US 2023/0267246 A1 · METHOD FOR CONSTRUCTING GENERAL PROBABILITY MODEL OF HARMONIC EMISSION LEVEL FOR INDUSTRIAL LOAD

Abstract

A method for constructing a general probability model of a harmonic emission level for an industrial load is provided. The method establishes, based on harmonic data monitored by a power quality monitoring system, a general probability model by combining a parametric estimation method based on a normal distribution function and a lognormal distribution function with a nonparametric estimation method represented by a kernel density estimation method, taking a degree of approximation between the general probability model and an actual probability distribution of each harmonic current as an objective function based on parameters required by the general probability model, and optimizing and solving the parameters of the proposed general probability model by using a multiplier method to determine parameters of the general probability model to finally obtain a general probability model applicable to different industrial loads.

A method for constructing a general probability model of a harmonic emission level for an industrial load, comprising the following steps:

step 1:
extracting harmonic monitoring data of the industrial load to obtain a harmonic characteristic dataset X of a user:
X
=
[
I
2
1
I
3
1
L
I
25
1
I
2
2
I
3
2
L
I
25
2
M
M
M
M
I
2
N
I
3
N
L
I
25
N
]
,
wherein N represents a total quantity of sampling points, each column vector in the X represents a harmonic current monitoring sequence, a subscript of the I represents a harmonic order, and a superscript of the I represents a quantity of sampling sequences;
step 2:
constructing the general probability model for an h-th harmonic Ih in the harmonic characteristic dataset:
f
⁡
(
I
h
)
=
∑
i
=
1
3
λ
i
⁢
f
i
(
I
h
)
,
wherein fi(.) represents a probability density subfunction and λi represents a weight coefficient of the probability density subfunction;
f1(.) represents a part that is of the Ih and obeys a normal distribution, f2(.) represents a part that is of the Ih and obeys a lognormal distribution, and f3(.) represents a part that is of the Ih and obeys another distribution;
and the fi(.) is expressed as follows:
f
1
(
I
h
)
=
1
2
⁢
π
⁢
σ
1
⁢
e
-
(
I
h
-
μ
1
)
2
⁢
σ
1
2
⁢
f
2
(
I
h
)
=
1
2
⁢
π
⁢
I
h
⁢
σ
2
⁢
e
-
(
lnI
h
-
μ
2
)
2
2
⁢
σ
2
2
⁢
f
3
(
I
h
)
=
1
nb
⁢
∑
j
=
1
n
K
⁡
(
I
h
-
I
h
j
b
)
wherein μ1 and μ2 represent mathematical expectations of the probability density subfunction;
σ1 and σ2 represent standard deviations of the probability density subfunction;
K(.) represents a kernel function, b>0, wherein b represents a smoothing parameter, which is referred to as a window;
Ihj represents a jth sample of the Ih in each window;
and n represents a total quantity of samples in each window;
the weight coefficient of the probability density subfunction meets the following formula:
∑
i
=
1
3
λ
i
=
1
⁢
(
0
≤
λ
i
≤
1
)
,
wherein λ1=1 represents that the Ih obeys a single normal distribution;
and A2=1 represents that the Ih obeys the lognormal distribution;
step 3:
discretizing the general probability model of the Ih, wherein
the Ih is discretized to discretize the f1(.) and the f2(.) to obtain the following discretized general probability model:
f
⁡
(
I
~
h
)
=
∑
i
=
1
3
λ
i
⁢
f
i
(
I
~
h
)
,
I
~
h
=
0
:
0.01
:
max
⁡
(
I
h
)
,
wherein max (Ih) represents a maximum current value of the h-th harmonic;
step 4:
constructing a parameter optimization model of the general probability model, which specifically comprises the following substeps:
step 4.1:
constructing an objective function, wherein
a degree of approximation between the general probability model and an actual probability distribution of the Ih is reflected by a difference between a mathematical expectation calculated by a general optimization model and an actual value, as well as a difference between a standard deviation calculated by the general optimization model and an actual value, wherein the objective function is as follows:
