2.3 Modelling the Electrical System
2.3.4 Grid Integration
As was shown in section 2.3.3, the DQ currents are controlled by manipulating the DQ controlling voltages vd(k) and vq(k). In a real life system, the DQ controlling voltages are created using pulsed signals that are produced using SVM, where the DQ voltages are limited toñv2d(k) + v2q(k) ≤ √23VDC(t). The SVM produces three phase controlling voltages vabc(t) that are directly dependent on VDC(t). Without a reliable constant VDC(t), the control of the LPMG can become difficult.
Controlling the VDC(t) voltage to be constant using the grid side converter, decouples the generation from the grid side (Lakshmi & Hemamalini 2016). As shown in Fig.
2.29, the grid and DC-link are represented by the following DQ domain continuous
Pgen(t)
VDC(t) Pgrid(t)
PDC(t)
DC
Link Grid side Inverter
Wave farm
network Grid
Machine side Inverter
vabc,g(t) v'abc,g(t)
PCC Q(t) 0 v'q,g(t) 0
Figure 2.29: A schematic of the overall system from wave to wire; starting from the WEC connected to the LPMG, which is controlled by the voltages generated from the machine side inverter that is limited by the voltage across the DC-link. The DC-link is controlled by the grid side inverter that interacts with (in this case) an ideal grid.
equation,
Pgrid(t) = Pgen(t) − PDC(t) (2.123) where
Pgrid(t) = vdg(t)idg(t) + vqg(t)iqg(t) Pgen(t) is the power generated from the WEC,
PDC(t) = CdcVDC(t)dVDC
dt , yielding (2.124),
CdcVDC(t)dVDC
dt = Pgen(t) −1vdg(t)idg(t) + vqg(t)iqg(t)2. (2.124)
2. WAVE-TO-WIREMODELLING OFGRID
CONNECTEDPOINTABSORBERS 2.3 Modelling the Electrical System
The active and reactive power flows at the point of common coupling (PCC) on the grid side are represented as (2.125) and (2.126),
P (t) = vd′gidg(t) + vq′giqg(t) (2.125) Q(t) = v′dgiqg(t) − v′qgidg(t), (2.126) where vd′g and vq′g are the DQ voltage at the PCC which are assumed constant. With the PLL connected to the PCC, as shown in Fig. 2.29, it becomes possible to align v′qg = 0 (Vittal & Ayyanar 2012). With vq′g = 0, the active and reactive powers are independently controlled by the D and Q grid side converter currents respectively, as shown in (2.127) and (2.128),
P (t) = vd′gidg(t) (2.127) Q(t) = vd′giqg(t). (2.128) The current dynamics of the grid-side converter are modelled by (2.129) and (2.130),
Lw
didg
dt = −Rwidg(t) + Lwω(t)iqg(t) + vdg(t) − vd′g (2.129) Lw
diqg
dt = −Rwiqg(t) − Lwω(t)idg(t) + vqg(t) − v′qg (2.130) where ω is the electrical speed (in this case ω = 314.159 rad/s because the grid fre-quency is 50 Hz); vd′g(t) and vq′g(t) are the voltages at the PCC and Rw and Lw are the values that respectively represent the impedance between the grid side converter and the PCC (Perera et al. 2013). Controlling iqg(t) = 0 so that Q(t) = 0 and assum-ing vq′g = 0, (2.124), (2.129) and (2.130) are combined, yielding (2.131), which is a non-linear system,
dVDC
dt = 1
CdcVDC(t)
C
Pgen(t) − Lwidg(t)didg
dt + Rwi2dg(t) + v′didg(t)
D
. (2.131) Linearising (2.131) yields (2.132),
VDC˜ (s) = Kgen
s P˜gen(s) +Kid(1 + sTid)
s I˜dg(s). (2.132) The DC-link voltage VDC(t) is controlled using a cascade controller as shown in Fig.
2.30. The outer VDC(t) controller sends i∗dg(t) reference signals to the inner loop of the cascade controller which controls the idg(t) current at a much higher bandwidth than the outer control loop. In respect to the outer control loop, the inner idg(t) controller
Electrical Power Optimisation of
Grid-connected Wave Energy Converters using Economic Predictive Control
67 Adrian C.M. O’Sullivan
loop is estimated as a first order transfer function (2.133) and the outer control loop utilises a PI controller (2.134).
