Optimization Strategy for Analysis of Power System Problems

Copyright © IFAC Control of Power Systems and Power Plants, Beijing, China, 1997

OPTIMIZATION STRATEGY FOR ANALYSIS OF POWER SYSTEM PROBLEMS Xianzhang Lei and Dusan Povh

Power Transmission and Distribution Group Siemens AG P. 0. Box 3220, D-9J050 Erlangen, Germany Email: [email protected]

Abstract: This paper presents an optimization strategy implemented in a simulation program. The motivation is to provide a flexible and efficient way for analysis of various problems arising in power systems. The most attractive features of this strategy are that 1). free formulation of an objective function and all constraints considered using the block-oriented simulation language, and 2). performance of various optimizations without modifications to the source program. Three improved optimization algorithms have been implemented and have shown efficient and robust in practical applications. Capabilities of the optimization tool implemented are demonstrated in this paper by solving four optimization problems arising in power systems. Copyright © 1998 IFAC Key Words: Optimization, identification, system analysis, simulation, dynamics, control.

1. INTRODUCTION Optimization I Identification :t i

Nowadays, power systems are growing rapidly and become more and more complex. In order to ensure operation safety, analyses of steady state and dynamics problems arising in public and industrial power systems are becoming imperative, for example, power flow analysis, dynamics of generators and behaviours of FACTS elements and other network elements after a fault event or transient stability etc ..

I[ • Simulation

IObjective function

~

operating points ~ loadftow

I .,stantaneous value mode us

ms

operating points

linearization !

:I stability mode . sec

I

measurements

•

frequency mode eigenvalues

m,n

I

Hz

Istress of equipment I stability analysis Idesign of controllers I

With the help of computation techniques, almost all problems mentioned above can be analyzed by simulating operation behaviours of the system involved. However, more important tasks in engineering aspects might be to achieve an optimal system performance under the given conditions through, for instance, minimizing power loses in steady state and quickly damping power swings to ensure transient stability etc ..

Fig. 1 Simulation system NETOMAC To solve this kind of problems, the optmuzation strategy has been implemented into the Siemens simulation program system NETOMAC ~twork Torsion Machine ~ontrol) . This program has been widely used for the simulation of electromechanical

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and electromagnetic transient phenomena as well as steady-state behaviors of a power system either in frequency or time domain (Kulicke, 1979). It contains transient mode calculation similar to the EMTP program and includes a full stability program mode under the same surface too. As illustrated in Fig. I, a broad range of tasks ari~ing in power systems can be performed by the program system. They are roughly summarized into three groups as follows: •

Calculation of transient values in order to study medium and high frequency transients, including HYDC transmission and FACTS as SVC, TCSC or PAR and UPFC,

•

Calculation of rms values for study system stability, including HYDC transmission and FACTS,

•

Calculation in the frequency domain to study resonance and stability of system, machine and control.

minimum by a quadratic function. The key feature of this algorithm is that a quadratic convergence can be achieved only using the information from the fIrstorder derivatives. The algorithm becomes efficient and robust mainly due to adaptive scaling, handling the bound constraints directly with an efficient quadratic programming approach etc. A typical convergence difficulty is ill-conditioned problem caused by some variables varying greatly in magnitude during the minimization process. To deal with this problem, we have proposed here a dynamic scaling strategy, in which the adjustment of the scaling is heuristic in the consideration of the convergence of the minimization. Once there is a change in variable with a quick convergence, the scaling will be adjusted based on the current operating point to in order adapt the minimization. This results in a better dynamic performance of the minimization, even if variables change greatly.

The implementation of the optimization strategy into this program system allows a general representation of non linear optimization problems with almost all kinds of constraints and offers users a flexible tool for power system analyses to achieve optimal system performances.

To handle bound constrained problems which very often arise in power system optimization tasks, many algorithms are time-consuming. Our idea is based on quadratic programming. In the minimization process, if any bound constraint becomes active at an iteration, we solve a quadratic programming problem (QPP) to determine a next feasible direction of search, then obtain a new feasible solution along this search direction with a line search. Otherwise, we search along the quasi-Newton. The most attractive feature of this strategy is that the algorithm allows several constraints to be dropped simultaneously by solving a quadratic programming problem, so that the algorithm is fast. For solving quadratic programming, we implemented in the program a dual algorithm which is quick and stable.

