Analysis of Numerical Methods to Include Dynamic Constraints in an Optimal Power Flow Model †
Abstract
:1. Introduction
2. Optimization Model
2.1. Network and Load Modeling
2.2. Modeling of Power Plants in the Optimization Model
2.3. Stability Limits
3. Implementation and Case Study
- A three-phase short-circuit occurs in the transmission line connecting buses 15 and 16—adjacent to bus 16. The fault is cleared by opening the circuit breakers at the two ends of this line after 300 ms.
- A three-phase short-circuit occurs in the transmission line connecting buses 3 and 4—adjacent to bus 3. The fault is cleared by opening the circuit breakers at the two ends of this line after 300 ms.
4. Results and Discussion
4.1. Comparison between Numerical Integration Methods
4.2. Simulation of Other Disturbances
- The power outage of power plant connected to bus 30 which is injecting 800 MW to the grid
- The loss of the load connected to bus 20 which is the largest of the system and consumes 680 MW.
- The loss of the line connecting buses 2 and 3.
4.3. Generalized Theta Method
4.4. Comparison with RK2 and RK4 Runge-Kutta Methods
4.5. Effect of the Use of a Variable Integration Time Step
5. Conclusions
- In terms of convergence, no problems were observed in the studied cases for time steps shorter than 0.05 s. However, the forward Euler method provides results that differ significantly from the reference solution when the integration time step is 0.05 s and fails to converge if larger integration steps are used. As expected, the numerical stability of the forward Euler method is known to be poor compared to implicit integration methods. Although the forward Euler method is viable in the studied case, in other cases numerical stability could be a problem depending on the stiffness of the set of differential-algebraic equations representing the dynamics of the power system.
- In terms of the numerical solution, it has been found that the backward Euler method and, in less degree, the trapezoidal rule, tend to damp the electromechanical oscillations. This effect increases with the length of the integration time step and can lead to dispatches that produce unstable cases in the real world. Operators assessing the stability of a system should be aware of this effect, and perhaps compensate it by introducing a security margin in the dynamic constraints of the optimization model.
- In terms of the speed of convergence, the trapezoidal rule or, more exactly, any generalized Theta method with values of Theta larger than zero and smaller than one, has been found to be faster than the forward and the backward Euler methods.
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Abbreviations | |
COI | Center of Inertia |
GAMS | General Algebraic Modeling System |
OPF | Optimal Power Flow |
PF | Pre-Fault |
PSSE | Power System Simulator for Engineering |
RK2 | 2nd order Runge-Kutta method. |
RK4 | 4th order Runge-Kutta method. |
TD | Time Domain |
TSCOPF | Transient Stability Constrained Optimal Power Flow. |
Indices and sets | |
i,j | Indices for nodes. |
t | Indices for time periods. |
Set of buses of the power system. | |
Set of the synchronous generation units. | |
Set of loads | |
Set of the time periods corresponding to the pre-fault, fault and post fault stages. | |
Set of the time periods corresponding to the fault and post fault stages. | |
Parameters | |
Fuel cost coefficients of the power plants. | |
Active power coefficients of the ZIP load model. | |
Reactive power coefficients of the ZIP load model. | |
Damping coefficient of the power plant [p.u.]. | |
Limits of the field voltage of the synchronous generator [p.u.]. | |
Inertia constant of the power plant [s]. | |
Upper limit of the current in lines and transformers. [p.u.]. | |
Active and reactive nominal load [p.u.]. | |
Active power limits of the generator [p.u.]. | |
Reactive power limits of the generator [p.u.]. | |
Armature resistance [p.u.]. | |
Generator transient time constants[s]. | |
Parameter that defines the numerical integration method. | |
Limits of the bus voltage [p.u.]. | |
Synchronous reactances of the power plant [p.u.]. | |
Transient synchronous reactances of the power plant [p.u.]. | |
Absolut value of the element (i,j) of the bus admittance matrix [p.u.]. | |
Integration time step [s]. | |
Phase of the element (i,j) of the bus admittance matrix [p.u.]. | |
Frequency reference | |
Limit of the rotor angle deviation [rad]. | |
Variables | |
