Modelling and Prediction of Reactive Power at Railway Stations Using Adaptive Neuro Fuzzy Inference Systems

Abstract

Electricity has become an important concern in today’s society. This is due to the fact that the electric grid now has a greater number of non-linear components. The AC-powered locomotive is one of these non-linear components. The aim of this paper was to model and predict the reactive power produced by an AC locomotive. This paper presents a study on the modelling and prediction of reactive power produced by AC-powered electric locomotives. Reactive power flow has a significant impact on network voltage levels and power efficiency. The research was conducted by using intelligent techniques—more precisely, by using the adaptive neuro fuzzy inference system (ANFIS). Several approaches to the ANFIS structure were used in the research. Of these, we mention the ANFIS-grid partition, ANFIS subtractive clustering and ANFIS fuzzy c-means (FCM) clustering. Thus; for modelling and predicting reactive power, ANFIS was trained, then tested. For the training of ANFIS, experimental data obtained from measurements performed in a train supply sub-station were used. The measurements were taken over a period of time when the locomotives were far away from the station, close to the station, and at the station, respectively. The currents and voltages from the supply substation, respectively the active, reactive, and distorted powers, were measured on the data acquisition board. With the measured data of the reactive power, the modelling with ANFIS was performed, and a prediction of the variation in the reactive power was made. The paper analysed the results of the modelling by comparing between several types of ANFIS architectures. The values of RMSE, RMS and the training time of ANFIS were compared for several structures of ANFIS.

1. Introduction

Electric rail transportation is a low-cost form of transportation that is used all over the world. Since automobiles pollute the environment, rail transportation is becoming more common. However, different issues occur in rail transport, including significant amounts of negative sequence currents (NSCs), harmonic currents, and reactive power [1,2].

The large-scale use of power converters in recent years, in which parameters such as voltage and frequency vary depending on the specific applications for which they were designed, has made these power converters the most widespread source of harmonics in distribution systems [1,2,3,4,5].

Current harmonics are becoming more widespread nowadays due to the growth of non-linear electrical equipment used in all fields, including in the household sector [6]. Basically, all modern electrical and electronic equipment has switching sources, including locomotives [7,8]. If they do not have switching sources, they control in some way the absorbed power, thus resulting in nonlinear loads [9].

A perfect power supply is one that is always available, that operates within permissible voltage and frequency limits, and that has a perfectly sinusoidal voltage curve. Voltage of poor quality is a hidden cost. Typically, it goes unnoticed and undiagnosed as long as no costly failures occur. The effect of nonlinear equipment on power quality increases in direct proportion to its nominal power. A nonlinear device of this kind is an electric arc furnace [10]. Additionally, computers and printers, fluorescent lighting, battery chargers, and variable-speed drives are all examples [11]. Any piece of equipment that contains switching equipment is a nonlinear load [9]. Static voltage and frequency converters are mounted on electric locomotives to supply auxiliary services [12,13]. Certain modern locomotives incorporate a computer system that monitors, operates, and controls the traction-braking regimes. Frequency converters dissipate energy by using direct current voltage produced by alternating voltage recovery [14]. This conversion results in energy losses and the generation of harmonics that can cause distribution networks to destabilize [15]. Simultaneously, reactive power flow toward the source of energy results in additional losses in the transmission and distribution networks [16]. Traction substations connect to the high-voltage electricity system or to the railway’s own power system, as well as adapt the electrical energy’s parameters (voltage, current, and frequency) to the standardized values required at the contact line. The contact line is an aerial electrical network that is positioned above the railway track and supplies power to the locomotive through a collector or current collector [17]. Alternating current is used to power the supply, which comes with a variety of voltages and frequencies. The traction substation is composed of low voltage transformers that are supplied with electricity by the electricity system. Due to the fact that the traction system is a single-phase consumer, connecting it to the three-phase system results in current and voltage imbalance [18]. To minimize them, substation transformers are connected in Scott, V/V, or cyclic configurations of single-phase transformers to three-phase transformers [18]. In comparison to other users, the 50 Hz single-phase alternating current electrified rail charges the energy system asymmetrically. Electric locomotives with installed power greater than 2000 kW are fitted with rectifiers, which introduce the effects of load asymmetry and higher harmonics of current caused by the deforming consumer [9,19]. For an analysis of these problems, in this paper a method was designed to measure electrical quantities that can influence the quality of electrical power. These measurements were made at a substation in Romania. The measurements were carried out continuously over a period of about 2 h, a period that included the function without load or under the load. The load was represented by an electrical locomotive that passes through the substation, the measuring point of a train. As a result of the measurements, one can see how the power quality is affected by an electric locomotive train moving through the station. We have performed such studies in the past [20,21], and we found that the power quality is affected in the substation. After analysing the results of the measurements, we conducted the reactive power modelling, which can propose solutions to these problems. Based on these measurements, a reactive power model and a prediction were made.

