Abstract: This study presents a critical evaluation of maximum loadability point estimation in a power system network. The critical point estimation involved maximum loadability estimation on individual load buses and simultaneous increase at several load buses. Different modes of evaluation determine the margin of load increase between the single and several loads. To obtain the optimum load values prior to voltage instability occurrence Evolutionary Programming (EP) technique is introduced. IEEE (30-Bus) Reliability Test System (RTS) was adopted for validation purposes, while comparative studies were performed with other techniques such as Artificial Immune System (AIS) and Automatic Voltage Stability Analysis (AVSA) to highlight the merit of EP.
INTRODUCTION
Power system growth has resulted in the increase of power system plant which needs to cope with the high demand. This has caused the power system to be more complex in the future. Power systems are becoming heavily stressed due to the difficulty in constructing new transmission systems as well as the complexity of building new generating plants near the load centres. Most of the load demands are in form of reactive power effect such as heavy machines confining windings, transformers and other elements rather than the real power effect.
There have been a number of incidents in the past few years which were diagnosed as voltage instability problem due to the increase in loading and decrement of stability margin. The stability margin can be defined as the distance between the base loading of the system and the maximum loading limit of the system. The contingencies occur in the system would lead to the decrease in stability margin and the system approaches a very critical stage, which may lead the system to a total collapse. Various techniques were reported in the literature to identify and estimate the maximum loadability (Musirin and Rahman, 2002a; Rahman and Jasmon, 1997; S-Yome et al., 2006; Kundur et al., 2004; Morison et al., 1993) to indicate its importance in power system studies.
One of the conventional techniques is the repetitive power flow in which load was increased in steps until load flow diverges (Semlyen et al., 1991; Obadina and Berg, 1998). At this point, it was assumed that the system is at its maximum loading point prior to the system collapse. The inaccuracy in determining the maximum loadability point through the conventional technique has been identified as the setback of the technique.
This technique estimates the maximum loadability through the implementation of automatic voltage stability assessment in estimating the maximum. Previous work on determining maximum loadability was proposed by Musirin and Rahman (2002a) using an index termed as Fast Voltage Stability Index (FVSI). This index is based on the evaluation of transmission line indices interconnected among buses in the system. Determination of Point of Collapse (POC) in the load flow studies can also be conducted using Automatic Voltage Stability Assessment (AVSA) as reported by Musirin et al. (2005a). However, this technique has the demerit in terms of inaccuracy of the collapse point. The implementation of optimization process could help identifying the accurate point of collapse.
One of the popular techniques and fast search techniques is by using the Artificial Intelligence (AI) search techniques. Musirin and Rahman (2002b) developed a new algorithm to execute the Evolutionary Programming (EP) based optimization technique for estimating maximum loadability or critical loading condition in power system for one load bus. Other optimization techniques which can also perform similar task are linear programming, Genetic Algorithm, quadratic programming, Ant Colony Optimization (ACO) (Kalil et al., 2002) and Artificial Immune System (AIS) (Musirin et al., 2005b).
This study presents the application of EP technique for searching single and multi-load optimal points critical loading condition utilizing a pre-developed voltage stability index as the measuring instrument. In this study, optimization engines for identifying single point maximum loadability and load increase at several buses were developed separately. This technique can assist the power system operators to plan and study the system capability in terms of incremental of loads in simultaneously. Comparative studies were performed with respect to AIS and AVSA. Results had indicated the merit of the proposed technique.
VOLTAGE STABILITY ASSESSMENT
Voltage stability is defined as the ability of a system to maintain its equilibrium condition when it is subjected to a disturbance. A system enters a state of voltage instability when a disturbance, increase in load demand, or change in system condition causes a progressive and uncontrollable decline in voltage (Kundur et al., 2004). The main factor which has profoundly caused instability condition is the constraint in reactive power support. Voltage stability problems normally occur in heavily stressed systems. While the disturbance leading to voltage collapse may be initiated by various causes, the underlying problem is an inherent weakness in the power system.
To ensure boundary of system in reliability and security; the incremental loads have to be monitored closely. The loads can be increased individually and also simultaneously for several chosen loads. Maximum loadability is one aspect which determines the load limit of a system prior to system instability. The determination of maximum loadability of one load or several chosen loads can be assessed by voltage stability analysis. This will require optimization technique to search the optimal point which may require an indicator. In this study a line-based voltage stability index termed as Fast Voltage Stability Index (FVSI) is utilised as the indicator.
The formulation of FVSI forms in as shown:
| (1) |
Where:
| Zij | = | Line impedance |
| Xij | = | Line reactance |
| Vij | = | Voltage at the sending end |
| Qj | = | Reactive power at the receiving end |
FVSI was developed by Musirin and Rahman (2002b) which could determine the voltage stability condition of all lines in a power system. This index has a range between 0 at no load and 1.0 at instability condition. To indicate voltage instability of the whole system, the maximum FVSI value for the system is indicative enough to imply the situation.