2. OPTIMZATION ALGORITHMS

Due to complexity of power systems, most problems to be solved are non-linear problems with high orders and multi-local minima. Such phenomena pose difficulties to solution procedures: there is no such a algorithm which might be efficient and robust for any kind of problems. Currently, three optimization algorithms have been implemented in the program system. They are quasi-Newton, conjugate direction and least-squares algorithm (Lei, 1993; Kulicke and Hinrichs, 1988). The fIrst two algorithms are used to solve general constrained optimization problems. The least-squares algorithm is developed for identifIcation problems. Because identifIcation problems can be formulated as a sum of the squares, the fIrst two algorithms can also be used to this kind of problems. Through combing different algorithms, the optimization mode can flexibly solve various optimization problems under consideration different operation and security constraints and fmd out an optimal solutions. To avoid getting lost in details, this paper will only present basic ideas in the algorithms. Detailed descriptions are given in (Lei, 1993; Kulicke and Hinrichs, 1988).

For handling functional constraints, a modifIed augmented Lagrange multiplier method has been proposed. With this method, an unconstrained Lagrange function will be formed with the original objective function and all related functional constraints. The modifIcation aims at adjusting the multiplier and the penalty factor in a suitable frequency by which the state of the constraint functions is considered instead of blind adjustment in order to avoid numerical infmity. This is achieved by employing a maximal constraint error which respects the magnitude of the most violating constraint term. If the minimization process fails to reduce this error, the penalty factor is increased. Otherwise with unchanged penalty factor, the multiplier is updated.

2. 1 Modified quasi-Newton algorithm

2.2 Modified Conjugate Direction Algorithm

The quasi-Newton technique assumes that the objective function can be adequately approximated near its

In practice, there are some optimization problems whose gradients may not be available. For solving

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such problems, the second algorithm without calculation of the gradients has been implemented in the program system as supplement to the quasi-Newton algorithm developed.

The Implementation of three different algorithms described above has an advantage in that, if one algorithm fails in convergence, another one can be applied for further minimization, starting from the current operating point, until an acceptable solution is achieved.

The algorithm is based on Powell's conjugate direction method (Fletcher, 1987) which has a quadratic convergent property. The basic concept of this algorithm is to search in n (number of the variables) orthogonal directions fIrst, and then create a new search direction by connecting the fIrst and last operating point. The most serious problem of this algorithm is that the search directions can become linearly dependent on account of a large number of variables or numerical operating. To overcome this weakness, we use the resultant step length uk * obtained by line search as a criterion to check the linear independence of the search directions. If a line search failed along a search direction with step length Uk = 0 or very small step length, then subsequent search directions will not be conjugate and may become linearly dependent. In this case, we replace this search direction with a new direction created. This modifIcation preserves the conjugate property of the search directions such that the convergent speed and reliability are improved.

3. OPTIMIZATION WITH THE SIMULATION PROGRAM

Three optimization algorithms have been implemented as a special mode in the simulation program system. In this program system, all aspects of the representation, solution and initialization as well as stopping of the optimization considered are under users control. Figure 2 illustrates briefly the principle of optimization processing with the simulation program system NETOMAC. NETOMAC

2.3 Least Squares Algorithm

Among power system optimization problems, there are a large number of identifIcation problems, whose objective function is usually defmed as a sum of the squares of the nonlinear functions. In practice, such problems can be solved using general purposeoriented optimization algorithms, but more effIcient would be Gauss-Newton techniques and their variants.

Fig. 2. Optimization procedure There are two very important features of the optimization with the program system. They are: l.) objective functions and constraints can be freely formulated in simple forms. Users do not need to establish explicitly the detailed mathematical relationship between all variables involved. This will be done automatically intern the program system; 2.) the system enables a free representation of optimization problems. Users only need to model the corresponding simulation system and describe the objective function as well as defme the state variables to be optimized in a input fIle. Any modifIcation to the source program is here not needed. Due to all special properties mentioned above the optimization mode in NETOMAC is very attractive in the practical applications: they offer the users a simple and flexible way to implement their practical application with various purposes.

In the program system, an improved least squares

algorithm has been implemented (Kulicke and Hinrichs, 1988). This algorithm is based on GaussNewton techniques. The main features are reducing automatically the order of a system, recognizing linear independence and the variable sensitivity analysis. Assuming J is the lacobian matrix, a nonsingular matrix JTJ will provide a descent direction for the next search. Such a non-singular matrix JTJ is built by means of variable sensitivity analysis. The partial derivatives with respect to non-dominant variables are removed if they might cause a singular matrix JTJ. In the next iteration, the non-dominant variables with removed partial derivatives will be maintained constant and all dominant variables whose partial derivatives remain in lacobian matrix J will move along a descent direction to a new operating point. Because this modifIcation guarantees that the algorithm will still take descent steps, the algorithm is robust.