Internal transient voltages of the synchronous generator [p.u.]. | |
Field voltage of the synchronous generator [p.u.]. | |
Output current d-q components of the synchronous generator [p.u.]. | |
Current between nodes (i,j) [p.u.]. | |
Intermediate variables for the calculation of the transient voltage in the d-axis when using the RK4 method | |
Electrical power in the rotor of the synchronous generator [p.u.] | |
Active and reactive power output of the synchronous generator [p.u.]. | |
Active and reactive load | |
Mechanical power input of the power plant [p.u.]. | |
Bus voltage magnitude [p.u.]. | |
Speed deviation of the synchronous generator [rad/s]. | |
Phase of the bus voltage [rad]. | |
Rotor angle [rad]. | |
Rotor angle of the center of inertia [rad]. |
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Network and load modeling | Equality constraints | PF | TD |
Power flow Equations (2) and (3) | Χ | Χ | |
Angle reference (4) | Χ | - | |
Load Equations (6) and (7) | Χ | Χ | |
Currents in branches and transformers (8) | Χ | - | |
Inequality constraints | PF | TD | |
Voltage limits at buses (5) | Χ | - | |
Active and reactive power generation limits (5) | Χ | - | |
Maximum currents through branches and transformers (9) | Χ | - | |
Power plant modeling | Equality constraints | PF | TD |
Discretized differential equations of power plants (14)–(17) | - | Χ | |
Initialization of the differential Equations (18)–(21) | Χ | - | |
Power plant static Equations (22)–(26) | Χ | Χ | |
Inequality constraints | PF | TD | |
Field voltage limits (27) | Χ | Χ | |
Stability Limits | Equality constraints | PF | TD |
Calculation of the center of inertia (28) | Χ | Χ | |
Inequality constraints | PF | TD | |
Rotor angle deviation limits (29) | Χ | Χ |
0.0005 | 0.001 | 0.002 | 0.003 | 0.005 | 0.01 | 0.02 | 0.05 | |
---|---|---|---|---|---|---|---|---|
Equations | 680,071 | 340,071 | 170,071 | 113,291 | 68,071 | 34,071 | 17,071 | 6871 |
Variables | 600,030 | 300,030 | 150,030 | 60,030 | 30,030 | 15,030 | 9930 | 6030 |
Forward Euler | ||||||||
Time (s) | 4211 | 796 | 394 | 188 | 157.6 | 23.48 | 9.95 | 2.84 |
Time/Equations | 6.2 × 10−3 | 2.3 × 10−3 | 2.3 × 10−3 | 1.7 × 10−3 | 2.3 × 10−3 | 7 × 10−4 | 6 × 10−4 | 4 × 10−4 |
Time/Variables | 7 × 10−3 | 2.7 × 10−3 | 2.6 × 10−3 | 1.9 × 10−3 | 2.6 × 10−3 | 8 × 10−4 | 7 × 10−4 | 5 × 10−4 |
Trapezoidal Rule | ||||||||
Time (s) | 3799 | 774 | 184 | 176 | 69.3 | 17.7 | 7.85 | 2.83 |
Time/Equations | 5.6 × 10−3 | 2.3 × 10−3 | 1.1 × 10−3 | 1.6 × 10−3 | 1 × 10−3 | 5 × 10−4 | 4.6 × 10−4 | 4 × 10−4 |
Time/Variables | 6.3 × 10−3 | 2.6 × 10−3 | 1.2 × 10−3 | 1.8 × 10−3 | 1.2 × 10−3 | 5.9 × 10−4 | 5.2 × 10−4 | 4.7 × 10−4 |
Backward Euler | ||||||||
Time (s) | 5739 | 1055 | 537 | 289 | 116 | 18 | 7.97 | 2.94 |
Time/Equations | 8.4 × 10−3 | 3.1 × 10−3 | 3.2 × 10−3 | 2.6 × 10−3 | 1.7 × 10−3 | 5.3 × 10−4 | 4.7 × 10−4 | 4.3 × 10−4 |
Time/Variables | 9.6 × 10−3 | 3.5 × 10−3 | 3.6 × 10−3 | 2.9 × 10−3 | 1.9 × 10−3 | 6 × 10−4 | 5.3 × 10−4 | 4.9 × 10−4 |
Method | Forward Euler | Trapezoidal Rule | Backward Euler | Runge Kutta 2 | Runge Kutta 4 |
---|---|---|---|---|---|
Number of variables | 150,030 | 150,030 | 150,030 | 459,950 | 619,870 |
Number of constraints | 170,071 | 170,071 | 170,071 | 499,991 | 659,911 |
Number of iterations | 50 | 49 | 56 | 55 | 54 |
CPU time (s) | 404 | 347 | 451 | 1463 | 1438 |
Size | Fixed Step | Variable Step | ∆Size (%) | ||
Equations | 68,071 | 51,071 | −24.97 | ||
Variables | 60,030 | 45,030 | −24.99 | ||
Method | Cost (M.U.) | Convergence (s) | Cost (M.U. 1) | Convergence (s) | ∆Convergence (%) |
F. Euler | 43,993 | 157.708 | 43,960 | 32.359 | −79.48 |
T. rule | 43,942 | 69.289 | 43,942 | 29.033 | −58.10 |
B. Euler | 43,893 | 116.569 | 43,925 | 29.947 | −74.31 |
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Arredondo, F.; Castronuovo, E.D.; Ledesma, P.; Leonowicz, Z. Analysis of Numerical Methods to Include Dynamic Constraints in an Optimal Power Flow Model. Energies 2019, 12, 885. https://doi.org/10.3390/en12050885
Arredondo F, Castronuovo ED, Ledesma P, Leonowicz Z. Analysis of Numerical Methods to Include Dynamic Constraints in an Optimal Power Flow Model. Energies. 2019; 12(5):885. https://doi.org/10.3390/en12050885
Chicago/Turabian StyleArredondo, Francisco, Edgardo Daniel Castronuovo, Pablo Ledesma, and Zbigniew Leonowicz. 2019. "Analysis of Numerical Methods to Include Dynamic Constraints in an Optimal Power Flow Model" Energies 12, no. 5: 885. https://doi.org/10.3390/en12050885