Low-observability matrix completion
An example device includes at least one processor configured to receive electrical parameter values corresponding to at least one first location within a power network. The at least one processor is further configured to determine, using matrix completion and based on the at least one electrical parameter value, an estimated value of at least one unknown electrical parameter. The at least one unknown electrical parameter corresponds to a second location within the power network. The at least one processor is also configured to cause at least one device within the power network to modify operation based on the estimated value of the at least one unknown electrical parameter.
1. A device comprising:
at least one processor configured to:
receive at least one electrical parameter value corresponding to at least one first location within a power distribution network at a first time;
determine, 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
cause 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 at 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
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,
subject to the following constraints:
∥X Ω −D Ω ∥ F ≤δ;
a constraint that correlates respective voltages, currents, and impedances for each line (f,t) within a first error tolerance, ϵ f,t ;
constraints that respective net real power flowing into and out of each location f be within a second error tolerance, τ f ;
ϵ f,t ≥0 for each line(f,t);
τ f ≥0 for each location f;
∥ν−Mx−w∥ ∞ ≤γ; and
∥|ν|−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;
n 1 represents a number of lines in the power distribution network;
n 2 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;
w 1 , w 2 , w 3 , and w 4 represent weighting parameters used to tune accuracy;
Ω represents observed values of electrical parameters, including the at least one electrical parameter value;
X Ω and D Ω represent X and D, respectively, with zero elements corresponding to the locations of unobserved electrical parameters;
δ represents a tolerance value for measurement noise;
νrepresents 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;
| 84 | represents a vector collecting voltage magnitudes at all locations;
K represents a constant matrix; and
|w| represents a norm of w.
2. The device of claim 1 , wherein determining the estimated value of the at least one unknown electrical parameter comprises:
determining a raw estimated value using matrix completion; and
applying a scaling factor to the raw estimated value to obtain the estimated value.
3. The device of claim 1 , wherein each electrical parameter value in the at least one electrical parameter value comprises a respective value of a parameter of the power distribution network that is correlated with a voltage parameter of the power distribution network.
4. The device of claim 3 , wherein the at least one electrical parameter value comprises at least one of:
a value of real voltage from a source bus of a line in the power distribution network;
a value of reactive power from the source bus;
a value of total real power entering the source bus from all sources;
a value of total reactive power entering the source bus from all sources;
a value of real current flowing through the line;
a value of reactive current flowing through the line;
a value of power flowing through the line;
a value of load at the source bus; or
a value of voltage magnitude at the source bus.
5. The device of claim 1 , wherein determining the estimated value of the at least one unknown electrical parameter does not rely on electrical parameter values from a second time that is different from the first time.
6. The device of claim 1 , wherein determining the estimated value of the at least one unknown electrical parameter comprises performing matrix completion on a matrix in which each row represents a line in the power distribution network, the matrix comprising the following columns:
a column representing, for lines in the power distribution network, a respective real voltage from a source bus of the line;
a column representing, for the lines, a respective reactive power from the source bus of the line;
a column representing, for the lines, a respective total real power entering the source bus of the line from all sources;
a column representing, for the lines, a respective total reactive power entering the source bus of the line from all sources;
a column representing, for the lines, a respective real current flowing through the line;
a column representing, for the lines, a respective reactive current flowing through the line;
a column representing, for the lines, a respective power flowing through the line;
a column representing, for the lines, a respective load at the source bus; and
a column representing, for the lines, a respective voltage magnitude at the source bus.
7. 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
∈
ℝ
n
1
×
n
2
,
{
ϵ
f
,
t
}
,
{
τ
f
}
X
*
+
w
1
Σ
(
f
,
t
)
∈
ℒ
ϵ
f
,
t
+
w
2
Σ
f
∈
𝒩
τ
f
+
w
3
γ
+
w
4
α
,
subject to the following constraints:
∥X Ω −D Ω ∥ f ≤δ;
a constraint that correlates respective voltages, currents, and impedances for each line (f,t) within a first error tolerance, ϵ f,t ;
constraints that respective net real power flowing into and out of each location f be within a second error tolerance, τ f ;
ϵ f,t ≥0 for each line (f,t);
τ f ≥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;
n 1 represents a number of lines in the power distribution network;
n 2 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;
w 1 , w 2 , w 3 , and w 4 represent weighting parameters used to tune accuracy;
Ω represents observed values of electrical parameters, including the at least one electrical parameter value;
X Ω and D Ω 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.
8. The method of claim 7 , wherein determining the estimated value of the at least one unknown electrical parameter comprises:
determining a raw estimated value using matrix completion; and
applying a scaling factor to the raw estimated value to obtain the estimated value.
9. The method of claim 7 , wherein each electrical parameter value in the at least one electrical parameter value comprises a respective value of a parameter of the power distribution network that is correlated with a voltage parameter of the power distribution network.
10. The method of claim 9 , wherein the at least one electrical parameter value comprises at least one of:
a value of real voltage from a source bus of a line in the power distribution network;
a value of reactive power from the source bus;
a value of total real power entering the source bus from all sources;
a value of total reactive power entering the source bus from all sources;
a value of real current flowing through the line;
a value of reactive current flowing through the line;
a value of power flowing through the line;
a value of load at the source bus; or
a value of voltage magnitude at the source bus.
11. The method of claim 7 , wherein determining the estimated value of the at least one unknown electrical parameter does not rely on electrical parameter values from a second time that is different from the first time.