KR20080047755A - The optimal power flow algorithm by nonlinear interior point method - Google Patents

Abstract

An optimal power flow algorithm by a nonlinear interior point method is provided to calculate optimal values for parameters of a dynamic model of a system element. An optimal power flow algorithm by a nonlinear interior point method includes the steps of: calculating an initial power flow to determine internal parameters of a dynamic model(S100); initializing variables through primal-dual variables for equilibrium optimization(S110); checking whether the initialized variables are converged to the equilibrium optimization and directly outputting the initialized variables when the initialized variables are converged to the equilibrium optimization(S120); selecting a barrier parameter when the initialized variables are not converged to the equilibrium optimization and indirectly calculating an optimal solution through a KKT(Karush-Kuhn-Tucker) first-order optimal necessary condition(S130); transforming the last two equations of the KKT condition into a Hessian matrix(S140); and calculating an optimal power flow of a power system through correction of the primal-dual variables by applying a Newton's method to calculate a solution satisfying the KKT first-order optimal necessary condition equation and the Hessian matrix equation of the last two equations of the KKT condition(S150).

Description

Power System Equilibrium Point Optimization Algorithm using Non-Linearized Point Method {THE OPTIMAL POWER FLOW ALGORITHM BY NONLINEAR INTERIOR POINT METHOD}

1 is a flow chart of a power system balance point optimization algorithm using a nonlinear visiting method according to the present invention,

2 is a block diagram showing an IEEE DC Type-I exciter model to which a power system equilibrium point optimization algorithm using a nonlinear visiting method according to the present invention is applied;

3 is a block diagram illustrating a simplified governor model to which a power system balance point optimization algorithm using a nonlinear branching method according to the present invention is applied.

4 is an exemplary diagram illustrating a six bus test system to which a power system balance point optimization algorithm using a nonlinear point method according to the present invention is applied.

In the present invention, the devices included in the power system are represented by a model having dynamic response characteristics. The equilibrium point equation for the dynamic model is solved by an optimization formula having a corresponding objective function along with transmission network constraints and operational constraints. In determining the solution of each device, the equilibrium points for the state variables of the dynamic model of the system elements are considered together. It is about.

With the introduction of the principle of competition in the electric power industry, the Korean electric power industry is now undergoing a privatization of the power generation business as a primary restructuring. Electricity is being supplied according to the principle, and the power industry will experience great changes until the two-way bidding, which is expected to be introduced in the future.

  In this environment, it operates with various power generation patterns according to the competition principle at each load level, thus increasing the uncertainty in the system operation. This increases the likelihood of a breach of operating constraints in the event of a system failure. Therefore, one of the key issues in power system operation in the power market environment is to maintain adequate system security that ensures that the system operates under normal and faulty operating constraints. It is closely related to the basic purpose.

To date, no research on the balance point optimization considering the DAE model of the power system has been conducted. The formulation of Trajectory Optimization based on power system model has been proposed, but Trial-and-error Discrete Control Technique based on Simo's method is applied to the decision of solution.

  In addition, a continuous method has been applied to track the path of the equilibrium solution for a given load increase considering the DAE model of the power system.In this regard, the saddle of the power system DAE model including the generator flux-decay model The direct method is proposed to directly obtain the -node bifurcation (SNB) point (the vertex in the continuous method), but it is not an optimization formulation.

  The spot method, which has been applied as a solution for linear programming (LP), has recently been in the spotlight as a solution for many application problems because of its superiority. Although the early papers of the Interior Point Method were published in the early 1950s and were studied in detail by Fiacco and McCormick, afterwards, the major development of the Interior Point Method was not until 1984, when Karmarer published his Polynomial-time Algorithm for LP. Was done.

  The application of Optimal Power Flow (OPF) to the nonlinear visiting method was initiated by Granville. Since then, the Sasaki Group of Japan has developed an optimal algae calculation based on the visit method using perturbed KKT (Karush-Kuhn-Tucker) conditions.