ALGORITHMS FOR MAXIMUM LOADING IDENTIFICATION
Identification of load flow analysis is used for searching the maximum loadability point of a particular load bus and also to compute the values of FVSI. The following procedures were implemented to identify the maximum loadability point in power system:
Choose load bus for the test.
| • | Run voltage stability analysis |
| • | Evaluate the FVSI values for all lines in the system using the load flow solution |
| • | Monitor the highest FVSI value for the system |
| • | If maximum FVSI is less than 0.95; increase the load at the selected bus and go to step (ii), otherwise go to (iv) |
| • | Record the loading conditions of the chosen load bus |
| • | For other load buses, repeat steps (i) to (vi). These are the maximum permissible load at the buses increased concurrently prior to system instability |
The steps described above are the conventional techniques which are computationally burdensome since heuristic technique is involved. In order to reduce the computation burden and to achieve more accurate results, optimization technique could be an effective technique. In this study Evolutionary Programming (EP) is proposed to alleviate the setback of the existing technique.
EVOLUTIONARY PROGRAMMING
Evolutionary Programming (EP) is a stochastic optimization technique based on the natural generation. It was invented by D. Fogel in 1962 and further extended for the optimization process by Burgin (Gomes and Saavedra, 1999). EP is a stochastic optimization technique based on the natural generation. The process involves random number generation at the initialization, followed by statistical evaluation, fitness calculation, mutation and finally the new generation created as a result of the selection (Lai and Ma, 1997; Ma and Lai, 1996). The generated random numbers represent the parameters which will responsible for the optimization of the fitness.
Evolutionary Programming Algorithm
The optimization process implemented using EP can be represented in the
flowchart as shown in Fig. 1.
Random Number Generation
In EP, initialization process was conducted by generating a series of random
number using a uniform distribution number generator. The random numbers represent
the reactive power loading at the chosen load buses for estimating the maximum
loadability. The number of variables depends on the number of buses chosen for
the simultaneous load increase. Since the objective of adopting EP is to estimate
the maximum loadability using accelerated search technique, therefore the parameters
would be only the reactive power on the chosen loads. Some constraints must
be set at the beginning so that the EP will only generate random numbers that
satisfy some pre-determined constraint. For the purpose of maximum loadability
estimation, only one constraint was identified i.e., the calculated FVSI must
be less than 0.95 and it is termed as FVSI_set. The FVSI value calculated using
the generated random numbers must be smaller than FVSI_set so that the fitness
could be optimized.
| Fig. 1: | Flow chart for the Evolutionary Programming (EP) |
Fitness Calculation and Statistical Evaluation
In this study FVSI is taken as the fitness equation, which needs to be maximized
and it was calculated by conducting the ac load flow program. It was done by
calling the load flow program into the EP main program. Thus in this problem,
the objective function was not going to be a single mathematical equation but
rather a subroutine which is executed accordingly in the EP main program.
Mutation
Mutation was performed on the generated random numbers, to produce the offsprings.
The mutation process was implemented based on the following equation:
| (2) |
| (3) |
Where:
| xi+m,j | = | Mutated parents (offspring) |
| xij | = | Parents |
| N | = | Gaussian random variable with mean μ and variance γ2 |
| β | = | Mutation scale, 0<β<1 |
| xj max | = | Maximum random number for every variable |
| xj min | = | Minimum random number for every variable |
| fi | = | Fitness for the random number |
| fmax | = | Maximum fitness |
The mutation scale β could be manually adjusted in order to achieve better convergence. Large value of β implies large search step, which causes slow convergence of the EP leading to large computation time and vice versa (Lai and Ma, 1997). The value of was determined by using the heuristic technique to produce the best results (Musirin, 2004).
Selection
The offsprings produced from the mutation process were combined with the
parents to undergo a selection process in order to identify the candidates to
be transcribed into next generation. In this study, elitism technique was performed
to select the candidates to be transcribed for the next generation.
Convergence Test
Convergence test is conducted to determine the stopping criterion of the
evolution. The pre-determined accuracy is normally dependent on the problem
orientation. In this study the convergence criterion is defined by the difference
between the maximum and minimum fitness 0.0001. The mathematical equation is
given by:
| (4) |
RESULTS AND DISCUSSION
In this study, maximum loadability estimation was conducted considering single load and multi-load bus increment. In realizing the effectiveness of the proposed technique, a reliability test system namely the IEEE 30-bus system was used as the test specimen. The IEEE 30-bus system has 6 generator buses and 25 load buses with 41 interconnected lines. To perform optimization of maximum loadability, the number of variables will represent the number of load buses which will be chosen for the optimization process. The results from this study are consequently compared with the results obtained from Artificial Immune System (AIS) and Automatic Voltage Stability Analysis (AVSA). Comparison is made in terms of the maximum loadability allowed with the highest fitness reached.
Optimization of Maximum Loadability
Maximum loadability of load buses is identified by increasing the reactive
power loading at particular load with the FVSI value set as 0.95. FVSI value
at 0.95 is chosen as the maximum limit to imply the cut off point prior to voltage
collapse occurrence. This is due to the fact that any lines connected to the
system will collapse when FVSI is reaching 1.0. This limit is specified in order
to search the Qmax before system loses its stability.
