Low-observability matrix completion

A method comprising:

• receiving, by a power network management system comprising at least one processor, at least one electrical parameter value corresponding to at least one first location within a power distribution network at a first time;
• determining, by the power distribution network management system, based on the at least one electrical parameter value, an estimated value of at least one unknown electrical parameter at the first time, wherein the at least one unknown electrical parameter corresponds to a second location within the power distribution network, and wherein determining the estimated value of the at least one unknown electrical parameter comprises performing matrix completion on a matrix of electrical parameters in the power distribution network corresponding to the first time, including the at least one electrical parameter value;
• and
• causing, by the power network management system, at least one device within the power distribution network to modify operation based on the estimated value of the at least one unknown electrical parameter,
• wherein:
• receiving the at least one electrical parameter value comprises receiving a first number of electrical parameter values corresponding to the first time;
• determining the estimated value of the at least one unknown electrical parameter comprises determining an estimated value of a second number of unknown electrical parameters as the first time, the second number being larger than the first;
• and
• wherein determining the estimated value of the at least one unknown electrical parameter comprises solving
• min
• X
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• ϵ
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• α
• ,
• subject to the following constraints:
• ∥XΩ−DΩ∥f≤δ;
• a constraint that correlates respective voltages, currents, and impedances for each line (f,t) within a first error tolerance, ϵf,t;
• constraints that respective net real power flowing into and out of each location f be within a second error tolerance, τf;
• ϵf,t≥0 for each line (f,t);
• τf≥0 for each location f;
• ∥v−Mx−w∥∞≤γ;
• and
• ∥|v|−Kx−|w|∥∞≤α,
• wherein:
• X represents a matrix of electrical parameter values including the estimated value of the at least one unknown electrical parameter;
• D represents a matrix of known electrical parameter values, including the at least one electrical parameter value;
• n1 represents a number of lines in the power distribution network;
• n2 represents a number of electrical parameters used for estimation;
• L represents the set of lines in the power distribution network;
• N represents the set of locations in the power distribution network;
• γ represents a third error tolerance;
• α represents a fourth error tolerance;
• w1, w2, w3, and w4 represent weighting parameters used to tune accuracy;
• Ω represents observed values of electrical parameters, including the at least one electrical parameter value;
• XΩ and DΩ represent X and D, respectively, with zero elements corresponding to the locations of unobserved electrical parameters;
• δ represents a tolerance value for measurement noise;
• v represents a vector collecting voltage phasors at all locations;
• M represents a first constant matrix, related to linear power flow equations;
• x represents a vector collecting active and reactive power injections at all locations;
• w represents a vector of constants;
• |v| represents a vector collecting voltage magnitudes at all locations;
• K represents a constant matrix;
• and
• |w| represents a norm of w.