CN105305440B - The hyperbolic cosine type maximal index absolute value Robust filter method of POWER SYSTEM STATE - Google Patents

Abstract

The invention discloses a hyperbolic cosine type maximum exponent absolute value robustness estimation method of the power system state, which belongs to the field of power system scheduling automation. The method includes the steps of: extracting active and reactive power injected into power system nodes, branch active and reactive power, and node voltage amplitude parameters; thereby establishing a hyperbolic cosine type maximum exponential absolute value robust state estimation model; Curvature cosine type maximum exponent absolute value robust state estimation basic model introduces auxiliary variable, obtains hyperbolic cosine type maximum exponent absolute value robust state estimation equivalent model; Solve the equivalent model for state estimation with maximum exponential absolute value robustness. The example analysis shows that the invention has strong error resistance and high calculation efficiency, and has good engineering application prospects.

Description

Hyperbolic Cosine Type Maximum Exponential Absolute Value Robust Estimation Method for Power System State

technical field

The invention belongs to the field of electric power system scheduling automation, in particular to a hyperbolic cosine type maximum exponent absolute value robustness estimation method for the state of the electric power system.

Background technique

Power system state estimation is the foundation and core of energy management systems. Now almost every large dispatch center has installed a state estimator, and state estimation has become the cornerstone of the safe operation of the power grid. Since state estimation was first proposed by foreign scholars in 1970, people have studied and applied state estimation for more than 40 years, and various state estimation methods have emerged during this period.

At present, the most widely used state estimation at home and abroad is the weighted least squares (WLS) method. The WLS model is simple and easy to solve, but its robustness is poor. In order to enhance the tolerance, there are generally two methods. The first is to add bad data identification after WLS estimation, such as the largest regularized residual test method (LNR) or estimation identification method, etc.; the other is to use the robust state estimation method. At present, the Robust state estimation methods (Robust state estimation) proposed by domestic and foreign scholars include weighted least absolute value estimation (Weighted least absolute value, WLAV), non-quadratic criterion methods (QL, QC, etc.), aiming at the maximum pass rate state estimation (Maximum normal measurement rate, MNMR) and exponential objective function state estimation (Maximum exponential square, MES). But the estimation performance of these robust state estimation methods still needs to be improved.

Contents of the invention

The object of the present invention is to propose a hyperbolic cosine type maximum exponent absolute value robust estimation method for the state of a power system, which is characterized in that the method is based on the hyperbolic cosine type maximum exponent absolute value with good robustness and high calculation efficiency Robust state estimation; includes steps:

Step A. Extract active power and reactive power injected into power system nodes, branch active and reactive power, and node voltage amplitude parameters; thus establish a hyperbolic cosine type maximum exponential absolute value robust state estimation model;

Step B. introducing auxiliary variables to the hyperbolic cosine type maximum exponential absolute value robust state estimation basic model, and transforming to obtain the hyperbolic cosine type maximum exponent absolute value robust state estimation equivalent model;

Step C. Using the primal-dual interior point algorithm to solve the hyperbolic cosine type maximum exponential absolute value robust state estimation equivalent model.

The hyperbolic cosine type maximum exponential absolute value robust state estimation basic model of the step A is: stg(x)=0, r=zh(x),

Among them: z∈R ^m is the measurement vector, including node injection active and reactive power, branch active and reactive power, and node voltage amplitude measurement; x∈R ⁿ is the state vector, including node voltage amplitude and balance node h:R ⁿ →R ^m is the nonlinear mapping from the state vector to the measurement vector; r _i is the i-th element of the residual vector r; g(x):R ⁿ →R ^c is the zero injection power equation constraint; w _i is the weight of the i-th quantity measurement, and σ ₀ and σ ₁ are window width parameters.

The step B includes: introducing a non-negative slack variable u,v∈R ^m , and transforming the hyperbolic cosine maximum exponent absolute value robust state estimation equivalent model into: stg(x)=0, zh(x)-u+v=0, u, v≥0.

