Class 15 - Q&A

[allensctong]

Forum for questions about power system background, dynamic control, and any other topic covered in lecture 15.

[andrewrosemberg]

From what I currently understand:

1) Swing equation at each generator bus.
The two ODEs describe frequency dynamics: rotor angle integrates speed deviation; speed acceleration equals mechanical–electrical power mismatch minus damping.

$$\forall i \in \{1,\dots,n\}:\qquad \dot{\delta}_i=\Delta\omega_i,\qquad \frac{2H_i}{\omega_s}\,\dot{\Delta\omega_i}=P_{m,i}-P_{e,i} (\boldsymbol{\theta},\boldsymbol{V})-D_i\,\Delta\omega_i - u(\Delta\omega_i)$$

$H_i$ is inertia, $\omega_s$ synchronous speed, $D_i$ damping/droop, $P_{m,i}$ mechanical input, $P_{e,i}$ electrical output (to the network), and $u(\Delta\omega_i)$ are things like inverter control (our online actuator).

2) Power injections from $(V,I)$.
Complex power $S_i=V_i I_i^\ast$ splits into active/reactive injections. The standard power–angle relations:

$$S_i=P_i^{\rm inj}+\mathbf{j}Q_i^{\rm inj}=V_i(I_{R,i}-\mathbf{j}I_{I,i})$$ $$P_i^{\rm inj}=V_{R,i}I_{R,i}+V_{I,i}I_{I,i},\qquad Q_i^{\rm inj}=V_{I,i}I_{R,i}-V_{R,i}I_{I,i}$$ $$P_i^{\rm inj}=\sum_{k=1}^n V_iV_k\!\left(G_{ik}\cos(\theta_i-\theta_k)+B_{ik}\sin(\theta_i-\theta_k)\right)$$ $$Q_i^{\rm inj}=\sum_{k=1}^n V_iV_k\!\left(G_{ik}\sin(\theta_i-\theta_k)-B_{ik}\cos(\theta_i-\theta_k)\right)$$

Electrical power in the swing equation is just the active injection:
$P_{e,i}=P_i^{\rm inj}$.

3) Injection‑type disturbance at bus $i$.
Any sudden change in load, generation, line-admittance happens at t=0 for the dynamic equations.

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What is not clear:

Q1) Let's start with the simplest event that changes the inputs to our dynamic equations: Active Load change $P_{m,j}$ at a node j. Let's just consider the continuous problem and ignore the fact that we can do things like line switching. We consider the damping of syncronous generators fixed and the $u$ inverter control is fixed in this analysis. I want just to understand the evolution of the dynamic equations.

At t=0 we have:

$$\delta_i(0) = 0 \; \forall i \; \text{:: frequency is nominal at the beginning.}$$ $$P_{m,j} \neq P_{e,j}(0) \; \text{:: unbalance between load draw and network injection .}$$

Now to see how the disturbance will change the states throughout the grid we need to model how the flow will progress. Does this mean that Power injections should have a $\theta_k + \delta_k$ instead of all $\theta_k$ for all $k$?

Q2) Are there reactive load disturbances? how would they affect frequency?

Q3) (WIP - early thoughts on voltage control) Voltage control seems to be viewed seperately from frequency. Are the swing equations also used when analysing voltage stability?

Q4) Do we care that power flow doesn't happen instantaneously? I.e. the above equations don't take into account any transient dynamics on the lines - e.g., inductance and capacitance effects, ... .

Q5) (Related to the above question) For frequency stability, the literature talks about a horizon of 10s and the time interval of the discretization is left to the modeler. While this sort of makes sense, the above equations are already implying some assumptions about what order of magnitude of discretization we care / make a difference.

References

(I would suggest that the other students ignore these to avoid all of us having the same gaps as I do - perhaps given some missing pieces in these references).

https://ietresearch.onlinelibrary.wiley.com/doi/full/10.1049/iet-gtd.2017.0345

https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9779512

https://dl.poweren.ir/downloads/PowerEn/Book/2019/Jun/Power%20System%20Stability%20and%20Control%20-%20Prabha%20Kundur%20(PowerEn.ir).pdf

[allensctong]

Re-Q2: there can be, e.g. a capacitor bank switching, especially when it has a large VAR capacity.

They would not directly affect frequency which is tied to active power balance, but they can affect it indirectly since the active power capacity could be reduced when the generator has to supply extra VAR due to an increase in load.

[allensctong]

Re-Q3: not that I know of. Reactive power has a direct relation with voltage and vice versa. Voltage too has an indirect relation with frequency that doesn't always have a closed form equation and it's very challenging to analyze their relation, if any. Operators rely on devices that can provide reactive power support for voltage control, and I heard they sometimes also analyze the power flow Jacobian for stability analysis