Maximum Loadability at Single Load
One load bus was chosen at a time randomly for this analysis. In this case,
four load buses namely buses 4, 14, 16 and 24 were chosen for the test. EP was
implemented to search the maximum loadability, Qmax for all these
load buses one at a time. Prior to this implementations AVSA was conducted to
monitor the voltage and FVSI profile with respect to the variation in reactive
power loading. In this study, it is obvious that only reactive power loading
was varied instead of the active/real power. This is due to the fact that, active
power is not significant in affecting the voltage stability. This statement
can also be referred to several previous works (Obadina and Berg, 1988; Gao
et al., 1992; Musirin et al., 2005b; Vournas, 1995).
From Fig. 2, it is observed that the voltage increases accordingly as reactive power loading at bus 4 increase. The minimum voltage; i.e., 0.80097 p.u is resulted when the load is subjected to 380 Mvar prior to the divergence of load flow.
| Fig. 2: | Max FVSI and voltage profile in p.u versus reactive power varied at bus 4 |
| Fig. 3: | Max FVSI and voltage profile in p.u versus reactive power varied at bus 4, 14 and 24 |
Therefore this value is identified as the maximum loadability for bus 4 with its corresponding FVSI of 0.92694. It is also observed that the FVSI value increases accordingly as Q increases. At the minimum voltage level, the computed FVSI value is closed to 0.95. This is the maximum FVSI value before system started to lose its stability.
Simultaneous Load Increase
In this study, simultaneous load increase at several load buses was also
conducted. Voltage and FVSI profiles were also monitored during this process.
The results for the voltage profile at buses 4, 14 and 24 when reactive powers
at these buses were increased are shown in Fig. 3. The maximum
FVSI value stopped at 0.92779 p.u with the respective Qmax of each
bus equal to 70 Mvar. From the figure, it is also observed that the Q value
for each bus is 70 Mvar. This implies that simultaneous load increase has caused
a low reactive power loading at the corresponding buses.
EP Implementation for Maximum Loadability Identification
From the Table 1, it is observed that the maximum loadability
value for bus 4 identified using EP is 386.5 Mvar with FVSI value of 0.94945,
which has been achieved in 76.781 sec. On the other hand, the maximum loadability
value for bus 14 identified using EP is 76.341 Mvar with FVSI value of 0.94178,
which has been achieved in 99.281 sec. The maximum loadability value for bus
16 identified using EP is 60.391 Mvar with FVSI value of 0.94271, which has
been achieved in 115.24 sec. The maximum loadability value for bus 24 identified
using EP is 66.031 Mvar with FVSI value of 0.9498, which has been achieved in
110.9 sec. From the results it is shown that EP managed to search the closest
point to 0.95.
From the Table 2, it is observed that the Qmax for buses 4, 14 and 24 increased simultaneously is 79.767, 71.656 and 68.47 Mvar, respectively. The corresponding FVSI value is 0.9424. This is achieved within 49.531 sec. In the second combination; reactive power loading at buses 4, 16 and 24 were increased simultaneously. The result is 77.032 Mvar, 63.92 Mvar, 95.308 Mvar for buses 4, 16 and 24, respectively with FVSI value of 0.94948. This is achieved within 175.859 sec. In the third combination reactive power loading at buses 14, 16 and 24 is 63.818, 76.877 and 63.478 Mvar, respectively with FVSI value of 0.94862. This is successfully optimized within 119.328 sec.
From the Table 3, it is observed that for bus 4; EP managed to search Qmax up to 386.5 Mvar, while AIS result is 385.51 Mvar and AVSA result is 380 Mvar. This implies that EP outperformed AIS and AVSA in terms of accuracy. Results for other load buses can be obtained from the same Table 3.
From the Table 4, it is observed that when reactive load at buses 4, 14 and 24 was increased simultaneously, EP technique managed to search up to 79.07, 71.656 and 68.47 Mvar, respectively with FVSI value of 0.9424. On the other hand, AIS only managed to search for 59.779, 73.009 and 36.318 Mvar with corresponding FVSI value of 0.93595. AVSA technique gives Qmax value of 70 Mvar at all the buses with its corresponding FVSI value of 0.89013. This shows the high accuracy achieved using EP over AIS and AVSA. Similar phenomenon can be observed from the same table for other load combinations.
| Table 1: | Maximum loadability using EP |
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| Table 2: | Several load increase using EP |
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| Table 3: | Results for comparative studies for single load Qmax (Mvar) |
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| Table 4: | Results for comparative studies for multiple load |
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CONCLUSION
Maximum loadability identification for single and multi load using Evolutionary Programming (EP) has been presented. In this study EP, was used as the optimization technique to optimize the exact reactive power loading values increased in several loads chosen randomly one at a time and several load simultaneously. In this study, EP technique was developed considering the optimized maximum loadability in particular loads as the objective function. Results obtained from the study utilizing EP were compared with the results using AIS and AVSA.
It was also found that EP outperformed AIS and AVSA in terms of accuracy on the maximum optimum loadability values and computation time. It can be concluded that EP technique is a better optimization technique as compared to AIS in searching the optimum value of reactive power loading at a single or multi-load. The developed EP engine could be beneficial for solving other optimization problems.