Systems modelling is important in all areas, because with a well-defined model, researchers will be able to better understand and predict the behaviour of a system [9]. Complex systems are more difficult to model because they are characterized by nonlinearities, and it is difficult to obtain a model that characterizes such a system [18,22].

Prediction of time series has an important theoretical significance, and there are many applications of this type in engineering [23,24]. Forecasting can be a solution in applications when estimation is not possible [25]; it consists of finding future values of time series data based only on past data [24,26]. The time series forecast is a current research topic because it has wide applications in many different fields [27,28]. In modelling time series, some data can be characterized by linear models, and other data must be characterized by nonlinear models [27].

Linear and/or nonlinear systems can be modelled using different methods. For the prediction of time series data, the study [23] proposes a hybrid model that combines a linear regression model and a deep belief network model. In order to make a short-term prediction, paper [24] proposes an alternative integrated model that is based on the improved empirical mode decomposition, autoregressive moving average with exogenous terms (ARMAX), and adaptive network-based fuzzy inference system (ANFIS).

For time series, there are also various forecasting methods. ANN-based methods have been proved to work well in nonlinear time series forecasting. The article [28] suggests three prediction techniques that can recognize the hidden connections between inputs and outputs, including artificial neural networks (ANN), deep neural networks (DNN), and models based on long- and short-term memories (LSTM). An efficient strategy to choose the number of clusters and their initial values for fuzzy modelling is presented in the study [29]. Forecasting power load is performed using chaos theory in the study [30]. A convolutional neural network is applied in [31] for prediction. ANFIS is used for modelling complex systems in studies [32,33,34,35]. For short-term forecasting, artificial neural networks (ANN) and hybrid time-series models were employed [36].

The short-term forecasting problem is addressed also in [37] using a method that combines wavelet transform, adaptive evolutionary algorithm, and fuzzy system with GNN. The study [38] proposes an effective method for forecasting short-term load using artificial neural networks.

Short-term prediction approaches available in the literature can be divided into two main categories: statistical methods and artificial intelligence-based methods. Artificial intelligence includes artificial neural networks [34,38], fuzzy inference systems [32,33,37] and expert systems [39,40]. In the statistical methods category are ARIMAX [35,37] support vector regression (SVR) [41,42,43], and stochastic time series [26,34]. ANN-based techniques are a non-linear modelling method and an efficient predictive technique. It has been demonstrated in many studies that ANNs can achieve the desired accuracy in prediction of nonlinear time series. In fact, there are thousands of published studies using ANN to model nonlinear systems and predict time series. During the last 10 to 20 years, ANN-based modelling techniques have been intensively developed [44].

2. Materials and Methods

2.1. Measurement Scheme

The measurements were made in a substation of the railway transportation network. The measurement scheme is shown in Figure 1. The supply was produced by means of a voltage transformer of 110 kV/27 kV. Acquisition of currents and voltages was achieved at the frequency of 5 KHz. The currents were acquired by means of a current transformer with a clamp meter that had the role of a current transformer (Figure 1). An adaptation block for high currents and voltages was used at values supported by data acquisition systems. In the same way, voltage was measured through the same adaptation block. The adaptation block also contained a numeric analogue converter, so the acquired data were in numeric format.

The variation in the current and voltage measured in a railway substation is shown in Figure 2a,b. Figure 2a depicts these measurements when the electric locomotive was passing very close to the railway station’s measuring point at the station, while Figure 2b depicts the variations in current and voltage when the locomotive was 50 m from the station. The current variation was distorted, as may be seen. Additionally, there was some distortion in the voltage variation. The aforementioned side effects, such as reactive power, harmonic currents, and low-power factor, were caused by these distortions.

The variations in active, reactive, and distorted forces at the measurement point, i.e., at the station, were calculated using data acquired with a DAQ board. Figure 3a,b illustrate these variations. As would be expected, the reactive power was very high. This indicated that the power output was compromised as a result of these distortions. To determine the actual effect of reactive power on power quality, power modelling was performed using intelligent computational techniques, specifically fuzzy neuro ANN. To validate the models, they were compared to the measurement results. In general, models aid in the analysis of structures, assist in the better understanding of a system’s behaviour, and contribute to the prediction or simulation of a system’s behaviour. Additionally, the models make it easier to apply and validate advanced control techniques. Power variation modelling is critical because excessive reactive and distorted power results in poor power quality and power loss.

