A Methodology is proposed for the online monitoring and assessment of voltage stability margins, using artificial neural networks and FACTS controllers with a reduced input data set from the power system. In this methodology, first the system model is reduced using self - organized artificial neural networks and an extended AESOPS algorithm. Then supervised learning of multilayered artificial neural networks is carried out on the basis of this reduced model. Finally, on the basis of trained network and the reduced set of system variables, monitoring is carried out along with the assessment of voltage stability margins. This methodology is tested comparatively with a methodology for monitoring and assessing voltage stability using a complete input data set. The tests were carried out on a real power system with 92 buses. The results obtained indicate the justifiability of using a reduced system because of the increased efficiency and accuracy of calculation, both in the learning stage and in the recall stage of the artificial neural network.
I) Introduction: Most of the large power systems are badly suffered due the phenomena of voltage instability. The basic method of preventing such severe system incidents is timely identification of voltage instability. In this sense, the development is made by using suitable computer producers
in online environment for monitoring and assessing the voltage stability margins. Two method are adopted to solve the stability problems I) static method ii) dynamic method the. This approach of static method is based on the steady state load flow model for calculating power flow and assessing how far the Jacobian matrix is true from the singularity function, that is, from classis static voltage collapse the practical implementation of this method is carried out while defining an entire series of simple indicator which show the static voltage stability margins. Some of this indicators can be used in an online environment, however such a simplified approach renders good results. The dynamic method of approach is based either on a linearised dynamic method of the power system and Eigen value analysis or on a non-linear dynamic model of the power system and direct methods. Such as dynamic bifurcation and numeric transient simulations. The method based on the dynamic approach give a much more real picture of voltage stability margins. However, methods based on the dynamic approach are time consuming in terms of computer time for the online environment. The most attractive means for solving this problem is found in artificial neural network (ANNs). These networks can be trained offline for complicated mapping, such as the difficult problem of determining voltage stability margin, and can be used in an efficient and timely way in the online environment to monitor and assess the same. In this paper, an entire series of efficient methods are proposed which are based on artificial neural networks (ANNs) and on various indicators of voltage stability however all the methods proposed are mostly based on the complete vector of system variables, which significantly diminishes their efficiency.
This paper deals with the development of a new methodology for monitoring and assessing voltage stability margins, it is a methodology based on artificial neural networks (ANNs) with a reduced input data set. The motive of this research is to establish what minimal necessary input data set is for the precise monitoring and assessment of voltage stability. The methodology proposed is based on static and dynamic stability margins, which are determined by eigen value analysis of the linearised static and dynamic models of a power system, and also on the use of multilayered and self organized artificial neural networks (ANNs). This methodology is composed, as in the practice of a learning stage and a recall stage. In the learning stage, first reduction of the model of the observed power system is carried out using self-organized artificial neural network (ANNs) and an extended AESOPS algorithm. This reduction is made for all relevant configurations, while defining the relevant configurations, all outages of lines and generators are considered. As well as the most portable combinations of double outages. Second, the training of multilayered feed forward artificial neural networks (ANNs) is carried out on this reduced model, by varying the load power over the range of the nighttime minimum and daytime peak value. During the training, mapping is established between the reduced input set of system variables and the voltage stability margin. Finally, during recall or exploitation based on reduced set of system variables, the monitoring and assessment of voltage stability margins of the considered operating regime is performed. The proposed methodology is tested with a methodology for monitoring and assessing voltage stability based on the complete input set. The results obtained indicate the justifiability of using a reduced system in the learning stage through the increased efficiency and quality of leaning of the artificial neural networks (ANNs) and in the exploitation stage through the increased efficiency and reduction of measurements required from the system.
2) Concept of Proposed Methodology:
(a)
(b)
Fig-1. Configuration of artificial neural network (ANNs) considered
a) UR b) NT
The proposed methodology for monitoring and assessment voltage stability is based on self-organized and multilayered feed forwarded artificial neural networks (ANNs). It consists of four stage: 1) configuration and design procedure. 2) Stability analysis. 3) Reduction of system model using a self organized artificial neural network (ANN) and an extended AESOPS algorithm, and training the multilayered artificial neural network (ANN) over such a reduced model. 4) Exploitation - that is the monitoring and assessment of voltage stability margins. In the text that follows the basic steps in the proposed methodology are briefly elaborated.