I˜dg(s)
I˜d∗g(s) = 1
1 + sτ (2.133)
I˜d∗g(s)
E(s)˜ = Kp(s + KT)
s . (2.134)
Kid(sTid+1) s
Kgen
s Kp(s+KT)
s
Idg(s) 1
~
*Idg(s)
~
P
~
gen(s)E(s)
~
+- ++
V
~
DC*(s) V~
DC(s)Faster current control loop
Figure 2.30: The cascade control scheme which shows the slower outer VDC(t) con-troller and a faster inner idg(t) controller which in this case is estimated as a first order system.
Ignoring the disturbances from (2.132), the open loop transfer function ˜VDC(s)/ ˜E(s) is found (2.135),
VDC˜ (s)
E(s)˜ = KpKid(s + KT) (sTid+ 1)
s2(sτ + 1) , (2.135)
where
Kid = −v′d⋆+ 2Rwi⋆dg
CdcVDC⋆ (2.136)
and
Tid = Lwi⋆dg
vd′⋆+ 2Rwi⋆dg, (2.137) where the systems operating point are denoted using ⋆.
The sensitivity of the closed loop system depends on the Pgen(t) level, where the cor-responding i⋆doperating point can become negative which consequently makes the sys-tems zero 1/Tid negative (2.135) where,
Tid < 0 : if {− vd′⋆ 2Rw
< i⋆d< 0} non-minimum phase system. (2.138)
2. WAVE-TO-WIREMODELLING OFGRID
CONNECTEDPOINTABSORBERS 2.3 Modelling the Electrical System
-5 -4 -3 -2 -1 0 1
Real Axis (seconds-1) Imaginary Axis (seconds-1)
Real Axis (seconds-1) Imaginary Axis (seconds-1)
Figure 2.31: The cascade control scheme which shows the slower outer VDC(t) con-troller and a faster inner idg(t) controller which in this case is estimated as a first order system.
If Tid ≥ 0, the closed loop system is an easily controlled minimum phase system, as shown in Fig. 2.31(a).If Tid < 0 (2.138), the closed loop system becomes a non-minimum phase system, which can lead to an unstable system if over-tuned (Dirscherl et al. 2017). An example of a non-minimum phase system occurring due to a negative Tid is shown in Fig. 2.31(b). It is demonstrated in Fig. 2.31 that during a minimum phase system, a stable system is feasible at a gain of Kp ≥ 0. However, the controller during a non-minimum phase has to be carefully detuned to maintain system stability.
Furthermore, by detuning the VDC(t) controller, the controller consequently increases the DC-link voltage VDC(t) variability which is undesirable as problems may arise on the generation side.
How easily controllable the VDC(t) link is depends on the power exported from the generation side Pgen(t) onto the grid side and the weakness of the grid. The most convenient method is to treat the grid network, as shown in Fig. 2.29, as an ideal network where the voltages from the grid vq‘(t) and v‘d(t) are assumed constant. This method would be sufficient when importing close to constant power and when the grid is strong. However, for weaker grids and fluctuating power levels, stability problems may transpire (Kundur et al. 1994). With wave energy resources located in remote ar-eas, there is a high probability that long transmission lines will be used, hence causing a weak grid (Huang et al. 2012).
Grid networks, as was estimated in Fig. 2.29, are modelled using a series impedance Zs = Rs+ jXs, as shown in Fig. 2.32, where Vabc(t) is the three phase source voltage which is constant. The series impedance Zsis found using a grid impedance angle Ψg
Electrical Power Optimisation of
Grid-connected Wave Energy Converters using Economic Predictive Control
69 Adrian C.M. O’Sullivan
Grid diagram
V
abc(t)
Rs Ls
Rs Ls
Rs Ls
v'a(t) v’b(t)
v’c(t)
Figure 2.32: A circuit diagram showing an equivalent grid impedance, selecting these values will define the grid strength.
and the short circuit ratio (SCR) (Drbal et al. 1996). The SCR is defined as (2.139),
SCR = Ssc
Sn
(2.139) where Ssc is the short circuit level of a certain point on the grid and Snis the nominal power level produced from the generation side. The grid is considered weak if SCR <
3 (Krishayya et al. 1997), although there have been cases where wind farms were able to operate during SCR = 2 (Diedrichs et al. 2012). By knowing the grid impedance angle Ψg and the SCR values of the grid the grid impedance can be found, as in (2.140) and (2.141),
tan (Ψg) = Xs
Rs (2.140)
Ssc = V2
ñR2s+ Xs2, (2.141)
where V is the line-to-line voltage of v′abcg(t). With these parameter found, the grid network can be modelled and simulated. However, in this thesis the main objective is maximising electrical power extraction from the energy source and only dealing with the wave to DC-link system, therefore assuming an ideal constant DC-link at all times.