2.2. Modelling the Reactive Power with ANFIS

Numerous experiments have demonstrated success when neural networks and fuzzy systems are applied to railway supply systems. The accuracy of the mathematical models proposed in [45] for the modelling of electric power consumption in railway facilities may be significantly improved by using artificial neural networks, fuzzy neural networks, and support vector machines. The paper [46] describes a program that employs neural network-based prediction algorithms and optimizes the reactive power mode. The study [47] describes how the adaptive neuro-fuzzy inference system (ANFIS), support vector machines (SVM), and artificial neural networks were used to determine the rail voltage for a 1500 V DC-fed rail system (ANN). The discussion of a detailed method for compensating for negative sequence and harmonic currents using PI and fuzzy controllers can be seen in [48,49]. ANFIS and MRAS-PI controller-based adaptive-UPQC for power quality enhancement applications are presented in [50].

The data shown in Figure 3b were utilized to train ANFIS in modelling the reactive power produced by electric locomotives. Understanding the phenomena related to the transportation of non-linear loads, such as electric locomotives, will be aided by the modelling results.

An ANFIS was trained and tested using the data sets. An ANFIS network [50,51,52,53,54] can be used to model complex systems that do not limit themselves to classical modelling. It is such a complex system in the case discussed in this paper, because of the fluctuations in the power consumption of the electric locomotives’ power station. Since currents and voltages are distorted, the device is difficult to mathematically model. As a result, intelligent techniques are a more suitable approach in this situation. The collected data for power generation were used to build this model of neuro fuzzy networks.

2.2.1. ANFIS

A neuro-fuzzy network is a smart hybrid system that integrates the characteristics of the two systems it is made of. As a result, the hybrid system outperforms all neural networks and fuzzy systems. In the development of intelligent systems, fuzzy logic and RNA are naturally complementary methods. Though neural networks are low-level computational systems that excel when dealing with raw data, fuzzy logic is associated with “high-level” reasoning and relies on linguistic information collected from human experts [55]. Fuzzy systems, on the other hand, lack the ability to learn and adapt to new situations. However, while neural networks are capable of learning, they are not transparent to the user. Integrated neuro-fuzzy systems combine neural networks’ parallel processing and learning capabilities with information representation and the ability of fuzzy systems to provide human-like explanations. As a result, neural networks have become more transparent, and fuzzy systems have gained learning capabilities.

With Takagi–Sugeno fuzzy systems, ANFIS hybrid neuro-fuzzy systems are functionally similar to adaptive neural networks [51]. In contrast to traditional fuzzy systems, ANFIS neuro-fuzzy systems can adapt throughout the learning process. The fuzzy membership functions can be adapted in this way by using an optimization approach [51].

Figure 4 depicts the ANFIS architecture. Several references, such as [32,50,51], offer information about the learning process in an ANFIS.

ANFIS is a tool for modelling and forecasting chaotic time series. According to the paper, power samples (active, reactive, or distorted) are samples collected at discrete time intervals using a data acquisition board. After testing, an ANN network is capable of predicting the value of a time series for which it has not been trained. As a result, the values in the time series represented by the reactive power were used to train ANFIS. K samples were used for training, and n-k samples were used for testing and validation from a total of n data taken. MATLAB was used to implement ANFIS. We used ANFIS based on grid partition, ANFIS based on subtractive clustering, and ANFIS based on FCM, all of which are available in MATLAB.

To make an ANFIS prediction, it must first know the values of the time series up to time t in order to predict the value at time t + p. To predict a future value x (t + p), the typical prediction method is to create a mapping from N samples sampled at d time units: x(t – (N – 1) d,…x (t – d), x (t).

For example, in the case presented in the paper, a vector with data will be created for N = 5, d = 3, and p = 3 to drive a vector with 5 columns:

To predict x(t + 3), use [x(t – 12) x(t – 9) x(t – 6) x(t – 3) x(t)].

The samples of the powers are represented by x in our paper.

2.2.2. ANFIS-Based GRID Partition

The grid partition technique is used on the second ANFIS layer to create the fuzzy sets, which are the fuzzy sets used for fuzzifying the inputs [26,33,54,56]. The input data space is divided into rectangles with sides parallel to the axes in this process, and each input is fuzzified with member functions of the same form. The number of fuzzy if-then rules is equal to M_n, where M is the number of fuzzy sets on each input, and n is the number of inputs. For two inputs and three fuzzy sets on each input, the partitioning shown in Figure 5a exists, and there are eight rules, with the ANFIS structure shown in Figure 5b.

The ANFIS grid partition has the disadvantage of producing a grid partition of the input space, which can result in a very high dimensionality, implying many rules in the third layer. When the number of inputs is relatively high, grid partition causes an explosion in the number of rules (more than four or five). It takes a long time to train the network. When using grid partition, the partitioned space, i.e., the area that defines the fuzzy rules, is uniformly generated, allowing for smooth analysis of the fuzzy rules. When the number of inputs is minimal, the grid partition method is used. For 10 inputs and 2 fuzzy member functions, for example, 1024 unique areas are produced [53]. A fuzzy rule is developed for each particular location, resulting in 1024 rules, a complex structure that requires extensive training time.

