Documentation: add current sensor + power sensor variance diff to ILSE docs

docs/user_manual/calculations.md

@@ -106,7 +106,11 @@ In observable systems this helps better outputting correct results. On the other
 Based on the requirements of observability mentioned above, users need to satisfy at least the following conditions for state estimation calculation in `power-grid-model`.
 
 - `n_voltage_sensor >= 1`
-- If no voltage phasor sensors are available, then the following conditions should be satisfied:  `n_unique_power_or_current_sensor >= n_bus - 1`. Otherwise: `n_unique_power_or_current_sensor + n_voltage_sensor_with_phasor >= n_bus`
+- If no voltage phasor sensors are available, then the following conditions should be satisfied:
+  - There are no global angle current sensors
+  - And:
+    - Either: `n_unique_power_or_current_sensor >= n_bus - 1`.
+    - And/or: `n_unique_power_or_current_sensor + n_voltage_sensor_with_phasor >= n_bus`
 
 `n_unique_power_or_current_sensor` can be calculated as sum of following:
 
@@ -474,8 +478,10 @@ $$
 $$
 
 - Branch current with global angle: Linear WLS requires a complex current phasor. The global angle current measurement captures
-the phase offset relative to the same predetermined reference phase against which the voltage angle is measured. As a result,
-at least one voltage measurement with angle measurement is required in the grid if a global angle current sensor is used.
+the phase offset relative to the same predetermined reference phase against which the voltage angle is measured. It is not
+sufficient to use the default zero-phase reference offset, because the reference point is not uniquely determined when there
+are no voltage angle measurements, resulting in ambiguities. As a result, using any global angle current sensor in the grid
+requires at least one voltage phasor measurement (with angle).
 
 $$
     \begin{eqnarray}

docs/user_manual/components.md

@@ -739,12 +739,12 @@ Due to the high admittance of a `link` it is chosen that a current sensor cannot
 
 ##### Input
 
-| name                     | data type                                                                     | unit       | description                                                                                                                               |              required               |  update  |                     valid values                     |
-| ------------------------ | ----------------------------------------------------------------------------- | ---------- | ----------------------------------------------------------------------------------------------------------------------------------------- | :---------------------------------: | :------: | :--------------------------------------------------: |
-| `measured_terminal_type` | {py:class}`MeasuredTerminalType <power_grid_model.enum.MeasuredTerminalType>` | -          | indicate the side of the `branch`                                                                                                         |              &#10004;               | &#10060; | the terminal type should match the `measured_object` |
-| `angle_measurement_type` | {py:class}`AngleMeasurementType <power_grid_model.enum.AngleMeasurementType>` | -          | indicate whether the measured angle is a global angle or a local angle                                                                    |              &#10004;               | &#10060; | the terminal type should match the `measured_object` |
-| `i_sigma`                | `double`                                                                      | ampere (A) | standard deviation of the current (`i`) measurement error. Usually this is the absolute measurement error range divided by 3.             | &#10060; see the explanation below. | &#10004; |                        `> 0`                         |
-| `i_angle_sigma`          | `double`                                                                      | rad        | standard deviation of the current (`i`) phase angle measurement error. Usually this is the absolute measurement error range divided by 3. | &#10060; see the explanation below. | &#10004; |                        `> 0`                         |
+| name                     | data type                                                                     | unit       | description                                                                                                                               | required |  update  |                     valid values                     |
+| ------------------------ | ----------------------------------------------------------------------------- | ---------- | ----------------------------------------------------------------------------------------------------------------------------------------- | :------: | :------: | :--------------------------------------------------: |
+| `measured_terminal_type` | {py:class}`MeasuredTerminalType <power_grid_model.enum.MeasuredTerminalType>` | -          | indicate the side of the `branch`                                                                                                         | &#10004; | &#10060; | the terminal type should match the `measured_object` |
+| `angle_measurement_type` | {py:class}`AngleMeasurementType <power_grid_model.enum.AngleMeasurementType>` | -          | indicate whether the measured angle is a global angle or a local angle                                                                    | &#10004; | &#10060; |                                                      |
+| `i_sigma`                | `double`                                                                      | ampere (A) | standard deviation of the current (`i`) measurement error. Usually this is the absolute measurement error range divided by 3.             | &#10060; | &#10004; |                        `> 0`                         |
+| `i_angle_sigma`          | `double`                                                                      | rad        | standard deviation of the current (`i`) phase angle measurement error. Usually this is the absolute measurement error range divided by 3. | &#10060; | &#10004; |                        `> 0`                         |
 
 #### Current Sensor Concrete Types
 
@@ -792,13 +792,25 @@ $$
 $$
 
 Internally, the current phasor measurement is decomposed into a separate real and imaginary component
-per phase, with each their own variances, which are estimated using the common linearization approximation:
+per phase, with each their own variances, which are estimated using the common linearization approximation.
+E.g., for a real random variable $X$, the random variable $Y = F\left(X\right)$, with $F$ sufficiently well-behaved, exists.
+The variances of $Y$ can then be estimated from the variance of $X$ using
+$\text{Var}\left(Y\right) \approx \text{Var}\left(X\right) \left\|\frac{\delta f}{\delta X}\right\|^2$
+
+This approach generalizes to extension fields and to more dimensions. Both generalizations are required for current sensors.
+The current phasor is described by a complex random variable, which can be decomposed into a real and an imaginary component, or into a magnitude and a phasor.
+Asymmetric sensors and asymmetric calculations, on the other hand, require multidimensional approaches.
+The generalization is done as follows.
+
+Let $A$ and $B$ be fields (or rings) that may be multidimensional and that may be extension fields.
+Let $\boldsymbol{X}$ be a random variable with codomain $A$ with components $X_i$.
+Let $\boldsymbol{Y} = \boldsymbol{F}\left(\boldsymbol{X}\right)$ be a random variable with codomain $B$, with $\boldsymbol{F}$ sufficiently well-behaved,
+so that its components $Y_j = F_j\left(\boldsymbol{X}\right)$ are well-behaved.
+Then the variances $\text{Var}\left(Y_i\right)$ of the components of $\boldsymbol{Y}$ can be estimated
+in terms of the variances $\text{Var}\left(X_j\right)$ of the components of $\boldsymbol{X}$ using the following function.
 
 $$
-   \begin{eqnarray}
-      \text{let } & \vec{F}(\vec{x}) \equiv \sum_i \vec{e}_i f_i(x_j) \\
-      \text{then } & \text{Var}(f_i) \approx \sum_j \text{Var}(x_j) \|\frac{\delta f_i}{\delta{x_j}}(\vec{x})\|^2
-   \end{eqnarray}
+\text{Var}\left(Y_j\right) = \text{Var}\left(F_j\left(\boldsymbol{X}\right)\right) \approx \sum_i \text{Var}\left(X_i\right) \left\|\frac{\delta F_j}{\delta x_i}\left(\boldsymbol{X}\right)\right\|^2
 $$
 
 The following illustrates how this works for `sym_current_sensor`s in symmetric calculations.