 Optimal Power Flow (OPF) is currently used for this purpose, and this optimal power flow process consists of determining the optimal static state operating point in the power generation-transmission system. It minimizes the selected objective function and satisfies physical constraints (network constraints) and operational constraints.

  However, the optimal bird calculation includes only static grid constraints and does not reflect the dynamic characteristics of the grid elements. Errors caused by these problems are even more severe, especially in line breakdown situations. According to the experience of the applicant, the system frequency is somewhat different from 60 Hz in the system fault state, and there is a problem that there is a significant error with the actual system state when considering only the power current calculation (transmission network) constraints such as the optimal bird calculation.

In the present invention devised to solve the above problems, through the equilibrium optimization of the power system expressed by the differential, logarithmic equation, nonlinear visit method to determine the optimal operating point according to the system operation and control purposes Develop an algorithm that applies. The purpose of this algorithm is to find the optimal solution for the state that is more systematic than the existing optimal algal calculation.

In order to achieve the above object, the power system balance point optimization algorithm using the nonlinear visiting method according to the present invention,

The objective function for minimizing the generation cost of the generator, the functions corresponding to the dynamic model rating equations and the electric network equations, and the inequality constraint functions representing the power system operating constraints are formulated to formulate various contract conditions in the power system. Expressed as a nonlinear optimization model, introduced slack variables to change the inequality constraints to equal constraints, constructs a Lagrangian function, and requires KKT (Karush-Kuhn-Tucker) first order optimization for the Lagrangian function. Construct the condition to find the optimal solution indirectly, construct the Hessian matrix in the last two expressions (L _SL , L _SU ) in the KKT condition, and finally, KKT (Karush-Kuhn-Tucker) 1 for the Lagrangian function from the given initial point. Find a solution that satisfies the Hessian matrix equation for the second best requirement equation and the last two equations (L _SL , L _SU ) under KKT Newton's method is applied to achieve this by modifying the main dual variables to find the optimal algal calculation in the power system.

Hereinafter, with reference to the accompanying drawings, the configuration and operation of the embodiment of the present invention will be described in detail.

First, the optimal algae calculation and balance point optimization algorithm according to the present invention will be described.

With the introduction of the principle of competition in the electric power industry, the Korean electric power industry is now undergoing a privatization of the power generation business as a primary restructuring. Electricity is being supplied according to the principle, and the electric power industry will experience great changes until the two-way bidding, which is expected to be introduced in the future.

  In this environment, it operates with various power generation patterns according to the competition principle at each load level, thus increasing the uncertainty in the system operation. This increases the likelihood of a breach of operating constraints in the event of a system failure. Therefore, one of the key issues in power system operation in the power market environment is to maintain adequate system security that ensures that the system operates under normal and faulty operating constraints. It is closely related to the basic purpose.

  Optimal Power Flow (OPF) is currently used for this purpose, and this OPF process consists of determining the optimal static operating point in the power generation-transmission system. This method minimizes the selected objective function and satisfies physical constraints (network constraints) and operational constraints.

  However, the optimal bird calculation includes only static grid constraints and does not reflect the dynamic characteristics of the grid elements. Errors caused by these problems are even more severe, especially in line breakdown situations. In the experience of the present inventors, in the system failure state, the system frequency is deviated to some extent from 60 Hz, and there may be a significant error with the actual system state when considering only the power current calculation (transmission network) constraints such as the optimum bird calculation.

  In the present invention, by including the dynamic models representing the grid elements in the optimization problem along with the transmission network constraints, we propose an algorithm that obtains an optimal solution closer to the actual system phenomena in the event of a system failure according to a given objective function.

In the present invention, the algorithm is called equilibrium optimization (EOPT).

  The equilibrium point optimization algorithm may be represented by a nonlinear optimization model as shown in Equation 1 below.