2.2.3. ANFIS-Based Subtractive Clustering (SC)

When the number of data distribution centres is unknown, this technique is useful. The subtractive clustering approach is a technique for overcoming the disadvantages of grid partitioning. The data set is divided into fuzzy clusters, and the different clusters are then mapped in the input space. As a result, the selected clusters’ centres could be considered a system behaviour, and a cluster could be thought of as an “if then” rule in the Takagi–Sugeno model [22,26,54]. Subtractive clustering [57] is an algorithm for estimating the number of clusters in a data set as well as the centre of each cluster. Based on the density of surrounding data points, the algorithm assumes that each point in the data set is a potential cluster centre and calculates the probability that each point in the data set determines the cluster centre.

For example, D_i, the density measure for each point x_i, is defined as [26] for n data (x₁, x₂,… x_k), according to (1):

$$D_i={\displaystyle \sum }_{j=1}^n{e}^{-\frac{\Vert x_i-x_j\Vert {}^2}{{\left(\frac{r_a}{2}\right)}^2}}$$

(1)

The radius of the cluster is defined by r_a, which is a constant positive number in (1). From relation (1), it can be seen that the density value D_i is higher when there are more data points in the radius-defined neighbourhood. A point outside this vicinity will have virtually no influence. Since any data point may be the centre, the density index of each point in the neighbourhood must be determined based on this assumption.

It is reasonable to state that a point’s density measurement is a function of its distance from all other data points. As a result, a data point with more neighbours has a greater chance of being a cluster centre. The number of clusters is influenced by the bad value. This is why the number of clusters is called the radius of influence. The number of data groups and, by extension, the number of fuzzy rules, is higher when the error is lower, and vice versa.

When it is unknown how many centres will be used to distribute the data, subtractive clustering is used. This algorithm evaluates each data point as a potential cluster centre candidate and then determines each data point’s potential by calculating the density of the data points around it. The approach is iterative, and it assumes that any point could act as the centre of a cluster depending on where it is in relation to other data points.

Even if the user does not define the number of clusters, the method divides the input space into them. Only the radius of each cluster is defined by the user. The radius is a number between 0 and 1 that represents the domain of control at the centre of each cluster. If the radius is too small, the cluster size is too small, and the number of clusters is too small, resulting in a larger number of fuzzy rules. The number of clusters and fuzzy rules decreases if the radius is too large, the cluster size is too large, and the radius is too large.

2.2.4. ANFIS-Based FCM

Fuzzy c-means (FCM) is a data clustering system in which each data point belongs to a cluster to a degree determined by membership [26,52,54].

FCM partitions a collection of n vectors x_i; i = 1, 2,…, n into fuzzy groups and determines a cluster centre for each group to minimize the non-similarity measure’s objective function [26]. The data patterns x₁, x₂,…, x_n are used to create the centres k = 1,…, c. The membership coefficient matrix is then calculated:

$${\mu }_{ik}=\frac{1}{{{\displaystyle \sum }}_{j=1}^c{\left(\frac{d_{ik}}{d_{jk}}\right)}^{\frac{2}{p-1}}}$$

(2)

The meaning μ_ik in relation (2) represents the degree to which the object k belongs to cluster i. The value above the degree of fuzziness, p > 1, is the Euclidean distance between the middle and the input data x.

After that, one may calculate the objective function:

$$J={\displaystyle \sum }_{i=1}^c{J}_i={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=1}^n{\mu }_{ik}^p{d}_{ik}^2$$

(3)

If the objective function’s value falls below a certain threshold, the process grinds to a stop.

The new centres are then calculated:

$$c_i=\frac{{{\displaystyle \sum }}_{k=1}^n{\mu }_{ik}^p{x}_k}{{{\displaystyle \sum }}_{k=1}^n{\mu }_{ik}^p}$$

(4)

3. Results

The three methods presented above were used to model and predict reactive power in this study, and comparisons were made in terms of RMS, RMSE, and training time. As shown in Figure 6, ANFIS was used to model and predict the measured reactive power. It is obvious that a number of previous samples can be used to predict a value in a reactive power time series.

3.1. Modelling and Predicting the Reactive Power Using ANFIS Grid Partition

As previously mentioned, this type of ANFIS architecture is best suited to situations with a reduced number of inputs. The ANFIS architectures for different numbers of inputs (N) and fuzzy sets are shown in Figure 7a–d (FS).

