Keywords

1 Introduction

HVDC transmission system is gathering attention of engineers and investors because of its promising attitude towards bulk power transfer with the aim of improving the standards of living. In addition to this, HVDC transmission system is superior to conventional AC transmission system because of its low losses, ability to interconnect unsynchronized grids, minimum right of way, capability of transferring bulk power, and installation as submarine cables. Therefore, HVDC transmission is basically the renaissance of DC currents in the war of currents [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

Researches have been conducted passionately over the decades for development of methods of fault detection, classification and location in HVDC transmission lines. It is equally important along with the growing concept of smart grids to build an intelligent system of fault monitoring and diagnosis (FMD). Protection system must be capable of detecting, classifying and locating faults in a better way [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41].

Detection, classification and finding location of faults in HVDC transmission are found to be the most blazing research areas in power system. This is entirely because of the intentions of interconnection of non-synchronized grids and bulk power transfer in a reliable fashion. Developments in digital signal processing techniques, artificial intelligence, global positioning system (GPS), communication and machine learning algorithms have opened doors for the researchers to carry out studies in a way so that the confinement associated with the traditional methods could be dealt and addressed successfully and effectively [24, 26, 28, 30,31,32,33,34,35,36,37,38,39,40,41].

When a DC fault occurs in HVDC transmission line, it grows rapidly and reaches a very high steady state value. Because of this sudden growth and higher steady state value, converter stations and equipment associated with HVDC transmission undergo severe danger of failures. In addition to this, this sudden growth and higher value of DC fault current are beyond the current interrupting and limiting capacity of conventional circuit breakers. Increase in DC fault current is required to be detected rapidly so that its effects could not be expanded to system. Therefore, in order to overcome this issue of DC fault, extensive studies are carried out over modified forms of relaying mechanisms. In short, there are basically two ways or strategies to counter DC fault current. In selective fault clearing strategy, only circuit breakers associated with the faulted line is used to interrupt fault current but in non-selective fault clearing strategy, fault current is interrupted by combined actions of multiple components such as the converters with the facilities of fault interrupting and limiting capability. Because of too much complexity in non-selective fault clearing strategy and fear of increasing capital expense of converter stations, it is practically not feasible to apply over large power systems [23, 24, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59].

DC fault at the terminals is detected by the rate of change of electrical parameters such as voltage and current. Sudden rise in current or sudden drop in voltage is a clear indication of origination of DC faults. Changes in voltage and currents are observed by optical transducers installed to measure these values. These values are converted to lower levels to make them applicable for relay sections. Relay senses them and prepares an action according to the rated capacity of flow of current or of voltage at terminals. This rate of change of voltage or current is compared with the threshold values of voltage or current to prepare a prompt action. When the fault is very closed to the converter stations or magnitude of fault is small, then this rate of change of voltage or current is not sufficient to prepare an interrupting action [42,43,44,45, 47, 55]. There must be additional techniques required to extract an in-depth knowledge available in these electrical parameters. This strikes an idea to implement analysis based on time and frequency domain. Fourier transform (FT) technique is applied to analyze the voltage and current signals in frequency domain [60,61,62,63,64,65,66,67]. Because of less information, wavelet transform (WT) technique is developed to observe time and frequency domain of electrical signals simultaneously. Fine analysis requires extensive computation in WT. In addition to this, although discrete WT is computationally efficient yet discretization involves reduction in efficiency of wavelets. Therefore, alternative techniques are taken into consideration or these techniques are supported with further advanced methods [47, 60,61,62,63,64,65,66,67,68,69,70,71,72,73,74].

Machine learning is an emerging approach for fault diagnosis in HVDC transmission lines. It is a technique in which information is acquired without the need of accurate programming. Hence, this reduces computational burden without questioning the accuracy and efficiency. Features are extracted from the voltage and current signals to prepare a testing and training data. In this way, algorithms are trained for a particular event so that they could respond to testing data accordingly [75,76,77,78,79,80,81,82,83,84]. In the literature, there are a vast number of machine learning algorithms. Broadly, they are classified into two techniques. Firstly, is the supervised learning, in which information is evaluated on the basis of available pre-defined tags. Secondly is the unsupervised learning in which unknown data (testing data) is evaluated without the existence of preset tags. Classification and regression are the examples of supervised learning. Clustering and association are the examples of unsupervised learning [85,86,87].

In this research, HVDC transmission line is simulated under normal and faulty conditions. Moreover, restricted Boltzmann machine learning (RBML) algorithm is developed and implemented. This algorithm is applied and tested for fault estimation in HVDC transmission lines. Matlab/Simulink is used for the simulation.