The step C uses the primal-dual interior point algorithm to solve the hyperbolic cosine type maximum exponential absolute value robust state estimation equivalent model, specifically, let x ⁽⁰⁾ ∈ R ⁿ represent the voltage amplitudes of all nodes The flat start state variable composed of and phase angle (except the reference node phase angle);

Step C includes:

Step C1: Introduce the Lagrangian function

In the formula: λ∈R ^c and π,α,β∈R ^m are Lagrangian multiplier vectors;

Let x be the flat start state variable; select λ ⁽⁰⁾ = π ⁽⁰⁾ = 0 and u ⁽⁰⁾ , v ⁽⁰⁾ , α ⁽⁰⁾ , β ⁽⁰⁾ >0, where λ∈R ^c and π ,α,β∈R ^m is the Lagrangian multiplier vector, m is the number of quantity measurements, and c is the number of zero injection power constraints; let the central parameter ρ∈(0,1) and the convergence criterion ε =10 ^-3 , set iteration counter k=0;

Step C2: Calculate the dual gap Gap=α ^T v+β ^T u, and judge whether it is converged. If Gap<ε, go to step C7, otherwise go to step C3;

Step C3: Solve the correction equation to complete the correction of the original variable and the dual variable, and obtain [dx ^T dλ ^T dπ ^T ] ^T , dv, du, dα and dβ; specifically include:

Step C31: Calculating the disturbance parameter μ=ρ·Gap/2m;

Step C32: Forming the measurement equation and the Jacobian matrix corresponding to the zero injection power constraint and Form the measurement equation and the Hessian matrix corresponding to the zero injection power constraint and Where h(x) is the mapping from the state vector to the measurement vector, that is, the measurement estimated value, and g(x)=0 is the zero injection power constraint;

Step C33: Calculate L _x =G ^T λ-H ^T π, L _λ =g(x), L _π =zh(x)-u+v, and

Step C34: calculate γ=z-h(x)-u+v+AA-BB,

where AA,BB∈R ^m ,

Step C35: Solve the equation get [dx ^T dλ ^T dπ ^T ] ^T ;

Step C36: Solve for dv _i =k _1i dπ _i +AA _i and du _i =k _2i dπ _i +BB _i , where k _1i =a _i v _i -b _i u _i , k _2i =c _i v _i -d _i u _i ; and step C37: solving and

Step C4: Calculate the modified step size θ _P and θ _D of the original problem and the dual problem, where:

Step C5: Correct the variables of the original problem and the dual problem as follows:

Step C6: set the iteration counter k=k+1, enter step C2;

Step C7: output the optimal solution, end.

The beneficial effect of the present invention is that the hyperbolic cosine type maximum exponential absolute value robustness state estimation can effectively suppress a plurality of bad data including poor consistency data in the estimation process, shows good tolerance, and has a high The calculation efficiency is very suitable for practical engineering applications. The hyperbolic cosine type maximum exponential absolute value robust state estimation equivalent model is solved. The example analysis shows that the invention has strong error resistance and high calculation efficiency, and has good engineering application prospects.

Detailed ways

The present invention proposes a hyperbolic cosine type maximum exponent absolute value robust estimation method for the state of a power system. The present invention will be described in detail below in conjunction with embodiments.

The hyperbolic cosine type maximum exponential absolute value robust estimation method of the power system state includes the following steps:

Step A: extract active and reactive power injected into power system nodes, branch active and reactive power, and node voltage amplitude parameters; in this way, hyperbolic cosine maximum exponential absolute value robust state estimation (Hyperbolic cosine maximum exponential absolute value state estimation) is established. estimation, COSH-MEAV) basic model.

The basic model of COSH-MEAV is shown below

s.t.g(x)=0 (2)

r=z-h(x) (3)

In the formula: z∈R ^m is the measurement vector, which often includes node injection active power and reactive power, branch active power and reactive power, and node voltage amplitude measurement; x∈R ⁿ is the measurement vector including node voltage amplitude and phase angle State vector (except the balance node phase angle); h: R ⁿ → R ^m is the nonlinear mapping from the state vector to the measurement vector; r _i is the i-th element of the residual vector r; g(x): R ⁿ → R ^c is the zero injection power equation constraint; w _i is the weight of the i-th quantity measurement, and σ ₀ and σ ₁ are window width parameters.

Step B: Introduce auxiliary variables to the basic model of hyperbolic cosine type maximum absolute value robust state estimation model, and transform to obtain an equivalent model of hyperbolic cosine type maximum exponent absolute value robust state estimation.