Here, f (·) denotes an objective function, and hD (·) and hA (·) denote functions corresponding to the equilibrium point equation and the electric network equation of the dynamic model, respectively. In the equation (1), g (·) represents an inequality constraint function representing a power system operating constraint. g min and g max are the vectors corresponding to the upper and lower limits for g . x and y are vectors representing state variables and algebraic variables, respectively, and u and p represent control variables and parameters, respectively.

Subsequently, in order to solve the optimization problem according to the present invention, a slack variable (s _L , s _U ) is first introduced to convert an inequality constraint into an equality constraint, and then a Lagrangian function is constructed as in Equation 2 below.

Here, λ _D and λ _A represent the Lagrangian multipliers of the constraints on the differential equation and the logarithmic equation among the equal signs. π _L and π _U are the Lagrangian multipliers for the function that converts the inequality constraint to the equality constraint, respectively. And μ corresponds to the log barrier parameter, allowing the solution to be determined within the execution domain.

  The visit method finds an optimal solution indirectly by constructing the KKT (Karush-Kuhn-Tucker) first order optimal condition for the Lagrangian function as shown in Equation 3 below.

  In the above KKT condition, the last two expressions are difficult to handle and are transformed as shown in Equation 4 below.

Finally, Newton's method is applied to obtain a solution satisfying Equations 3 and 4 from a given initial point, and a modified equation corresponding thereto may be expressed as Equation 5 below.

Hereinafter, the present invention will be described by applying a synchronous generator two-axis model, an exciter and a simplified governor model, a ULTC model, and a static / dynamic load model through a power system balance point optimization algorithm using a nonlinear point method according to the present invention.

First, the balance point optimization of the dynamic model for the power system can be summarized as follows.

First, the model included in the balance point optimization formulation is defined as the generator two-axis model, the basic exciter model, the simplified governor model, and the load model.

Second, we use the main-twin nonlinear visiting method (Jacobian and Hessian matrix), which is a nonlinear optimization technique applied to formulation.

Third, set an objective function that minimizes control parameter deviation.

Through the content of the balance point optimization of the DAE model for the power system will be described the manufacturing process of the balance point optimization program based on the non-linear point method according to the present invention.

First, an initial algae calculation for determining a dynamic model internal parameter is performed (S100).

At this time, network data, dynamic data, assumed accident data on a real power system, and the like are input.

Here, the dynamic model data including the balance point optimization (EOPT) is as follows.

First, the generator two-axis model is expressed as in Equation 6.

Next, the IEEE DC Type-I exciter model shown in FIG. 2 is expressed by Equation 7.

Next, the simplified governor model shown in FIG. 3 is represented by Equation (8).

The load model is represented by Equation (9).

Subsequently, the variable is initialized through the main dual variable for the balance point optimization (S110).

It is checked whether the variables initialized as described above converge to the balance point optimization (S120).

If converged to the balance point optimization, the output is immediately output. If not, the barrier parameter is selected.

Subsequently, a Krush (Karush-Kuhn-Tucker) first optimal requirement is constructed to indirectly obtain an optimal solution (S130).

Subsequently, the final two expressions (LSL and LSU) are transformed into a Hessian matrix under KKT conditions (S140).

Subsequently, the Newton's method is applied to find a solution that satisfies the Karush-Kuhn-Tucker (KKT) first order optimal requirement equation and the Hessian matrix equation for the last two expressions (LSL, LSU) in the KKT condition. By correcting the optimal algae on the power system is calculated (S150).

Hereinafter, in order to explain the usefulness of the power system balance point optimization algorithm using the nonlinear spot method according to the present invention, the results of applying the present invention to a simple 6 bus test system will be described as an example. .

Network and dynamic model data applied to the present invention is as shown in Figs. The dynamic model data includes the generator 2-axis model of Equation 6, the exciter model of Equation 7, and the governor model parameters of Equation 8 described above.