This research paper consists of following sections: Sect. 1 covers the introduction and literature review regarding fault estimation in HVDC transmission system. Section 2 consists of explanation of HVDC transmission system. Restricted Boltzmann machine is explained in Sect. 3. Contrastive divergence based learning approach of restricted Boltzmann machine is covered in Sect. 4. Steps involved in methodology of fault estimation in three terminal HVDC transmission system is added in Sect. 5. Section 6 contains simulation analysis of test model of three terminal HVDC transmission system. Conclusion is given in Sect. 7.

2 HVDC Transmission System

In high voltage direct current (HVDC) transmission system, direct current (DC) is used for transmission of bulk power. This transmission system possesses significantly low losses in the case of long distanced transmission but has an expensive infrastructure. This system is used as submarine cable transmission without taking into consideration of charging and discharging of capacitance of line in each cycle with heavy currents. In case of short distances, its high cost of deployment is justified because of numerous advantages of HVDC transmission system. Voltage range between 100 kV to 800 kV is usually utilized in HVDC system. In the end of 2019, a new voltage value of 1100 kV will be operational for HVDC system in China [2,3,4,5,6, 8, 9, 15, 18, 20, 23, 24].

Non synchronized alternating current (AC) transmission systems can be interconnected with HVDC transmission link. AC transmission systems working at different frequencies like at 50 Hz and 60 Hz can be integrated with help of HVDC link. Hence, stabilization can also be achieved with HVDC transmission systems in the case of incompatible networks. Control of HVDC link is independent of phase angle between load and source. This concept helps in stabilization of power networks against any disturbances resulted from the change in power [2, 20, 23, 24].

Practically, most of HVDC transmission technology deployed around the globe is the technology developed in 1930s in Sweden and in Germany. In 1951, commercial installation was made between Moscow and Kashira in Russia. In 1954, 100 kV and 20 MW system was installed between Gotland and Mainland Sweden. Rio Madeira link is the longest link established in Brazil consisting of two bipoles of ±600 kV, 3150 MW. Porto Velho in the Rondonia is connected to Sao Paulo Area with long DC link of 2,375 km. ABB group is going to build an ultra-high voltage direct current (UHVDC) land link of 1100 kV and 12 GW for a distance of 3000 km. It is going to be the largest DC link with the highest bulk power capacity [2,3,4,5,6, 18, 20].

The idea of transmission at high voltage is driven because of significant losses and reduction in the lives of equipment deployed in the case of transmission at high currents. High voltages are beneficial for long distance transmission. Because of AC generation systems, three phase AC voltages are converted to high DC voltages. Due to advancement in power electronics in 1970s, it is practically feasible to convert AC to DC and vice versa with the semiconductor components like thyristors, insulated gate bipolar transistors (IGBT), etc. Therefore, based on the semi-conductor technology, HVDC transmission system is broadly classified into two categories: First is Line Commutated Converter in which thyristor technology is deployed for conversion. External AC circuit is provided for turning off and on the thyristor. Second is the voltage source converter (VSC) in which IGBT based converter stations are developed. IGBT technology provides more variants than thyristor based technology. Multiple grid connections, effective controlling, renewable energy integration and self-commutation are the notable features of VSC based HVDC system. Recently, researchers are very much attracted towards development of VSC based HVDC system as an advantageous substitute of thyristor based HVDC system [2,3,4, 17,18,19,20,21,22]. Comparison of LCC - HVDC and VSC - HVDC transmission system is presented in Table 1.

Table 1. Comparison of LCC- HVDC and VSC – HVDC transmission system
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2.1 VSC Based HVDC Transmission System

Self-commutating switches, e.g. gate turn-off thyristors (GTOs) or insulated gate bipolar transistors (IGBTs) are used in voltage source converters (VSC). These switches can be turned on or off in a controlled manner. High switching frequency based pulse width modulation is utilized in the operation of voltage source converters [23, 25, 27, 28, 31, 40, 47, 56, 58, 62, 64, 69, 78].

Typical configuration of VSC based HVDC system is presented in Fig. 1.

Fig. 1.
figure 1

Typical three terminal VSC – HVDC transmission system

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VSC-based HVDC transmission system is composed of two VSCs, transformers, phase reactors, ac filters, dc-link capacitors and dc cables [90,91,92,93].