Specifically, although the objective function of the COSH-MEAV basic model is continuous everywhere, it is not differentiable at 0, so it is difficult to solve it directly. Models (1)-(3) can be transformed into an equivalent model that can be derived everywhere.

Introduce variable ξ∈R ^m to satisfy

|r _i |≤ξ _i i=1,...,m (4)

From formulas (1) and (4), we can get, The maximum equivalent is maximum.

Introduce non-negative slack variable l,k∈R ^m , transform inequality (4) into two equality constraints as

r _i -l _i =-ξ _i i = 1,...,m (5)

r _i +k _i =ξ _i i=1,...,m (6)

Introduce non-negative slack variables u,v∈R ^m to satisfy

u _i = l _i /2 i = 1,...,m (7)

v _i =k _i /2 i=1,...,m (8)

From formulas (5) to (8), we can get

r _i =u _i -v _i i=1,...,m (9)

ξ _i =u _i +v _i i=1,...,m (10)

Putting Equation (9) into Equation (3), the equivalent measurement constraints can be obtained as

z-h(x)-u+v=0 (11)

Then the equivalent model of COSH-MEAV basic model given by formulas (1)~(3) is

s.t.g(x)=0 (13)

z-h(x)-u+v=0 (14)

u,v≥0 (15)

Models (12)-(15) are COSH-MEAV equivalent models, which are continuously differentiable everywhere and can be solved by gradient-based methods.

Step C: Using the primal-dual interior point algorithm to solve the hyperbolic cosine type maximum exponential absolute value robust state estimation equivalent model.

(1) Solution method of COSH-MEAV equivalent model

Note that the COSH-MEAV equivalence model (12)~(15) is an optimization problem containing equality constraints and inequality constraints, and it is suitable to be solved by the primal-dual interior point algorithm. In order to make those skilled in the art understand the present invention better, at first provide detailed derivation process as follows:

Introducing the Lagrange function

In the formula: λ∈R ^c and π,α,β∈R ^m are Lagrange multiplier vectors.

In order to obtain the optimal value, according to the KKT condition, we can get

In the formula:

In order to effectively solve the above problems, the modern interior point method introduces the disturbance parameter μ > 0 to relax equations (22) and (23), thus obtaining

The above equations can be solved by Newton's method to get

Gdx＝ _-Lλ (27)

-Hdx-du+dv=-L _π (28)

in,

From equations (29) and (30), we can get

Substitute (33), (34) into (31), (32) to get

make From formula (35) and (36) can get

dv _i =k _1i dπ _i +AA _i (37)

du _i =k _2i dπ _i +BB _i (38)

In the formula: k _1i = a _i v _i -b _i u _i , k _2i = c _i v _i -d _i u _i ,

Substitute (37), (38) into (28) to get

Hdx+Qdπ=γ (39)

In the formula: Q is a diagonal matrix of R ^m×m , and its diagonal elements are Q _ii =-k _1i +k _2i ; γ=zh(x)-u+v+AA-BB, AA,BB∈R ^m , AA _i , BB _i are the same as those in formulas (37) and (38).

According to equations (39), (26) and (27), the modified equation can be obtained as

dx, dλ and dπ can be obtained by solving formula (40); dv and du can be obtained from formulas (37) and (38); dα and dβ can be obtained by bringing the obtained results into formulas (33) and (34), then the iteration is sustainable.

(2) The solution steps of the COSH-MEAV equivalent model

After introducing the solution and derivation process of the COSH-MEAV equivalent model, the inventor summarizes the solution steps as follows:

Step C1: initialize, let x be the state variable of flat start; select λ ⁽⁰⁾ = π ⁽⁰⁾ = 0 and u ⁽⁰⁾ , v ⁽⁰⁾ , α ⁽⁰⁾ , β ⁽⁰⁾ >0; make The central parameter ρ∈(0,1) determines the value of the convergence criterion, and sets the iteration counter to zero.

Specifically, let x ⁽⁰⁾ ∈ R ⁿ represent the flat start state variable composed of all node voltage amplitudes and phase angles (except the reference node phase angle); choose λ ⁽⁰⁾ = π ⁽⁰⁾ = 0 and u ^{( 0)} ,v ⁽⁰⁾ ,α ⁽⁰⁾ ,β ⁽⁰⁾ >0, where λ∈R ^c and π,α,β∈R ^m are Lagrangian multiplier vectors, and m is the number of measurements , and c is the number of zero injection power constraints; let the central parameter ρ∈(0,1) and the convergence criterion ε=10 ^-3 , set the iteration counter k=0.