When analysing the modelling results based on ANFIS grid partition, it is noticeable that in the testing process, there are no significant differences in ANFIS performance. There have been experiments with various architectures.

Table 1. ANFIS training performance using grid partition method.

Crt. No.	Model Parameters			Training				Testing
	N	FS	Number of Rules	Training Time [s]	MSE	RMSE	R	MSE	RMSE	R
1	2	2	4	0.27028	0.000212	0.014554	0.97105	0.000154	0.012408	0.9578
2	2	3	9	0.63031	0.000201	0.014173	0.97256	0.000151	0.0123	0.95809
3	2	4	16	13.604	0.000192	0.013843	0.97384	0.000154	0.012394	0.95763
4	2	5	25	28.133	0.000181	0.013436	0.97538	0.000164	0.012787	0.95454
5	2	6	36	53.957	0.000172	0.013101	0.9766	0.000204	0.014289	0.94334
6	2	7	49	99.179	0.000158	0.012568	0.97849	0.000183	0.013512	0.94852
7	2	8	64	18.164	0.000153	0.01235	0.97924	0.000966	0.031076	0.7767
8	2	9	81	28.607	0.000143	0.01195	0.98057	0.000417	0.020429	0.8859
9	3	2	8	0.6962	0.000197	0.014029	0.97308	0.000154	0.012429	0.95933
10	3	3	27	50.373	0.000174	0.013202	0.9762	0.000261	0.016159	0.92919
11	3	4	64	26.357	0.000156	0.012485	0.97874	0.000428	0.020684	0.87967
12	3	5	125	106.89	0.000128	0.011312	0.98258	0.000938	0.03063	0.78754
13	4	2	16	30.238	0.000175	0.013247	0.97597	0.000295	0.01718	0.91542
14	4	3	81	68.548	0.000146	0.012091	0.98002	0.000245	0.015658	0.93317
15	5	2	32	14.849	0.00016	0.012638	0.97808	0.000226	0.01503	0.93793
16	5	2	32	14.735	0.00016	0.012638	0.97808	0.000226	0.01503	0.93793

shows the results. In Table 1, it is noted that FS represents the number of fuzzy sets and N represents the number of inputs, which is actually the number of previous samples. When the ANFIS parameters are set so that the ANFIS structure has many rules (for example, N > 3, FS > 3), the phase testing results are not any better. On the other hand, overtraining occurs at some point, and ANFIS loses the ability to generalize. This can be shown with N = 4 and FS = 4. It was found that the training time increased excessively. As a result, if a large number of fuzzy sets are selected, this type of architecture is not very suitable for time series prediction.

As shown in Figure 7, the grid partition method generates a large number of rules in the rule layer. Many rules are generated when the input space is partitioned. The training time is increased in this way. Line 12 of Table 1 illustrates this. In fact, for a number of previous samples N = 6, these models from Table 1 are the only ones with an acceptable training time. Other models, for example, for N = 5, FS = 4, have been tested, but they failed, necessitating a large amount of training time. The training time for this type of ANFIS based on grid partition models is excessively long due to the large number of neurons that result from the partitions made by the fuzzy sets, resulting in a large number of neurons in layer 3 and 4, and thus a large number of laws. Of course, a larger number of rules leads to smoother modelling in general, but if there are too many rules, the model cannot be used in practice due to the lengthy training period.

Figure 8 depicts the modelling and prediction results for an ANFIS grid partition architecture (N = 4, FS = 2) chosen from Table 1. Figure 8 depicts the difference between the calculated reactive power and the one obtained by simulation, as well as the error. Figure 9 depicts the differences between measured and predicted reactive power, as well as the test phase error. Figure 10 depicts the variance in the regression coefficients during the training and testing phases.

3.2. Experimental Results with the Subtractive Clustering Method

Table 2. Results of modelling and prediction using ANFIS subtractive clustering.