VSCs are connected at the sending and receiving ends of transmission line. Both VSCs have same configuration. One of the VSCs works as a rectifier and the other VSC works as an inverter. Therefore, it can be said that these converters are back to back converters. Normally, transformers are used to connect AC system to converters. In addition to this, transformer also transforms the AC voltage level to DC voltage level. Three winding type transformers are usually deployed and configuration of transformer can be altered with respect to rated power and transformation requirement. Phase reactors act as current regulators for controlling the flow of active and reactive power. These phase reactors also work as AC filters to mitigate the effects of high frequency harmonic contents developed in AC currents because of the switching operation of IGBTs. Two equally sized capacitors are installed at the DC side of converter stations. Sizes of these capacitors are dependent on DC voltage. DC capacitors provide a low inductance path for the turned off current. These capacitors also act as energy storage components in order to control the flow of power. Moreover, these capacitors also decay down the voltage ripples on DC side. Harmonics are blocked by AC filters from entering the AC system. Insulation of DC cable used in VSC based HVDC system is composed of extruded polymer. This material is resistant to DC voltage. Mechanical strength, flexibility and low weight are the notable features of this polymer cable. AC side of VSC station works as a constant current source. Therefore, inductor is used as an energy storage component. Elimination of harmonics is required which is carried out by small AC filters. VSC acts as a constant voltage source on DC side of converter station. Therefore, capacitors are installed which work as energy storage components. In addition to this, these capacitors provide the facility of DC filters. High switching losses are associated with VSC based HVDC system. These losses are reduced by using commutation scheme of soft switching. Controlling of power is done in the same way in VSC based HVDC system as in the case of LCC based HVDC system. Active power is controlled at inverter side and DC voltage is controlled at rectifier station. Fast inner current control loop accompanied by several outer control loops makes the realization of control system of VSC based HVDC system [23, 24, 88, 92,93,94,95].

AC currents are controlled by rapid inner current control located at the base level of control system of VSC based HVDC system. Reference AC currents are obtained by outer controllers. These outer controllers are relatively slow and include DC voltage controller, AC voltage controller, Active power controller, reactive power controller and frequency controller. Reference of active current is obtained from DC voltage controller, active power controller or from frequency controller whereas, reference of reactive current can be extracted from reactive power controller or from AC power controller [24, 88,89,90, 96, 97].

3 Restricted Boltzmann Machine (RBM)

Deep belief networks (DBNs) was introduced by Hinton along with his team at University of Toronto in 2006. It was proposed with a learning algorithm. In DBNs, one layer is trained greedily at a time and unsupervised learning algorithm known as Restricted Boltzmann Machine (RBM) is exploited for each layer. Architectural depth of brain has been working as a source of motivation for neural network researchers to train deep multi-layer networks for decades. Machine learning model is built with multiple hidden layers. Layer feature transformation is applied for training samples layer, for training the samples in the feature space for representation of new features in another space in order to achieve desired results of experiment. Classification, regression, information retrieval, dimensionality reduction, modeling motion, modeling textures, object segmentation, robotics, natural language processing and collaborative filtering are the emerging examples solved by DBNs [98,99,100,101,102,103,104].

Restricted Boltzmann machine (RBM) is composed of two layers of units. One is visible unit represented by \( V = \left( {V_{1} , V_{2} , \ldots , V_{m} } \right) \) and other is hidden unit denoted by \( H = \left( {H_{1} , H_{2} , \ldots , H_{m} } \right) \). RBM is a probabilistic model in which hidden binary variables are utilized to model the distribution of variables of a visible layer as shown in Fig. 2.

Fig. 2.
figure 2

Graphical representation of Restricted Boltzmann Machine with hidden and visible layers.

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Observed data is represented by visible units. Relation between two observed variables is captured by hidden units. Since all the units are composed of binary variables,

$$ \left( {v,h} \right) \in \left\{ {0,1} \right\}^{m + n} $$
(1)

Energy function of RBM is defined as:

$$ E\left( {v,h |\theta } \right) = \mathop \sum \limits_{i = 1}^{n} a_{i} v_{i} - \mathop \sum \limits_{j = 1}^{m} b_{j} h_{j} - \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1}^{m} v_{i} w_{ij} h_{j} $$
(2)

Where \( \theta = \left\{ {w,a,b} \right\}, a = \{ a_{i} , i = i, 2, \ldots , n\} \;{\text{and}}\;b = \left\{ {b_{j} , j = 1, 2, \ldots ., m} \right\} \) are the biases term associated to visible and hidden units respectively. Conditional probability function is given as:

$$ P\left( {v,h |\theta } \right) = \frac{{{ \exp }\left( { - E\left( {v,h |\theta } \right)} \right)}}{Z\left( \theta \right)} $$
(3)

Where \( Z\left( \theta \right) \) is a partition function or normalization constant and it ensures the validity of probability distribution given as:

$$ Z\left( \theta \right) = \sum\nolimits_{v,k} {exp \left( { - E\left( {v,h |\theta } \right)} \right)} $$
(4)

Likelihood function provides an information about how well the data summarizes these parameters and given as:

$$ P\left( {v |\theta } \right) = \frac{{\mathop \sum \nolimits_{h} { \exp }\left( { - E\left( {v,h |\theta } \right)} \right)}}{Z\left( \theta \right)} $$
(5)