2.1) Configuration and design procedure of the artificial neural networks (ANNs)
(a): NY
(b): NR
Fig-2. Configuration of artificial neural networks (ANNs) considered
In the learning stage, four artificial neural network (ANN) configurations are considered as in figures 1(a2b), 2(a2b)
i) UR-is a self-organized ANN intended for the reduction of the vector of system variables the input to this ANN is the complete vector of system variables (Y) and the out put is the reduced vector of system variables (YR) reduction of the vector of system variables is carried out for each relevant topology.
ii) NT-is a multilayered ANN for detecting system topology, the input to this ANN is the complete vector of system variables, and the out put is binary identification code of the topology.
iii) NY-is a unified multilayered ANN for evaluating voltage stability margins for each considered topology of the power system. The input to this ANN is the complete vector of system variables and the identification code of the topology. The output is the real part of the critical Eigen value (Sc) and also the voltage stability margin M.
iv) NR- is a multilayered ANN for evaluation of the voltage stability margin for the considered topology of the power system. The input to this ANNs is a reduced vector of system variables and the output is the real part of the critical Eigen value and also the voltage stability margin.
By combining these four configurations two global algorithms for analyzing stability are obtained.
AY-an algorithm based on a combination of 'NT' and 'NY' configurations of the neural network as in fig-3.
AR-an algorithm based on a configuration of 'NT' 'UR' and 'NR' configurations of neural network as in fig-4
Fig-3. Algorithm 'AY' based on the complete input set of system
Fig-4. Algorithm 'AR' based on the reduced input set of system variables
All the artificial neural networks (ANNs) mention ed were configured with one input layer, one out put layer, and one hidden layer each. If the number of hidden layers is increased to two, the learning time is increase significantly and there is no increase in accuracy. The number of neurons in the hidden layer for all absorbed networks was designed so that a minimal square error was obtained. For most of the artificial neural network observed in this paper, the number of neurons in the hidden layer is 30% less than the number of neurons in the input layer. The artificial neural networks (ANNs) for detecting system topology had a sigmoid activation function for the neurons in the hidden layer and for the neurons at the output layers the activation function was a pure linear function. The (ANNs) for evaluating system stability had a tangent hyperbolic activation function for the neurons in the hidden layer, and for the neurons at the out put layer the activation function is a pure linear function.
2.2) Analysis of stability:
The basic stability analysis model is referred in this paper. In addition, this model is extended to accept the following components. Standard types of turbines and governing systems, basic types of automatic voltage regulators (AVRs) including the power system stabilizer (PSS) and finally an aggregate model of an induction machine are also included. In this model the fast transient processes in all machines stator/network is ignored. The elimination of these transient leads to standard singular perturbation form of the description:
x=f(x,y) ………… (1)
0=-g(x,y) …………(2)
Where x & y are the vectors of dynamic state variables and algebraic system variables, respectively.
x1=(xM, xAVR, xG) ... (3)
y=(ï±,V,P,Q) ………(4)
Where the meaning of following symbols is i.e., xM, xAVR, xG - the sub vector of dynamic state variables of synchronous and induction machines, the sub vector of AVR dynamic state variables including the pss and dynamic state variables of the turbine and governing system respectively;
ï±,V,P,Q - the sub vector of voltage phase angles, the sub vector of voltage magnitudes, the sub vector of injected active powers and the sub vector injected reactive powers of all nodes in the observed power system, respectively. Monitor and assessment of voltage stability in this paper are actually based on the algebraic system variable vector and on various combinations of its sub vectors-(ï±,V,P,Q).
For analysis of stability, the algorithm of fig-5 is used. Both the static and dynamic aspects of voltage stability are considered. The static aspect is based on the distance of operating point from the saddle node bifurcation point which corresponds to the singularity of the linearised matrix of the static power system model, while the dynamic aspect is based on the distance of the operating point from the critical node bifurcation point which corresponds to the null real parts of the pair of complex eigen values linearised the matrix of the dynamic power system model.