4. FLEXIBLE OPTIMIZATION

As examples, the optimization are demonstrated on following problems which are in the aspect of control system design, power flow analysis and parameter identifIcation respectively.

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H = 3.55 Sr= 1200 MVA

2

6QAE) dt

(1)

~:::rr

B

A

2

o where Kp and K, as well as Td are used as variables to be optimized and Kw is a weight factor for reactive power oscillation. Fig. 4 shows the active power flow between node A and E. A comparison of the active power flow before and after the optimization demonstrates the effectiveness of the TCSC with the controller optimized in damping power oscillation. Note that it is also possible to achieve the maximum oscillation damping capability of the system through optimizing all controllers of the FACTS elements (e.g. TCSC, SVC, etc.) in the system simultaneously using different locally measured signals.

In recent years thyristor-controlled series compensation TCSC has been introduced to effectively damp power swings and increase the transmission capability of a power system. The following example demonstrates the optimal employment of the TCSC in a 3-area meshed system depicted in Fig. 3 (all data are given in (Lerch and Li, 1994). The series compensation is varied from 25% and 50% of the line impedance of the line A-E, controlled by dP AE/dt, where PAE is active power flow between node A and E. H = 65 Sr= 6000 MVA

t

= f (M'AE + Kw

min Z

4.1 Optimization ofa controlled series compensation for power oscillation damping

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Fig. 4. System dynamics with and without the optimization of the TCSC controller

-1 TCSC Controller

4.2 Minimization ofpower losses

Fig. 3.400 kV meshed network with a TCSC and a SVC

As the second application example, the optimization was performed for minimizing the power loss on the 110 kV side of a 110 / 60 kV -Test power system given in Fig. 5. The solution of the given problem should be achieved through optimal adjustment of the active and reactive power flow (PUW1 , QUWJ) at the slack node UWI as well as the transformer tap (VTH) on the 110 kV side. By solving this problem, the voltage constraints at nodes, such as at node UWI and UW2, must be simultaneously taken into account.

The TCSC is active only during the fIrst few second. The TCSC controller used is of the PI type, whose proportional and integral gains are Kp and KJ• To obtain a better damping capability of the system, beside the employment of PS Ss in each subsystem and a SVC in the system, the TCSC control system is to be optimized. In case of the optimization of the TCSC controller, a defmed fault, a 100 ms 3-phase short-circuit at location "a" in one of the lines between node E-F (critical fault time until system instability is 265 ms) is taken. The optimal TCSC controller is obtained through minimizing the objective function

The details of the objective function in regard to the complete description of relationship between power flows and voltages are not particularly relevant to this paper since only the applicability of the optimization program is to be shown. Thus, the correspond-

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ing objective function for this problem can be defmed in a simplified form as follows min f{PUW1 ' QUW1' VTH } = (Pg - p,)2 Vmin(i) ~

subject to

V(i) ~ VmaX(i) ,

shows the structure of this typical voltage control system. In case of a disturbance, the terminal voltage of the generator is stabilized by adjusting the generator exciting voltage EFD . The control signal is provided by the controlling error defmed as the difference between the reference and actual value of the terminal voltage of the generator as follows:

(2)

; = 1, .,2. , ... ,m

where Pg, Qg are total injection of the generators connected to the 110 kV bus, PI , QI are total load of the 110 kV bus in MW and MY AR, Vi 0= I ,.2, ...,m) are voltages at nodes considered respectively.

(3)

In case of loss of one transmission line (a 100 ms long three-phase line-to-earth fault at point A in Fig. 6) the standard settings of the control gain destabilizes the system. By this fault two voltage control systems were optimized simultaneously.

The objective function is defmed as the sum of the integration for the quadratic controlling error ei of each generator. That means that over a certain time the quadratic error ei is to be minimized. For the twomachine system, the corresponding objective function is formulated as follows t

f

min Z = (ef + e~ ) dt o

(4)

Altogether 6 parameters are optimized, namely, the derivative times TFI , TF2 , the controller gains KFI> KF2 as well as the time constants TAl, T A2, where the index 1 and 2 indicate the generator GEN I and GEN2 respectively. The optimization adjusts parameter settings of each controller with different values. Through minimizing the objective function (4), the system is stabilized. The terminal voltage curves of the generators GENI and GEN2, each before and after optimization with 18 iterations, are given in Fig. 8 and in Fig. 9 respectively.