Step C2: Calculate the dual gap Gap=α ^T v+β ^T u, and judge whether it is converged. Specifically, if Gap<ε, it is considered to be converged and can directly go to step C7; otherwise, it is not converged and goes to step C3.

Step C3: Solve the correction equation to complete the correction of the original variable and the dual variable, and obtain [dx ^T dλ ^T dπ ^T ] ^T , dv,du, dα and dβ.

Specifically, first calculate the disturbance parameter μ=ρ·Gap/2m, then solve formula (40) to get [dx ^T dλ ^T dπ ^T ] ^T ; solve formulas (37), (38) to get dv,du; solve (33) , (34) get dα, dβ.

Step C4: Calculate the modified step size θ _P and θ _D of the original problem and the dual problem, where:

Step C5: Correct the variables of the original problem and the dual problem as follows:

Step C6: set iteration counter k=k+1, enter step C2; and

Step C7: output the optimal solution, end.

In order to enable those skilled in the art to better understand the present invention and understand the advantages of the present invention over the prior art, the applicant further explains in conjunction with specific embodiments.

The performance of COSH-MEAV based on primal-dual interior point algorithm is tested by using IEEE standard system. The test adopts full measurement, and the measurement value is obtained by superimposing white noise (mean value is 0, standard deviation is τ) on the result of power flow calculation. For voltage measurement, take τ _V =0.005pu; for power measurement, take τ _PQ =1MW/MVar. The test environment is a PC, the CPU is Intel(R) Core(TM) i3M370, the main frequency is 2.40GHz, and the memory is 2.00GB.

1. Comparison of tolerance performance

The inventors compared COSH-MEAV of the present invention with other state estimators to test the robustness of COSH-MEAV.

Set four inconsistent data (P _1-2 , Q _1-2 , P ₁ , Q ₁ ) on the IEEE-14 system. Table 1 shows the set bad measurement value and the correct value of measurement.

Table 1 COSH-MEAV identification of poor consistency data of IEEE 14 system

As a comparison, the widely used WLS is used to estimate first, and LNR is used to identify bad data (abbreviated as WLS+LNR). The result of the first identification is: the standardized residuals of 10 quantity measurements are greater than the threshold value (3.0), and these 10 quantity measurements are considered suspicious data; among them, the quantity measurement with the largest standardized residual error is P _2-1 , and this quantity is deleted Re-run WLS after the test; at this time, it is found that _P2 has the largest standardized residual. The above process is cycled 4 times, and 4 good measurements are mistaken for suspicious data by LNR and deleted, but the real bad data still exists. It can be seen that WLS+LNR cannot identify data with poor consistency.

The estimated results using the COSH-MEAV method are shown in Table 1. It can be found that COSH-MEAV's estimated and true values agree well even if there are poor consistency data in the quantity measurement. Many experiments in other IEEE systems also show that COSH-MEAV can automatically suppress bad data in the estimation process, and has good tolerance.

2. Comparison of computational efficiency

In order to compare the efficiency, the inventor tested four state estimators WLS, WLAV, MNMR and COSH-MEAV respectively under normal measurement conditions, among which the latter three are robust state estimators. In the experiment, WLS is solved by Newton's method, and the other three states are estimated by interior point method; and MNMR adopts a two-stage method, that is, WLS is estimated in the first stage, and the estimated value of WLS is used as the initial value of MNMR in the second stage Calculation.

A total of 50 simulation tests were carried out. The number of iterations and the average calculation time when the state estimation converges are shown in Table 2. It can be seen from Table 2 that among the four state estimators, WLS has the highest computational efficiency; among the latter three robust state estimators, COSH-MEAV has the highest computational efficiency; and with the increase of the system scale, COSH -The number of iterations and calculation time of MEAV grows very slowly, so COSH-MEAV is suitable for the estimation of actual large-scale systems.

In summary, the COSH-MEAV proposed by the present invention can effectively suppress multiple bad data including bad consistency data in the estimation process, shows good tolerance, and has high computational efficiency, which is very suitable for in practical engineering applications.