Fig-5. Flow chart
Stability analysis is basically differentiated for the 'AR' algorithm that is based on a reduced system model and for the 'AY' algorithm, which is based on the complete system model. Stability analysis for the 'AR' algorithm is carried out in six steps as in fig-6. The calculation of stability analysis begins by taking the night minimum as the starting system load, this load is then distributed to all loaded buses according to the specified distribution co-efficient and a random number generators. In the first step the observed load level 'P' determine the stationary point (x0,y0). In the second step, linearisation of the dynamic model is carried out using the defined stationary point. Then, elimination is carried out of the algebraic system variable vector (y0) for dynamic stability analysis, and then the elimination of the dynamic state variable vector (x0) for the analysis of the static stability. In this way, linear models are obtained for dynamic and voltage stability. In this way linear models are obtained for dynamic and static voltage stability analysis. In the third step system reduction is carried out first by applying the self organized ANN from the vector of dynamic and algebraic variables (x,y) the reduce dynamic and algebraic vectors (xR, yR ) are obtained, and then by applying an
Fig-6. Measurement of the real part of the critical Fig-8. Measurement of the voltage stability margin for
Eigen values for the algorithms AY and AR the algorithm AY and AR
a) AY-(ï±,V) (----), AR-(ï±,V),(----), simulation a) AY-(ï±,V) (----), AR-(ï±,V),(----), simulation
b) AY-(P,Q) (----), AR-(P,Q),(----), simulation b) AY-(P,Q) (----), AR-(P,Q),(----), simulation
Fig-7. Measurement of the real part of the critical Fig-9. Measurement of the voltage stability margin Eigen values for the algorithms Ay & AR for the algorithm AY & AR
a) AY-(V,P,Q) (----), AR-(V.P,Q),(----), simulation a) AY-(V,P,Q) (----), AR-(V,P,Q),(----), simulation
b) AY-(ï±,V,P,Q) (----), AR-(ï±,V,P,Q),(----), simulation b) AY-(ï±,V,P,Q) (----), AR-(ï±,V,P,Q),(----), simulation
Extended AESOPS algorithm from the dynamic and static system matrix E & F, a reduced system matrices ER & FR are obtained. In this way reduced dynamic and static models are obtained. The application of self organized ANNs for the reducing vector x & y is described in the next section, while the application of the extended AESOPS algorithm for system reduction is described in the section 4.1. In the fourth step by applying QR algorithm to the reduced static and dynamic models the Eigen values (ï¬i, vi) are determined. In the fifth step the smallest Eigen values (ï¹f) are determined and then the following variables are recorded: actual load level (p), the reduced vector of algebraic system variables (yR) and the smallest critical Eigen value (ï¹f ). In the final step testing is done to see if the critical load level (Pcrit) has been reached and if all condition for completing the calculation have been fulfilled.
When all condition for completing the calculation have been fulfilled the voltage stability margin is determined for all operating points using the following equation:
M = Pcrit - p ………(5)
Pcrit
Calculation of the stability analysis for the AY algorithm is standard and is different from the described calculation in that it does not contain step 3 for system reduction. In this way stability analysis and determination of Eigen values is done for the entire system model.
2.3) Supervised and unsupervised learning of Artificial Neural Networks (ANNs):
The multilayered and self-organized artificial neural networks are used in the proposed methodology. Self-organized ANNs are used for reduction of the system model and of the input set for multilayered artificial neural network learning, while multilayered ANNs are used for detecting topology and for monitoring and assessment of voltage stability.
i) System reduction using self organize ANN: The self organized ANN is introduce in order to reduce the learning time of the ANN and to reduce the number of system variables needed for voltage stability margin monitoring and assessment. According to this method the problem of reducing the vector of system variables from the ANN can be presented as a problem of extracting characteristics variables and the variables, which differ from one another. In other words the algorithm developed for the reduction of vector system variables groups; its coordinates, which are sufficiently, approximate in time, and separate the dissimilar once. The straight variable, which is closes to the calculated center of group, is taken as the characteristic representatives of the group. Finally by grouping all characteristics representative of the group in one vector, the reduced system vector (yR) is obtained. If only the chosen representative of each group is observed during the formation of the system model, the dimension of the system description can be decrease. A more detail review of the ANN algorithm used is to be found in the, section 4.2.
ii) Supervised learning of the ANNs:
After system reduction using the result of stability analysis on a reduced set of system variables learning is carried our. Supervise learning of multilayered ANN is a process which the correction of synaptic weights between neurons in proximate layers to learn carry out. The error back propagation algorithm with a variable learning rate is used for these purposes. The general expression for the correction of the synaptic weights is well known as the generalized delta rule.