Crt. No.	Model Parameters		Results		Training			Testing
	N	IR	Training Time	Rules (Clusters)	MSE	RMSE	R	MSE	RMSE	R
1	2	0.05	2.6366	19	0.000164	0.012797	0.97769	0.000175	0.013212	0.95115
2	2	0.07	1.2458	11	0.000189	0.01373	0.97427	0.000169	0.012988	0.95439
3	2	0.1	0.99668	9	0.000189	0.013749	0.9742	0.000157	0.012533	0.95648
4	2	0.2	0.36192	4	0.000216	0.014681	0.97053	0.000143	0.011957	0.96026
5	2	0.3	0.27515	3	0.00021	0.014493	0.97129	0.000161	0.012684	0.95557
6	2	0.4	0.26771	3	0.000209	0.014447	0.97147	0.000159	0.012627	0.95601
7	2	0.5	0.18152	2	0.000232	0.015229	0.96825	0.000153	0.012367	0.95812
8	2	0.6	0.19457	2	0.000232	0.015228	0.96826	0.000153	0.012368	0.95813
9	3	0.05	6.8699	25	0.00014	0.011816	0.98098	0.00025	0.015821	0.93191
10	3	0.07	2.9711	15	0.000164	0.012822	0.97757	0.00018	0.013433	0.95043
11	3	0.1	1.6277	10	0.000168	0.012966	0.97705	0.000193	0.013895	0.9469
12	3	0.2	0.68568	5	0.000197	0.014045	0.97302	0.00015	0.012258	0.95935
13	3	0.3	0.38495	3	0.000206	0.014368	0.97175	0.000145	0.012047	0.96081
14	3	0.4	0.3931	3	0.000205	0.014308	0.97198	0.000143	0.011964	0.96112
15	3	0.5	0.24955	2	0.000213	0.014597	0.97082	0.00015	0.012242	0.96041
16	3	0.6	0.25627	2	0.000214	0.014645	0.97063	0.000154	0.012418	0.95936
17	4	0.05	11.83	27	0.000103	0.010159	0.98594	0.000703	0.026509	0.80328
18	4	0.07	4.7695	16	0.000145	0.012044	0.98018	0.000263	0.016218	0.92704
19	4	0.1	2.6738	11	0.000161	0.012704	0.97792	0.000218	0.014777	0.94118
20	4	0.2	0.92833	5	0.000197	0.01403	0.973	0.000209	0.014447	0.94156
21	4	0.3	0.50229	3	0.000215	0.014665	0.97047	0.000152	0.01231	0.95909
22	4	0.4	0.51696	3	0.000204	0.014278	0.97203	0.000157	0.012518	0.95738
23	4	0.5	0.50626	3	0.000205	0.014312	0.97189	0.000151	0.012301	0.95872
24	4	0.6	0.32582	2	0.000211	0.014513	0.97108	0.000167	0.012931	0.95591
25	4	0.7	0.32674	2	0.000211	0.01453	0.97101	0.00017	0.013053	0.9548
26	5	0.05	81.067	65	5.29E-05	0.007276	0.99279	0.000339	0.018403	0.91364
27	5	0.07	6.9995	17	0.000114	0.010675	0.98441	0.000203	0.014232	0.94381
28	5	0.1	3.5888	11	0.000165	0.012832	0.97739	0.000165	0.012826	0.95632
29	5	0.2	1.1733	5	0.000184	0.013579	0.97465	0.000185	0.013611	0.94962
30	5	0.3	0.88894	4	0.000206	0.014364	0.97159	0.000154	0.012413	0.95943
31	5	0.4	0.64202	3	0.000213	0.014597	0.97065	0.00016	0.012656	0.95706
32	5	0.5	0.62356	3	0.000203	0.01426	0.97201	0.00015	0.012255	0.95936
33	5	0.6	0.39809	2	0.00021	0.01448	0.97112	0.000169	0.012992	0.95559
34	5	0.7	0.40196	2	0.000209	0.014466	0.97118	0.000168	0.012951	0.95578

shows the results of modelling and prediction using the ANFIS subtractive clustering process. In addition, four ANFIS models using this approach are depicted in Figure 11a–d. The parameters used in these architectures are N, the number of inputs represented as in the previous method, the number of samples and the influence radius, IR, which gives the number of clusters, C. This type of ANFIS model differs from those obtained using the grid partition method. There are no very large numbers of neurons in layers 3 and 4, nor are there many rules. This ensures that the training time will not increase significantly.

Analysing the modelling results, it can be concluded that the ANFIS architecture will not produce many rules in layer 3 using this method, subtractive clustering, unless the radius of influence is very small. When using this method, on the other hand, the smaller the radius of influence, the faster the number of fuzzy sets increases. The number of fuzzy sets in layer 2 does not increase beyond a certain radius of influence. Thus, over a radius of influence of 0.7, the ANFIS architecture is practically difficult to realize since it will result in less than two fuzzy sets, which is impossible. However, if the radius of influence is set to less than 0.1, the number of clusters, or fuzzy sets on each input, may increase. The training time increases with N = 5 and r = 0.05, as seen in Table 2, and the network architecture contains 65 clusters, or 65 fuzzy sets. However, there are practically not as many rules in this case as there are in ANFIS grid partition.

The modelling and prediction results for the architecture in line 31 in Table 2 are shown in Figure 12a,b for the training and testing stages, respectively. As a result, in Figure 12a for training and Figure 12b for testing, the calculated active power and that obtained by training, as well as the errors, are shown. Figure 13 shows the regression coefficients for training and testing, respectively.

