1.01 1.02 1.03 1.04 1.05 1.06 0 10 20 30 40 50 60 DC Voltage (pu) Time (s) Vmax T1 Vmax T2 Vmax T4 T1 T2 T4
Figure 5.18: Scenario 5: MTDC grid voltages
-800 -600 -400 -200 0 200 400 600 0 10 20 30 40 50 60 DC Power (MW) Time (s) T1 T2 T3 T4
Figure 5.19: Scenario 5: DC power of VSCs
5.4
Scheme 2 - Frequency containment
5.4.1 Overall controller description
The second option for frequency support approaches the problem from a different perspective. Despite the various implementation differences of the methods for primary frequency support in the literature, the main idea remains the same: a frequency droop gain is selected relating
the power of the VSC with the AC frequency deviation, sometimes complemented by an inertia emulation gain to provide some derivative response [FPB17]. This is inherited from AC systems and power plants, where the droop method has been the norm for several decades. However, although the droop control has proven indispensable for continuous regulation of frequency by conventional units (if the small frequency deadband present in many speed governors is ignored), the same does not necessarily hold true for the VSCs. In fact, simply specifying a frequency droop gain prevents from utilizing the VSC maximum capacity in emergency cases, e.g. when an Under-Frequency Load Shedding (UFLS) threshold is approached. Instead, we can think of a more adaptive control scheme that will provide as much support as possible in stressed situations, especially when operating the future AC systems with a lower number of synchronous machines. A novel possibility for provision of frequency support by VSCs is proposed in this section, refrai- ning from the requirement to select a frequency droop gain, and exploiting the almost instantane- ous response of VSCs. The main idea is to provide as much power as required (or as possible) to prevent the triggering of UFLS relays or at least reduce the amount of load shedding. Typi- cally, these relays have multiple shedding steps with various frequency thresholds. For example, in [EE14b] UFLS schemes with a maximum of ten shedding steps between 49 and 48 Hz are described. Avoiding or reducing UFLS can be translated into keeping frequency above the first frequency threshold of the relays. To achieve this, an MPC-based scheme is proposed that chan- ges the VSC power setpointPset of itsP -V characteristic (2.35), as soon as such a violation is predicted.
Clearly, the added control should not jeopardize the operation of the MTDC grid as well the other AC areas. This imposes to obey constraints on the DC voltages, on the rate of change of powers, etc. As for Scheme 1, only local measurements readily available to each VSC are used.
5.4.2 Constrained optimization problem
Instead of a reference trajectory, the objective here is to minimize the total control effort, while satisfying DC voltage, power and frequency constraints. The complete quadratic programing pro- blem at the heart of the proposed controller is described by Eqs. (5.24)-(5.31) hereafter:
min ∆Pset,,ζ,V,P,f Nc−1 X j=0 [∆Pset(k + j)]2+ w1 Nc X j=1 2(k + j) + w2 Nc X j=1 ζ2(k + j) (5.24) subject to the following linear constraints, forj = 1, . . . , Nc:
5.4. Scheme 2 - Frequency containment 117 Pmin≤ P (k + j) ≤ Pmax (5.26) fmin− ζ(k + j) ≤ f(k + j) ≤ fmax+ ζ(k + j) (5.27) (k + j), ζ(k + j)≥ 0 (5.28) P (k + j) = P (k + j− 1) + ∆Pset(k + j− 1) − Kv(V (k + j)− V (k + j − 1)) (5.29) V (k + j) = V (k + j− 1) + sv∆Pset(k + j− 1) (5.30) f (k + j) = f (k + j− 1) + rf(k)Ts+ [P (k)− P (k + j)] sfTs (5.31)
The first term in the objective function (5.24) aims to minimize the total control effort. By mini- mizing theL2norm, the overall control effort is distributed throughout the whole control horizon
Ncand a smooth response is achieved. Again, the prediction horizon has been taken equal to the
control horizon.
Constraint (5.25) specifies that the DC voltage should remain between some security limitsVlow andVup. These limits evolve with timek + j to bring the voltage progressively inside the desi- red range defined byVminandVmax, as for the first scheme (see also (5.11)). Constraint (5.26)
specifies that the VSC power must remain between the valuesPmin, Pmaxcorresponding to the capability of the VSC. Constraint (5.27) keeps the AC frequency inside the limitsfminandfmax.
Both constraints (5.25) and (5.27) can be relaxed by making variables and ζ larger than zero in case of infeasibility. Progressive constraint tightening could be also applied to constraint (5.27), but is not considered here. However, both and ζ are kept to small values by selecting large weig- hting factorsw1andw2in the objective function (5.24). Since the DC voltage constraint is critical
to avoid VSC tripping or damage, the weighting factors are chosen such thatw1 w2 1.
Equations (5.29)-(5.31) make up the prediction model. The same model as for the first option is considered here.
The controller operation forNc= 3 steps is illustrated in Fig.5.20. As long as the MPC does not
predict any frequency violation, the first term in the objective function (5.24) ensures that no action is taken. Suppose now that, at timek, the controller receives measurements and identifies that the minimum frequency limit is going to be violated, unless corrective actions are taken. The solution of the quadratic programming problem (5.24)-(5.31) is a sequence of control actions∆Pset(k),
∆Pset(k + 1), . . . , ∆Pset(k + Nc− 1) required to keep the frequency above its minimum value,
if possible. The controller then applies the first action ∆Pset(k) and discards the rest of the sequence. At timek + 1, new measurements are received and the whole procedure is repeated for the updated control horizon. As shown in Fig.5.20, the updated frequency measurement atk + 1 differs from the one predicted at timek due to model simplifications, uncertainties, etc. However, the closed-loop nature of the MPC can compensate for such modeling errors.
k k + 1 k + 2 k + 3 f t freq. measurement at k fmin k + 4 freq. measurement at k + 1 + + +
freq. evolution until k + 1 (after first action) freq. evolution until k
Pset
Psetevolution calculated at k
Psetevolution calculated at k + 1
pred. evolution without controller actions at k pred. evolution with controller actions at k
pred. evolution without controller actions at k + 1
+
pred. evolution with controller actions at k + 1
Figure 5.20: Illustrative example of proposed controller operation
It has to be emphasized that the success of the VSC in keeping frequency above this limit cannot be ensured. Instead, the various constraints, in particular (5.25) and (5.26) may prevent this. Nevertheless, the proposed scheme is expected to provide the maximum possible support to the system.
It is also noted that other constraints could be also accommodated by the formulation, such as maximum power available for frequency support, maximum rate of change of frequency, maxi- mum power requested from the other AC area, etc. For simplicity, the formulation has been kept to the minimal set given by (5.24)-(5.31).
5.4.3 Discussion
The first scheme for frequency support required a deadband (see Fig.5.1) to prevent continuous interactions between the AC areas, and avoid activation of frequency support on all VSCs. The size of the deadband should be sufficiently large to satisfy the above requirements. However, too large a deadband would decrease the effectiveness of the controller, since it would be activated with a larger delay. Thus, a compromise has to be found regarding the size of the deadband. The formulation of the second scheme removes the need for a deadband. The first term in the objective function (5.24) ensures that no action is taken, unless a violation of a frequency threshold