2.4) Exploitation of artificial neural network (ANNs):
After selecting required ANN configuration with the most suitable characteristics, it is possible to move on to the next stage is called exploitation. First, the system topology is detected.
Next, and evaluation of the stability of the power system with the identified topology on the basis of complete or reduced vectors of the system variables is carried out. A schematic over view of the considered variants of the exploitation stages shown in fig-3 and fig-4.
Monitoring of voltage stability margins is performed by introducing measured and estimated system variable values into the input of the corresponding neural network with algorithm 'AY' and 'AR'. Voltage stability assessment is carried out by testing, if there is an outage of any single element that is lines or generators or if any of there probable combinations leads to a disruption of the voltage stability margins. For the simulation of the considered outages two approaches are followed.
i) Indirect stability assessment (ISA):
The first approach is indirect and is performed so that for each outage the evaluation is first done for the vector of system variables by applying the fast-decoupled load flow calculation. Then on the basis of the vector of system variables and the specified identification code for the topology of the outage in question, the voltage stability margin of the observed outage is tested by applying one of the suggested ANNs.
ii) Direct stability assessment (DSA):
The second approach is direct, and is performed so that the input of the ANN is the input vector y = (P,Q) consisting of actual injected active and reactive powers in all buses and the identification code of the outage. In these way voltage stability margins of the observed outage is directly evaluated with out the fast-decoupled load flow calculation.
3) Application of the methodology:
The proposed methodology for stability analysis is based on ANNs with complete and reduced sets of input data. It can be tested on high voltage power system about 92 buses 12 number of generator buses and 174 lines. In doing so the following four cases may be considered for the complete input vector of system variables.
Vector y=(ï±,V), state variable system
Vector y=(P,Q), directly measured system variables
Vector y=(V,P,Q), directly measured variables
Vector y=(ï±,V,P,Q), all system variables.
By combining this input vector by combining this input vector with the proposed algorithm or detecting the topology 'NT' and monitoring and evaluating voltage stability 'AY' and 'AR' the following configurations/algorithms are obtained which are tested in this paper:
NT-(ï±,V), AY-(ï±,V), AR-(ï±,V),
NT-(P,Q), AY-(P,Q), AR-(P,Q),
NT-(V,P,Q), AY-(V,P,Q), AR-(V,P,Q),
NT-ï±,V,P,Q), AY-(ï±,V,P,Q), AR-(ï±,V,P,Q),
As, such, of special interest are the algorithms which are based on the vector of system variables y=(V,P,Q) which are directly measurable from the system, while the remaining input vectors require estimated values of the sub vector of the voltage phase angles of the buses (ï±).
3.1) System reduction and ANN learning on a test sample:
The following relevant configurations are considered in this paper: base configuration, all single outages of lines and generators, and most probable combinations of double outages. For each relevant configuration, first the reduced set of system variables is determined using the self-organized neural network UR. This reduction renders an average reduction of the system variable vector by 95%, so that the reduced vector represents only about 5% of the whole vector. In forming these vectors as separate groups, the following are usually set apart:
i) Voltage phase angles and magnitudes in buses where low, rated power generators are connected;
ii) Active and reactive powers in generators buses;
iii) active and reactive powers in loaded buses of a single region;
iv) voltages in load buses of single compact region.
It is interesting to note that it was actually those buses which would be assigned as pilot buses, according to the secondary voltage regulation algorithm (7,16), which were chosen as characteristic representatives of the group with the voltages of load buses. In this way, the proposed algorithm for reduction of the sys tem variable set can also be used for selecting a candidate for the pilot buses of secondary voltage regulation.