Basic bird calculation data Mothership (Bus #) Volt [pu] Angle [degree] P _L [MW] Q _L [MVAr] PG [MW] QG [MVAr] One 1.0500 0.00 · · 96.98 62.39 2 1.0000 -0.42 · · 50.00 7.43 3 0.9510 -13.27 55.00 13.00 · · 4 0.8926 -9.98 · · · · 5 0.8478 -12.52 30.00 18.00 · · 6 0.8753 -12.43 50.00 5.00 · ·

Generator data (including exciter and governor models) Generator (Gen #) Xd Xq Xd ' Xq ' Rs Tdo ' Tqo ' One 0.995 0.568 0.195 0.568 0 10.8 0.5 2 0.619 0.134 0.238 0.365 0 7.4 0.5 Generator (Gen #) D H MVA Ke Te Se Ka One 0 6.41 200 One 0.25 0.1 20 2 0 3.8 113 One 0.25 0.1 20 Generator (Gen #) Ta Kf Tf Tch Tg Rg Vrmax One 0.06 0.04 One 1.6 0.2 0.05 3 2 0.06 0.04 One 1.6 0.2 0.05 3

Also, the effective load multipliers α and β corresponding to the load model parameters were all zeros and the constant power load model was used.

The objective function of the applied equilibrium point optimization is as shown in Equation 10 below.

는 EOPT 알고리즘 수행전의 초기값을 의미한다. P _gsi in Equation 10 represents an active power designation value of the governor model of the i th generator,

Means the initial value before executing the EOPT algorithm

는 그 초기값을 나타낸다. V _refi corresponds to the generator terminal voltage specified in the exciter model included in the i th generator,

Indicates its initial value.

W _Pgsi and W _Vrefi mean weights for the amount of change from the initial values of P _gsi and V _refi , respectively.

As shown in Table 1, since the initial voltage of buses 4, 5, and 6 is less than 0.9 pu, in the embodiment of the present invention, first, the reactive power dispatch problem is applied so that all bus voltages are within the range of 0.9 pu to 1.1 pu. In order to exist at, the optimum value of Vref corresponding to the reference voltage of the generator bus voltage of the exciter is obtained through EOPT. In the previous objective function, W _Vrefi corresponding to the weight of the second component is _set to 100.0. The other parameters related to the objective function are listed in Table 3 below.

Objective function parameter when applying active power dispatch problem Generator (Gen #)

One 0.9698 0.9598 0.9798 1.127693 1.0 1.2 2 0.5000 0.4900 0.5100 1.058023 1.0 1.2

In Table 3 above, the initial value and the upper and lower limit values of Pgs corresponding to the parameters of the governor model are included. The EOPT problem includes system frequency constraints, which range from 0.999 pu to 1.001 pu. In order to find the optimal solution, the EOPT was repeated 11 times, and the optimal solution was determined such that the complementary gap of constraints and the mismatch of network equality constraints were in the range of 10-5 and 10-4, respectively. In the 6 bus system, there are two generators, and the optimum solution for the Vref is 1.16671285 pu for the generator of bus 1 and 1.1023135 pu for the generator of bus 2. It is a slightly higher value than the initial value of.

Table 4, below, shows the network data for the optimal solution of the reactive power dispatch problem. From this, we can see that the proper solution is determined as the voltage of all buses is 0.9 pu or more. Due to the EOPT result, the voltage angle of the slack bus (bus 1) is no longer 0, and the rotor angle of bus 2, which corresponds to the reference generator phase angle, is the reference angle of the EOPT formula.

Network data for optimal solution of reactive power dispatch problem Mothership (Bus #) Volt [pu] Angle [degree] P _L [MW] Q _L [MVAr] P _G [MW] Q _G [MVAr] One 1.0907 1.74 · · 96.98 50.36 2 0.0411 -0.03 · · 48.07 11.51 3 0.9390 -10.77 55.00 13.00 · · 4 0.9633 -7.42 · · · · 5 0.9000 -10.13 30.00 18.00 · · 6 0.9426 -9.72 50.00 5.00 · ·

In addition, the system frequency of the optimal solution determined through the execution of EOPT is 1.0004988 pu, and the system frequency increases slightly when controlling the Vref value to the value corresponding to the optimal solution. Exists in

Next, an example of application of the EOPT program to the active power dispatch problem using the parameter Ggs of the governor will be described.

