Pitfalls of Zero Voltage Values in
Optimal Power Flow Problems

The complex voltage vector $\mathbf{U}_{i}$ at bus $i$, a.k.a. phasor, stacks the values of the nodes at buses $i$,

For a two-wire set of electromagnetically coupled conductors, we end up with a $2\times 2$ impedance matrix, capturing the self-impedance (diagonal) and mutual impedance (off-diagonal). By taking the matrix inverse, we obtain the admittance,

Note that, based on the physics (Carson/Pollaczek eq.), we expect that $\mathbf{R}_{l}$ and $\mathbf{X}_{l}$ are symmetric and positive definite. We define the complex current vector in branch $l$ in the direction of $i$ to $j$ to stack the individual values for each of the wires,

The branch currents satisfy,

The multiconductor ac Ohm’s law is,

The power flow into the branch is,

where ‘$\circ$’ indicates the element-wise product. The current vector for a generator is,

and by convention flows from the generator to the bus it is connected to, in each of its wires. Note that conservation of current applies through the generator,

The power from the generator through each of the wires is,

For loads, the current flows from the bus to the load,

Conservation of current through the load is,

The power flowing through each of the wires to the load is,

To force the load value (i.e. constant-power set point), we account for the voltage drop over the load, i.e. $U_{i,a}-U_{i,n}$,

We now continue to expand the RHS, and use (11) and (12),

and finally obtain,

Note that this does not imply conservation of current,

The effective dispatch value for a generator is,

Finally, KCL at bus $i$ is,

We state Ohm’s law (5) in admittance form,

and substitute this into (6) to obtain power flow vector into the branch $l$ in the direction of $i$ to $j$,

We lift KCL (18) to power variables (vectors with the same sizes as the original current vectors in (18)), by taking the conjugate and multiplying element-wise with $\mathbf{U}_{i}$,

When we allow 0 V entries in $\mathbf{U}_{i}$, the lifted KCL is a relaxation, as it does not imply current-based KCL (18), due to (possible) multiplication with 0 V. This is equivalent to adding a slack variable to the original KCL, e.g. when $U_{i,n}=$ 0 V:

As we are working with complex values, $I^{\text{slack}}_{i}$ actually represents two degrees of freedom, which can be exploited by optimization solvers.

Even though SVR-1 has proven to work well to optimize three-phase Kron-reduced four-wire networks with only wye loads³³3In this case conservation of current can be arbitrarily satisfied, as the resulting neutral current residual can flow into the ground unimpeded. , we need to strengthen the feasible set when we model loads/generators between phases (‘delta’) or phase-to-neutral⁴⁴4Neutral that is not exactly 0 V.. Constraints (15),(17) are insufficient in general, as they do not imply (8) or (11). To resolve this, we use the variables $\mathbf{I}_{d},\mathbf{I}_{g}$ and add the constraints (8), (9), (11), (12).

We define lifted variables for voltage, $\mathbf{W}_{i}$, current, $\mathbf{L}_{l}$, and power, $\mathbf{\bar{S}}_{lij}$,

where superscript ‘H’ indicates the conjugate transpose. Note that the substitutions don’t destroy any information⁵⁵5Up to a complex scale factor, see [15]., as long as $\mathbf{U}_{i},\mathbf{I}_{lij}$ are not the zero vector - individual entries being zero is fine. The power variable $\mathbf{\bar{S}}_{lij}$ is a square matrix, and therefore distinct from the vector $\mathbf{S}_{lij}$, however they are related,

We can construct a block matrix based on the lifted variables,

In convex relaxation schemes, this constraint gets cast as,

and the rank constraint is dropped to obtain a semi-definite programming formulation. Ohm’s law (5) is multiplied with its own conjugate transpose to obtain,

Furthermore, by multiplying Ohm’s law (5) on the right with $\mathbf{I}_{lij}^{\text{H}}$ and using (4), we obtain,

Combining (21) with (24), we have the BFM formulation with a diagonal KCL expression, as proposed by Gan and Low [13].

We now generalize the definitions of the load and generator power to matrix variables as well⁶⁶6Matrix generalizations add a bar above the original symbol.,

We can partition based on the rows,

and recognize that (11) implies,

Similarly, for generators we obtain,

We note valid equalities of the matrix variables w.r.t. the vector-style ones previously defined,

Note that we do not project KCL on the rows of $\mathbf{\bar{S}}_{lij}$ or $\mathbf{L}_{l}$ in a similar fashion to (31)-(32). Due capacitive coupling to earth as well as grounding of the neutral, the current through a branch does not in general sum to zero. Finally we generalize KCL, using the matrix variables $\mathbf{\bar{S}}$,

For visualizing the feasible sets we enforce voltage bounds,

By setting $U^{\text{min}}_{i,a}$ for instance to 0.9 pu, we typically exclude the low-voltage solution to quadratic equality (13).

Table I summarizes the variables and constraints for five optimization formulation variants, using the previously-defined constraints.

Implementation of these equations in nonlinear optimization frameworks typically requires translating the complex-valued expressions to sets of equivalent real-valued ones. Furthermore, lines should be modelled with shunt admittances, typically used to represent capacitive behavior w.r.t earth. Note that in general grid codes specify phase-to-neutral voltage magnitude limits, not phase-to-ground. Geth and Ergun [14] present the corresponding real-value expressions, and derive expressions for a variety of bounds. Claeys et al. [17] propose a framework to represent unbalanced transformers across variable spaces.

The exclusion of the 0 V solution in SWR-2 is due to the combination of 1) KCL as a matrix equality 2) conservation of current through loads/generators. Adding either constraint individually to SWR-1 is not sufficient to obtain a 0 gap. That suggests the spurious solution in SVR-2 can be excluded by also using the matrix KCL (34) and generalizing (20) to the outer product (instead of element-wise). Alternatively, as voltages are inherently relative, we can also choose to redefine the ground voltage value, at for instance 1 V.

Modelling loads with a nonzero voltage at both terminals requires additional constraints. In SVR-2, the current variables could still be eliminated in favor of power variables (i.e. as $(S/U)^{*}$), following the approach in [11], but this may result in higher order polynomial expressions. Furthermore, constant-power three-phase loads have multiple solutions, with the amount depending on the configuration and neutral grounding impedance [19, 20]. Deriving valid bounds to exclude the ‘low-voltage solution’ may speed up the computation.