3.3. Experimental Results with ANFIS FCM

The ANFIS FCM method, like the methods shown above, evaluated multiple ANFIS models for a number of previous samples (N = 2, 3, 4, 5).

Table 3. Results of modelling and prediction using the ANFIS FCM.

Model Parameters			Training Time	Training			Testing
Nr. Crt	N	C		MSE	RMSE	R	MSE	RMSE	R
1	2	2	0.18246	0.000212	0.014566	0.971	0.000163	0.012774	0.95502
2	2	3	0.26939	0.000219	0.014804	0.97003	0.000152	0.012332	0.95856
3	2	4	0.36268	0.000213	0.014606	0.97083	0.000154	0.012402	0.95787
4	2	5	0.4642	0.00022	0.014847	0.96985	0.000146	0.012062	0.96012
5	2	6	0.56686	0.000204	0.014285	0.97212	0.000151	0.012291	0.95913
6	2	7	0.68551	0.000199	0.014095	0.97287	0.000161	0.012679	0.95627
7	2	9	0.94153	0.000191	0.013825	0.97391	0.000146	0.012074	0.95988
8	2	10	1.0583	0.000186	0.013642	0.97461	0.00017	0.01305	0.95276
9	2	11	1.1927	0.000184	0.013553	0.97494	0.000155	0.012437	0.95739
10	2	12	1.325	0.000183	0.013528	0.97504	0.000157	0.012519	0.95648
11	2	14	1.6644	0.00018	0.013416	0.97545	0.000174	0.013174	0.95164
12	2	15	1.8287	0.000176	0.01326	0.97603	0.000194	0.013933	0.94554
13	3	2	0.25381	0.000226	0.015024	0.96906	0.000145	0.012034	0.96073
14	3	3	0.38316	0.00021	0.014508	0.97118	0.000148	0.012178	0.96031
15	3	4	0.52299	0.000216	0.014703	0.97039	0.000155	0.012461	0.95803
16	3	5	0.67229	0.000201	0.01417	0.97253	0.000147	0.012106	0.96111
17	3	6	0.83768	0.000197	0.014041	0.97303	0.000174	0.013204	0.95264
18	3	7	1.0275	0.000194	0.01394	0.97343	0.000163	0.012782	0.95514
19	3	9	1.3958	0.000178	0.013334	0.97572	0.000231	0.015208	0.93711
20	3	10	1.5929	0.000169	0.013003	0.97692	0.000219	0.014794	0.93913
21	3	11	1.8357	0.00017	0.013043	0.97683	0.000179	0.013384	0.93892
22	3	12	2.0515	0.000167	0.012923	0.97721	0.000222	0.014888	0.93988
23	3	14	2.5393	0.00015	0.012242	0.97957	0.000274	0.016567	0.92246
24	3	15	2.8494	0.000159	0.012601	0.97834	0.000186	0.013638	0.94894
25	4	2	0.33254	0.000222	0.014903	0.96948	0.000156	0.012475	0.95761
26	4	3	0.50614	0.000206	0.014352	0.97173	0.000145	0.012058	0.96075
27	4	4	0.70743	0.000211	0.014518	0.97106	0.000152	0.01234	0.96
28	4	5	0.92273	0.0002	0.014145	0.97255	0.000156	0.012505	0.95851
29	4	6	1.1462	0.000192	0.013842	0.97373	0.000153	0.012387	0.95944
30	4	7	1.4101	0.000189	0.013731	0.97416	0.000166	0.01287	0.95709
31	4	9	1.9151	0.000177	0.013288	0.97582	0.000211	0.014541	0.94267
32	4	10	2.2161	0.000174	0.013179	0.97622	0.000208	0.014409	0.94369
33	4	11	2.5697	0.00016	0.01266	0.97808	0.000219	0.014809	0.9395
34	4	12	2.91	0.000158	0.012588	0.97833	0.000242	0.015555	0.93197
35	4	14	3.6893	0.000139	0.011788	0.98102	0.000238	0.015436	0.93395
36	4	15	4.1067	0.000127	0.011258	0.9827	0.002198	0.016886	0.50124
37	5	2	0.4067	0.000221	0.014864	0.96954	0.000157	0.012537	0.95728
38	5	3	0.63525	0.000208	0.014419	0.97137	0.000154	0.012406	0.95852
39	5	4	0.87649	0.000212	0.014547	0.97085	0.000164	0.012789	0.95718
40	5	5	1.1637	0.000203	0.014261	0.972	0.000161	0.012694	0.95738
41	5	6	1.4853	0.000184	0.013572	0.97468	0.000154	0.012421	0.95935
42	5	7	1.7939	0.000186	0.013649	0.97438	0.000167	0.012937	0.95655
43	5	10	3.0054	0.000157	0.012531	0.97845	0.000227	0.015052	0.93749
44	5	9	2.5105	0.00016	0.012657	0.97801	0.000226	0.01502	0.93802
45	5	11	3.3503	0.000154	0.01241	0.97887	0.000248	0.015741	0.9317
46	5	12	3.885	0.000147	0.012107	0.9799	0.000261	0.016166	0.92824
47	5	14	4.8099	0.000138	0.011733	0.98113	0.000276	0.016606	0.92385
48	5	15	5.4863	0.000124	0.011137	0.98302	0.000317	0.017812	0.9135