Table 5 shows the parameters related to the objective function used in the active power dispatch problem. Unlike the reactive power problem, it has a wider upper and lower range of Pgs, and the constraint of the exciter parameter Vref is reduced.

Objective function parameter when applying active power dispatch problem Generator (Gen #)

One 0.9698 0.6000 1.5000 1.127693 1.166 1.168 2 0.5000 0.3000 1.0000 1.058023 1.102 1.104

Table 6 shows the upper and lower limits of the active power currents for the three lines selected in the embodiment of the present invention.

Line tide constraint applied when applying active power dispatch problem Line Fmin [MW] Fmax [MW] Initial flow [MW] 1-4 -40 40 50.8 1-6 -40 40 44.5 2-5 -50 50 31.9

In the embodiment of the present invention, the constraint of the system frequency of the EOPT is set to 0.999 to 1.001 pu. The optimal solution was obtained after 15 iterations using EOPT, and the optimal solution satisfies the line bird constraint of Table 6. In the optimal solution, the Pgs of generators 1 and 2 are 0.727692 pu and 0.742784 pu, respectively. The system frequency was determined as the lower limit and calculated as 0.999 pu.

 Table 7 shows the network for the optimal solution of the active power dispatch problem. Compared with the initial solution in Table 1, reactive power generation does not change much, while active power generation amounts to 74.77 MW and 76.28 MW. We can see that we are out of the sun

Network data for optimal solution of reactive power dispatch problem Mothership (Bus #) Volt [pu] Angle [degree] P _L [MW] Q _L [MVAr] PG [MW] QG [MVAr] One 1.0912 -13.19 · · 74.77 58.93 2 1.0417 -2.51 · · 76.28 11.76 3 0.9133 -22.58 55.00 13.00 · · 4 0.9488 -19.83 · · · · 5 0.8665 -19.22 30.00 18.00 · · 6 0.9298 -21.46 50.00 5.00 · ·

As described above, in the present invention, the optimal algal calculation currently applied for systematic analysis includes only static systematic equations, thereby improving the fact that the dynamic characteristics of the system elements are not reflected and developed through the present invention. By applying the equilibrium balance optimization, it is a good effect to find the optimal operating point to get closer to the actual system phenomenon.

In addition, through the present invention it is possible to obtain the optimal value for each parameter of the dynamic model of the system element, and to secure the basic technology to consider the dynamic stability constraints in terms of optimization, it is possible to present an effective control plan during operation This will contribute to the efficient operation of the power market. It also has a good effect that can be used for various purposes in the system operation and planning stage.

Claims (1)

The objective function for minimizing the generation cost of the generator, the functions corresponding to the dynamic model rating equations and the electric network equations, and the inequality constraint functions representing the power system operating constraints are formulated to formulate various contract conditions in the power system. Expressed as a nonlinear optimization model, introduced slack variables to change the inequality constraints to equal constraints, constructs a Lagrangian function, and requires KKT (Karush-Kuhn-Tucker) first order optimization for the Lagrangian function. Construct the condition to find the optimal solution indirectly, construct the Hessian matrix in the last two expressions (L _SL , L _SU ) in the KKT condition, and finally, KKT (Karush-Kuhn-Tucker) 1 for the Lagrangian function from the given initial point. Find a solution that satisfies the Hessian matrix equation for the second best requirement equation and the last two equations (L _SL , L _SU ) under KKT Power system equilibrium point optimization algorithm using a nonlinear visiting method characterized in that the Newton's method is applied to the main dual variable is modified to enable the optimal algae calculation in the power system.

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