Table 2 The number of iterations and calculation time of the four state estimators

Claims (1)

1. The hyperbolic cosine type maximum exponent absolute value robust estimation method of the state of the electric power system is based on hyperbolic cosine type maximum exponent absolute value robust state estimation with good robust performance and high calculation efficiency; the method comprises the following steps:

a, extracting active power and reactive power injected into a node of a power system, branch active power and reactive power, and a node voltage amplitude parameter; the method for establishing the hyperbolic cosine maximum exponential absolute value robust state estimation basic model comprises the following steps:

wherein: z is equal to R^mMeasuring vectors, including node injection active and reactive power, branch active and reactive power and node voltage amplitude measurement; x is formed by RⁿThe state vector comprises the voltage amplitude of the node and the phase angles of other nodes except the balanced node; h is Rⁿ→R^mIs a non-linear mapping from the state vector to the measurement vector; r is_iIs the ith element of the residual vector r; g (x) Rⁿ→R^cConstraint for zero injection power equality; w is a_iThe weight, σ, measured for the ith quantity₀And σ₁Is a window width parameter;

b, introducing auxiliary variables into the hyperbolic cosine type maximum exponent absolute value robust state estimation basic model, and transforming to obtain a hyperbolic cosine type maximum exponent absolute value robust state estimation equivalent model; introducing non-negative relaxation variables u, v epsilon R^mThe hyperbolic cosine maximum exponential absolute value robust state estimation equivalent model obtained through transformation is as follows:s.t. g (x) 0, z-h (x) -u + v 0, u, v ≧ 0; it is characterized in that the preparation method is characterized in that,

c, solving the hyperbolic cosine maximum exponent absolute value robust state estimation equivalent model by using a primary-dual interior point algorithm; specifically, let x⁽⁰⁾∈RⁿRepresenting a flat start state variable consisting of all node voltage magnitudes and phase angles, with the exception of the reference node phase angle; the step C comprises the following steps:

step C1: introducing Lagrangian functions

In the formula: λ ∈ R^cAnd pi, α e R^mIs a lagrange multiplier vector;

let x be a flat start state variable; selection of lambda⁽⁰⁾＝π⁽⁰⁾0 and u⁽⁰⁾,v⁽⁰⁾,α⁽⁰⁾,β⁽⁰⁾>0, where λ ∈ R^cAnd pi, α e R^mIs a lagrange multiplier vector, m is the number of measurements and c is the number of zero injection power constraints; let the central parameter rho epsilon (0,1) and the convergence criterion epsilon be 10^-3Setting an iteration counter k to be 0;

step C2, calculating the dual Gap (Gap) α^Tv+β^Tu, judging whether convergence occurs or not, if Gap<E, turning to the step C7, otherwise, entering the step C3;

step C3: solving a correction equation to finish the correction of the original variable and the dual variable to obtain [ dx ]^Tdλ^Tdπ^T]^TDv, du, d α and d β, specifically including:

step C31: calculating a disturbance parameter mu which is rho. Gap/2 m;

step C32: forming a Jacobian matrix corresponding to the measurement equation and the zero injection power constraintAndforming a Hessian matrix corresponding to the measurement equation and the zero injection power constraintAndwherein h (x) is the mapping from the state vector to the measurement vector, i.e. the measurement estimation value, and g (x) is 0, i.e. the zero injection power constraint;

step C33: calculating L_x＝G^Tλ-H^Tπ，L_λ＝g(x)，L_π＝z-h(x)-u+v， And

step C34 calculating gamma-z-h (x) -u + v + AA-BB, where AA, BB e R^m，

Step C35: solving equationsTo obtain [ dx ]^Tdλ^Tdπ^T]^T；

Step C36: solving for dv_i＝k_1idπ_i+AA_iAnd du_i＝k_2idπ_i+BB_iWherein k is_1i＝a_iv_i-b_iu_i，k_2i＝c_iv_i-d_iu_i；

Step C37: solving forAnd

step C4: calculating the correction step length theta of the original problem and the dual problem_PAnd theta_DWherein:

step C5: the variables that correct the original problem and the dual problem respectively are:

step C6: let the iteration counter k be k +1, go to step C2;

step C7: and outputting the optimal solution, and ending.