[FEATURE] Line drop compensation based on Active Power

[mgovers]

This ticket is an incomplete placeholder and should be refined together with the feature requestor first.

Currently, we model line drop compensation based on the cable impedance and current.

$$ (\Delta V)_{\text{compensation}} (I) = (R + i X) I $$

New advancements in research about tap changers have indicated that a modeling of the line drop compensation with the following form produces a better representation of the grid:

$$ (\Delta V)_{\text{compensation}} (P) = \begin{cases} -(\Delta V)_0 , && P <= -P_0 \ a P , && - P_0 < P < P_0 \ (\Delta V)_0 , && P >= P_0 \ \end{cases} $$

for some coefficient $a$.

Vision goes even further and defines P<<, P<, P0, P> and P>>, and similarly V<<, V<, V0, V> and V>> which provide additional points for piece-wise linear line-drop compensation. If we define $P_a' =$ P<<, $V_a' =$ V<<, $P_b' =$ P<, $V_b' =$ V<, $P_0 =$ P0, $V_0 =$ V0, $P_b =$ P>, $V_b =$ V>, $P_a =$ P>> and $V_a =$ V>>

$$ V_{\text{measure}} (P) = \begin{cases} V_a' , && P <= P_a' \ V_a' + \frac{V_b' - V_a'}{P_b' - P_a'} (P - P_a') , && P_a' < P < P_b' \ V_b' , && P = P_b' \ V_0 + \frac{V_b' - V_0}{P_b' - P_0} (P - P_0) , && P_b' < P < P_0 \ V_0 , && P = P_0 \ V_0 + \frac{V_b - V_0}{P_b - P_0} (P - P_0) , && P_0 < P < P_b \ V_b , && P = P_b \ V_a + \frac{V_b - V_a}{P_b - P_a} (P - P_a) , && P_b < P < P_a \ V_a , && P >= P_a \ \end{cases} $$

(note that here, $V_0 =$ V0 is the setpoint in point $P_0 =$ P0)

TBD: Design

Should we:

• [ ] create a flag to enable this type of behavior like Vision does? if so: how?
  • In the current description, this does not allow for reactance
• [ ] add support for arbitrary (piecewise) functions? If so, that decision would be a blocker for this ticket and would also be an enabler for #169
• [ ] only provide support for cutoff points $P_a' =$ P<< and $P_a =$ P>>.
  • While not as general as the Vision equivalent, this does give the opportunity to keep the existing line_drop_compensation_r and line_drop_compensation_x and let the user only provide a single slope, rather than 4 (!) different piecewise linear sections. The symmetry would also keep the implementation much easier, but it may not be general enough (see below)
• [ ] provide cutoff points $P_a' =$ P<< and $P_a =$ P>>, as well as line_drop_compensation_r_negative and line_drop_compensation_x_negative. This would be general enough for the below mentioned voltage controller manuals, as well as reactance that may be different on both sides of the origin, but runs into the same extensibility issues as the current implementation.

Examples existing voltage controller functions

The Siemens SIPROTEC 5 Universal Protection 7SX82/7SX85 Manual and a-eberle REG-D Relay for Voltage Control & Transformer Monitoring manual both describe a characteristic curve with different values for $P_a'$ and $P_a$, as well as different values for $V_a'$ and $V_a$. See p.2710 of the former and p.158 of latter.