summarizes these results. ANFIS models are shown in Figure 14a–d. The network architecture is directly influenced by the number of clusters. Figure 10 from the ANFIS subtractive clustering method shows that the architectures are comparable. In fact, the method is very similar to the one described in the previous paragraph.

The number of clusters in this architecture, which is also based on the clustering technique, is specified at the beginning, and is, therefore, not determined by other parameters such as the radius of influence. The network design is similar to the subtractive clustering technique in that it is based on the same principles. Similarly, the network’s architecture is highly dependent on the number of previous training samples and the number of clusters.

As compared to the subtractive clustering method, it is clear that the two methods are very close. This is because, in addition to the number of previous samples, the model contains a parameter with a very similar influence: in the FCM method, this parameter is the number of clusters, which is equal to the number of neurons in the rule layer, and thus to the number of rules. While the previous method used the term “influence radius”, this term actually refers to the number of clusters (or rules). When the structures of the models for the two methods are compared, as shown in Figure 10 and Figure 14, as well as in Table 2 and Table 3, it is clear that the structures are very similar and that some of the parameters are identical. However, when either of the ANFIS methods—subtractive clustering or ANFIS FCM—is compared to the method based on input space partitioning, in which the ANFIS model’s parameter is the number of fuzzy sets for each input, it is found that there is no similarity between the models. Even if we choose a small number of fuzzy sets for each input, we still end up with a large number of rules. To summarize, the ANFIS subtractive clustering and ANFIS FCM methods are more appropriate for this type of modelling and prediction. The modelling and prediction results for the model for N = 3 and C = 11 are shown in Figure 15 and Figure 16.

4. Conclusions

Analysing the modelling by the ANFIS method, it can be seen that the modelling results depend on the type of architecture used, but not very much. When using the ANFIS grid partition, the ANFIS architecture is highly dependent on the number of previous samples used in modelling. If more than 5 previous samples are used, many rules will result in the rule layer, even for a small number of fuzzy sets in the input layer. Regarding the subtractive clustering method and the FCM method, it can be seen that the results are similar. The difference is that the FCM specifies the number of clusters in advance, while in the subtractive clustering method, the number of clusters results after establishing the value for the influence radius parameter. In any case, in both the subtractive clustering method and FCM method, the training time is not high. That is because there are not many rules in the rule layer.

It can be said that modelling and prediction with ANFIS works well and that it is useful to better understand how reactive power varies in a power station of electric locomotives. A better understanding of these phenomena also facilitates actions that can be taken to limit the factors that lead to decreased power quality.

The RMSE value during testing is another indicator that the modelling produced good results. For modelling with the ANFIS grid partition, this value falls within the ranges of 0.012 and 0.015, and for this range, the regression coefficient is between 0.92 and 0.95. Further, the modelling is efficient for ANFIS subtractive clustering, obtaining RMSE between 0.011 and 0.02 during the testing phase. Additionally, the RMSE for modelling with ANFIS FCM is between 0.012 and 0.018. These demonstrations are for configurations that avoid overtraining.

Considering Table 1, Table 2 and Table 3, it is possible to say that there are ANFIS configurations in which the phenomenon of overtraining can occur. This happens particularly when too many rules are implemented, which results in good performance during the training stage but limits the system’s ability to generalize. Due to the ability to produce numerous rules even for a small number of fuzzy sets in this architecture, the ANFIS grid partition structure is particularly susceptible to the overtraining phenomenon.

The regression coefficient does not drop below 0.9 for the parameters at which the phenomenon of overtraining does not occur, according to an analysis of all three approaches for ANFIS. This supports the earlier conclusion that using ANFIS, one may achieve an accurate modelling of reactive power variation that is helpful for making short-term predictions.

Future modelling efforts will make use of trendy ANN architectures, such as convolutional ANNs developed on deep learning.

A general conclusion about modelling with ANFIS, regardless of the chosen architecture, is that it works well. The parameters of the models must be chosen so that the training time does not increase too much because, in addition to this increase in training time, most of the time there is an overtraining phenomenon that causes the network to not work properly during the testing phase and lose its ability to generalize.