UW5 -(11 .7 + j 5.667) (P in MW, Q In MVAR)

Fig. 5. 110 / 60 kV - Test power system The minimization process began by the initial power loss PLQSS(O) = 78.6 MW. After 19 iterations the minimization converged to the minimal value of the power loss at PLOSS* = 4.9 MW, where all voltage limits at the nodes considered were also satisfied.

4.3 Optimizing Voltage Controllers o/Generators With the optimization tool, all generator voltage controllers of a system can be optimized simultaneously. The third example will demonstrate how it works, here only a simplified system of two generators is taken into account (Fig. 6).

GEN1

A}

®mh . !

Icontroller I

-

~

380/11kV

UMSOKm

Fig. 7. Structure of the voltage control system

GEN2

I I~ • Icontroller I 0 .00 L' _ _ _ _ _ _ _ __

Fig. 6. 380 / 11 kV - Two-machines power system

0 .0

1 .0

2 .0

3 .0

nm.

The structure of the voltage control system of the two machines is assumed to be the same. The parameter values of the controllers are standard settings. Fig. 7

_

_ __ _ __ 4 .0

5 .0

(MIC..)

Fig. 8. Terminal voltage curves before optimization - - Generator GENI, - - - Generator GEN2

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e .o

1_ ____________________________ ~

such as coordination of power equipment, control and also protection problems can be effectively solved.

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650

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600

- - - identified curve - - - - - initial curve - - - measured curve

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Fig. 9. Terminal voltage curves after optimization - - Generator GEN 1, •• - Generator GEN2

500+---~----~--~----~--~~--~

o

As the fourth example (an identification task), parameters of a steam turbine-generator model with a speed controller were identified by the use of measured response of mechanical power of the turbine. A simple model is illustrated in Fig. 10, where the elements whose parameters will be identified are marked in dark color. They are an integral which represents the valve and two first order lag elements which represent the high pressure part and a summary of the intermediate and the low pressure parts of the turbine respectively. The objective function is defmed as a sum of error squares between the modeled and the measured mechanical power: .

2

t

8

10

Fig. 11 . Result of the identification

4.4 Identification ofparameters of a turbine speed controller

mm Z = J(mmeas -mmodel) dt

2 time 4 (sec.J3

5. CONCLUSIONS An optimization strategy in a simulation program has been presented. Several modifications have been made to enhance the effectiveness and the reliability of optimization performance. Implementing different optimization algorithms offers users a great flexibility to perform various optimization problems concerned with the power system dynamics and control because of: I) a free formulation of an objective function and all related constraints of a linear or nonlinear system considered using the block-oriented simulation language, and 2) the performance of optimizations without modifications to the source program. Three optimization tasks as examples were performed with desired results.

(5)

o

where mm... and mmodel are curves of the measured and mode led mechanical power respectively. 6. REFERENCE speed contToller

steam

pftSSU~

Kulicke, B. (1979). NETOMAC digital program for simulating electromechanical and electromagnetic transient phenomena in a.c. systems, (in German) Elektrizitatwirtschaft, Heft 1, pp. 18 23 Lei, X. (1993). Robust optimization of problems in power systems with a simulation program, (in German), ISBN-3-86111-348-1, Verlag Shaker, Germany.

Fig. 10. A simple model of a steam turbine block Fig. 11 illustrates the result of the identification. It is shown clearly that as the result the difference between the identified and measured curves is very small.

Kulicke, B. and H. Hinrichs (1988). Parameteridentification und Ordnungsreduction mit Hilfe des Simulationsprogrammes NETOMC, (in German) etzArchiev Bd. 10. H. 7. pp. 207 - 213

In solving real problems, many optimizations with a large number of variables have been also performed. These applications demonstrated the efficiency and flexibility of the optimization strategy implemented in the simulation program. It is worthwhile to be pointed out that by means of the optimization strategy many problems arising in power engineering,

Fletcher, R. (1987). Practical Methods for Optimization, John Wiley and Sons. Lerch, E. and W. Li (1994). Application of FACTSElements in Meshed Systems, ICPST'94 Beijing, pp. 963 - 969

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