On the basis of the complete and reduced vectors of system variables, supervised learning of ANNs was carried out. The ANNs were trained using either one network for all relevant configurations (algorithm AY), or a single network for each relevant configuration AY), or a single network for each relevant configuration (algorithm AR). For each relevant configuration, the load was varied over the range of the nighttime minimum to the daytime peak value, during which 100 different margins were observed. The basic characteristics are presented in Table-1.
Table-1. Basic learning characteristics of the observed ANNs
In Table 2, the relationship is presented comparatively of the time needed for stability analysis, system reduction and ANN learning for algorithms AY and AR. Learning was done on a workstation ALPHA 4100/300 MHz. From the Table it can be seen that the AR algorithm is significantly more efficient in the learning phase. This efficiency is achieved because of the shorter time necessary for calculating the stability analysis for the reduced system model. The other more significant reduction
in computer time was achieved in the shorter learning time for several smaller ANNs that for the one large ANN. However, the obstacle is the application of algorithm AR that is necessary for the configuration of the large number of smaller ANNs. This number is approximately equivalent to the number of relevant configurations.
Table-2. Comparative computer learning time for algorithms AY and AR
3.2) Testing the methodology in exploitation:
The testing of ANNs on a real power system was carried out for the all configurations for topology detection, and for monitoring and assessing voltage stability margins. Testing of the trained ANNs for the detection of system topology was carried out on an unknown set of 12960 samples (40 each for every topology observed). In doing so, the following four ANNs for topology detection were considered:
NT-(ï±,V), NT-(P,Q), NT-(V,P,Q), NT-(ï±,V,P,Q). On the observed test set, all four networks detected all the considered topologies with a high degree of accuracy. Namely, the output neurons of neural network NT were so designed that they give a binary code for each topology considered. The mean square errors calculated from the real values at the output neurons are given in table-3.
Table-3. Mean square errors for output neurons and for the considered power system topology:
The test results of the neural networks for the monitoring and assessment of the voltage stability margin are shown in figs. 6-9. In figs. 6 & 7 there is a comparative presentation of the dependence of the real part of the critical Eigen values on the system load level for algorithms AY and AR. In figs. 8 & 9 there is comparative presentation of the dependence of the voltage stability margin on the system load level for algorithms AY and AR. From Fig. 6a and Fig. 8a, it can be seen that in the situation where the input set contains only system state variables (ï±,V), greater precision is achieved by using algorithm AY with the whole input set, than by using algorithm AR with a reduced input set.
When the input set consist of input vectors (P,Q), the precision achieved with reduced or complete input set, is almost equal with reference to fig-6(b) and fig-8(b). Finally from fig-7 (a), 9(a) and fig-9(b). It can b e seen that when the input set contains enough data input vectors (V,P,Q) and (ï±,V,P,Q), greater precision is achieved by using algorithm AR with a reduced input set then by using algorithm AY with the entire input set. In addition algorithm AR is one order of magnitude faster in exploitation than algorithm AY in terms of ms, which is exceptionally important because exploitation is done in an online environment.
As mention above, when assessing voltage stability, both the ISA and DSA approaches were analyzed the computer time needed for analyzing stability in both approach are given in table-4. From the table it can be seen that the direct calculation of stability using DSA and AR-(P,Q), decreases computer time remarkably without significant loss in precision.
Table-4. Computer time for assessing voltage stability using the indirect approach ISA and the Direct approach DSA
4) Details of Model:
4.1) System Reduction
The basic model for system reduction is a linearised mathematical model of power system
x=Ax+By ………….. (1)
0=cx+Dy …………….(2)
It the vector of state x is divided into two sub vectors xR is the sub vector which contains the variables of state whose dynamics are under consideration and x0 is the sub vector which contains the variables of the state whose dynamics are not under consideration the equations above obtain the following form:
xR= ARRxR+ AROx0+ BR .y ……. (3)
xO= AORxR+ AOOxO+ BO .y …..... (4)
0= CRxR+ COxO + D.y ……… (5)
Where
ARR, ARO, AOR, AOO, correspond to the sub matrix of matrix A
BR, BO correspond to the sub matrix of matrix B
CR, CO, correspond to the sub matrix of matrix C
That is the system of equations 3 - 5 in the Laplace frequency domain are
sxR= ARRxR+ AROx0+ BR .y ……. (6)
sxO= AORxR+ AOOxO+ BO .y …..... (7)
0= CRxR+ COxO + D.y …………. (8)
In order to observed the dynamics of only relevant state variables, which are contained in sub vector xR, it is necessary to eliminate the sub vector of variables of state x0 from models equations 6-8, and the vector of algebraic variable y. After these elimination, which are realized using the gauss method, a reduced system model is obtained
sx=[ARR-B1.D1-1(s)C1] xR ……….. (9)
Where the following symbols are used:
B1=[ARO BR], D1(s)= A RR-SI Bo
Co D
C1= AOR
CR …………………….. (10)
To analyze the dynamic reduce model equation-9 it is necessary to detetemine its Eigen values. Determining these Eigen values is done by an iterative approach (1) individual or (2) simultaneous.