min
⁢
y
1
=
(
∑
i
=
1
3
λ
i
⁢
E
i
(
I
~
h
)
-
E
0
(
I
~
h
)
)
2
⁢
min
⁢
y
2
=
(
∑
i
=
1
3
λ
i
2
⁢
Var
i
(
I
~
h
)
-
Var
0
(
I
~
h
)
)
2
wherein y1 and y2 respectively represent mean square errors of a mathematical expectation and a standard deviation of the general probability model;
Ei(Īh) and E0(Īh) respectively represent a mathematical expectation that is of the Ih and calculated by the probability density subfunction, and an actual mathematical expectation of the Ih;
and Vari(Īh) and Var0(Īh) respectively represent a standard deviation that is of the Ih and calculated by the probability density subfunction, and an actual standard deviation of the Ih;
and
min y1 and min y2 are combined into a minimum objective function, and the combined minimum objective function is as follows:
min
⁢
y
=
y
1
+
y
2
2
step 4.2:
determining constraints, wherein
the constraints comprise an equality constraint and an inequality constraint, wherein
1) an equality constraint for optimizing the weight coefficient λi of the probability density subfunction is determined according to the following formula, and is expressed by l:
l
=
∑
i
=
1
3
λ
i
-
1
=
0
2) the inequality constraint comprises a value range of the weight coefficient λi and a value range that is of the random variable Ih and determined by numerical characteristics (μ1, σ1) and (μ2, σ2) when the single probability density subfunction takes effect;
an inequality constraint of the λi is as follows:
{
g
1
=
λ
1
≥
0
g
2
=
λ
2
≥
0
g
3
=
λ
3
≥
0
,
wherein assuming that 95% confidence intervals of {μ1, μ2, σ1, σ2} are [{circumflex over (θ)}1, {circumflex over (θ)}2], [{circumflex over (θ)}3, {circumflex over (θ)}4], [{circumflex over (θ)}5, {circumflex over (θ)}6], and [{circumflex over (θ)}7, {circumflex over (θ)}8] respectively, inequality constraints of the optimization variables {μ1, μ2, σ1, σ2} are as follows:
{
g
4
=
μ
1
-
θ
^
1
≥
0
g
5
=
θ
^
2
-
μ
1
≥
0
,
{
g
6
=
μ
2
-
θ
^
3
≥
0
g
7
=
θ
^
4
-
μ
2
≥
0
,
{
g
8
=
σ
1
-
θ
^
5
≥
0
g
9
=
θ
^
6
-
σ
1
≥
0
,
and
⁢
{
g
10
=
σ
2
-
θ
^
7
≥
0
g
11
=
θ
^
8
-
σ
2
≥
0
,
wherein gq represents the inequality constraint and q=1, 2, . . . , 11;
step 5:
solving parameters {λ1, λ2, λ3, μ1, μ2, σ1, σ2} of the general probability model, wherein
a constrained problem is converted into an unconstrained problem, and a multiplier method is used for solving, that is, an optimization variable set is defined as γ={λ1, λ2, λ3, μ1, μ2, σ1, σ2}, and an augmented Lagrange function is defined as J and is expressed is as follows:
J
⁡
(
γ
,
ω
,
ν
,
ρ
)
=
y
⁡
(
γ
)
-
ν
⁢
l
⁡
(
γ
)
+
ρ
2
⁢
l
2
(
γ
)
+
1
2
⁢
ρ
⁢
∑
q
=
1
11
{
[
max
⁡
(
0
,
ω
q
-
ρ
⁢
g
q
(
γ
)
)
]
2
-
ω
q
2
}
,
wherein y(γ) represents the objective function, l(γ) represents the equality constraint, gq(γ) represents the inequality constraint, ωq represents a Lagrange multiplier of the inequality constraint, and ν represents a Lagrange multiplier of the equality constraint;
and
for the J(γ, ω, ν, φ, the sufficiently large parameter ρ is taken, and the multipliers ω and ν are continuously corrected to obtain a local optimal solution by minimizing the J(γ, ω, ν, φ, wherein correction formulas of the multipliers ω and ν are as follows:
{
ω
q
(
k
+
1
)
=
max
⁡
(
0
,
ω
q
(
k
)
-
ρ
⁢
g
q
(
γ
(
k
)
)
)
,
q
=
1
,
2
,
…
,
11
v
(
k
+
1
)
=
v
(
k
)
-
ρ
⁢
l
⁡
(
γ
(
k
)
)
,
wherein k in a superscript represents a quantity of corrections;
and
step 6:
obtaining the general probability model of the Ih.

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Method for constructing general probability model of harmonic emission level for industrial load

Life of this patent

Method for constructing general probability model of harmonic emission level for industrial load

Life of this patent

Abstract

Claims3