned to the actual main voltage vector by means of suitable adaptation laws. It will then be showed how, in this case, an easy tuning can be performed in order to guarantee both high selectiveness and robustness to negative sequence.
The chapter is organized as follows. In Section 7.2 the line voltage models are recalled, adding the negative sequence representing unbalanced conditions, and the observer objec- tive are defined. In Section 7.3 the first adaptive solution, based on stationary reference-
frame representation is presented along with a detailed analysis on its selectiveness and robustness with respect to the above defined voltage disturbances. In Section 7.4 the nonlinear adaptive solution, referred to synchronous reference frame representation of line voltage, is presented with design and stability analysis details. Also in this case the ro- bustness and selectiveness properties are carefully discussed via analysis and simulations.