In this paper the individual method is used which is based on the following iterative approach
sK+1=Aii-B1D-1(SK)C1, k=1,2, ….. n (11)
Where SK SK+1 is the approximation of the required Eigen values in the and k and k+1 iteration. For improving the convergence of these iterative methods product linearisation B1D-1(SK) is used.
With this iterative method the Eigen value is determined which predominantly associated to the ith component of the sub vectors of xR. By varying the index I=1, ….., R it is possible to determine all Eigen values of the reduced model as in equation 9 with great efficiency.
4.2) Artificial neural network (ANN) algorithm:
The goal of unsupervised classification is to group the state variables of the system which are characteristic by similar dynamic in the process the cosine of angle covered by vectors of individual state variables at all interval is taken as the measure of similarity between them. If two arbitrary state variables are observed over a period of time, for example xi=[x1i, x2i, …. xmi]T and xj=[x1j, x2j,…. Xmj]T, They will be similar if they both cover a small angle. In order the measure there similarity the following express is used
cosï±ij = xiTxj (1)
xi xi
If equation (1) is larger than the given value, represented by the similarity threshold (cosï±0), that is if the condition ï±ij < ï±0 then the vectors xi and xij are similar and belong to the same group. Keeping the criterion for defining similar vector as in equation (1) it is necessary to normalize them before hand by using the formula
i = xi I=1,2, …. n (2)
xi
The grouping of similar variables is achieved in the following way. The arbitrary group g(g=1, … , G) is observed consisting of k variables. The arbitrary ith variable xi, that is its normalized value i, will belong to this group if it satisfies the condition that ï±ig < ï±0 where ï±0 is the threshold of group membership, ï±ig is the angle between ith vector i and the center of the gth group Cg.
As the new element enters the group, the center of group moves toward the position of the variable i according to the formula
Cgk+1= Cgk+áï„ Cgk …….. (3)
Where Cgk and Cgk+1 are the centers of g group before and after vector i enters this group. The coefficient 'a' is a factor of proportionality define by the intuitive expression 'a'=1/k+1, where k is the number of variables making of the group, and the coefficient ï„Cgk is the correction of the position of the group center (ï„Cgk=xi- Cgk). By replacing 'a' and ï„Cgk finally we get Cgk+1=ai+ Cgk(1-a) i.e.,
Cgk+1= KCgk+i ….. (4)
K+1
The calculated group centers of equation 4 serve for the basis for finding the closest state variables for them, which become the characteristic representatives of the group.
5) Conclusion & Result:
These paper deals with the developments of new methodology for monitoring and assessment the voltage stability, which is based on artificial neural networks (ANNs) with a reduced input system data set. The input data set is reduced using a self-organized neural network, and then a multilayered feed forward neural network is trained using this reduced data set to monitor and assess the voltage stability margins. The proposed methodology tested comparatively with a method for monitoring and assessing the voltage stability, which is based on a complete input data set. Tests are carried out on a power system with 92 buses. The obtained results indicate that, both in the learning and exploitation stages the efficiency of the methodology is significantly increase by reducing the input data set. More over, when the input data set is representative enough, the ANNs based on the reduced data set achieved greater precision in calculating voltage stability than did the ANNs, which is based on the complete input data set. In addition, by using the reduction of input data, there is a significant reduction also in the demand for the measurements within the power system.