Training of restricted Boltzmann machine is done to evaluate the unknown parameters. The derivatives or gradient of log likelihood function with respect to model parameters θ is calculated to minimize the log likelihood function as:

$$ \frac{\partial \;log\;P(v|\theta )}{\partial \theta } = \mathop \sum \limits_{t = 1}^{T} \{ \frac{{\partial \left( { - E\left( {v^{\left( t \right)} ,h |\theta } \right)} \right)}}{\partial \theta }\}_{{P(h|v^{\left( t \right)} ,\theta )}} - \mathop \sum \limits_{t = 1}^{T} \{ \frac{{\partial \left( { - E\left( {v,h |\theta } \right)} \right)}}{\partial \theta }\}_{P(v,h|\theta )} $$
(6)

Parameters \( w, a \) and \( b \) are calculated by finding the gradient of log likelihood function as:

$$ \begin{aligned} & \frac{\partial \;log\;P(v|\theta )}{{\partial w_{ij} }} = \{ v_{i} h_{j} \}_{data} - \{ v_{i} h_{j} \}_{model} \\ & \frac{\partial \;log\;P(v|\theta )}{{\partial b_{j} }} = \{ h_{j} \}_{data} - \{ h_{j} \}_{model} \\ & \frac{\partial \;log\;P(v|\theta )}{{\partial a_{j} }} = \{ v_{i} \}_{data} - \{ v_{i} \}_{model} \\ \end{aligned} $$
(7)

4 Contrastive Divergence (CD) Algorithm

Contrastive divergence (CD) is an approximate maximum likelihood learning algorithm proposed by Geoffery Hinton. This algorithm follows the gradient of the differences of two divergences. This algorithm helps in reducing difficulty of computing log likelihood function.

Training of data is usually conducted by this CD algorithm in RBM [105, 106]. In this algorithm, sample for training, number of hidden layers, learning rate and maximum training cycle are specified and are taken as input. Outputs are usually weight w, bias of hidden layer b, bias of visible layer a. Weights are optimized to train product of expert models. Gibbs sampling is performed in this algorithm and is used inside a gradient descent procedure for the evaluation of weights. Visible layer is usually taken as initialization unit \( V_{1} = X \) in training. Minimum values of \( w, a \) and \( b \) are selected randomly.

figure a

In this research, classifier of RBM is constructed. In this classifier model, there are two layers. Lower layer is stacked by a number of layers of RBMs. Upper layer is added containing the desired output variable. This is the classification layer. In the top level units, softmax classifier outputs are used. It is a sample in which probability of different states are added and maximum probability of the state is classified. Proposed flow diagram of CD learning based RBM is shown in Fig. 3.

Fig. 3.
figure 3

Contrastive Divergence Learning based Restricted Boltzmann Machine

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5 Methodology

Following are the steps involved in the implementation of contrastive divergence learning based restricted Boltzmann machine for fault estimation in MT- HVDC transmission system.

5.1 Data Selection

Voltage and current samples are prepared based on the observations made at converter stations, working as rectifier or inverter, in VSC based HVDC transmission system. Voltage and current values are recorded before and after the occurrence of faults. This data is untagged and it can be employed as pertaining sample. Tuning is done by the application of small amount of tagged samples of current and voltage values recorded at converter stations of VSC based HVDC transmission system.

5.2 Characteristics of Selected Variables

Variables are selected as features of model’s input. These variables are basically DC value of the fault, its dominating frequency component and value of total harmonic distortion (THD). These variables depict the normal and faulty state of multi terminal HVDC transmission system. Contrastive divergence (CD) trains the Restricted Boltzmann machine (RBM) on the characteristics of these variables.

5.3 VSC Based HVDC System Status Codes

Fault estimation is basically a multi classification task. This diagnostic technique is divided into following categories based on fault type and fault distance.

  1. 1.

    Pole to Ground Fault

  2. 2.

    Pole to Pole Fault

  3. 3.

    Pole to Pole and Ground Fault

  4. 4.

    AC Fault

  5. 5.

    Fault at Converter Station

  6. 6.

    Fault at 50 km

  7. 7.

    Fault at 100 km

  8. 8.

    Fault at 200 km

In this research, fault distance and measurement points are employed for fault estimation in multi terminal HVDC transmission system.

5.4 Implementation of Fault Estimation Process

Sample and characteristics variables are selected. Classification model is built based on RBM in which training of untagged sample is done through contrastive divergence (CD) algorithm.

6 Simulation Results

Simulation environment is created in Matlab/Simulink. Three terminal VSC – HVDC model is developed as a test specimen. Parameters associated with this test model of VSC HVDC system is given in Table 2.

Table 2. Parameters of proposed three terminal HVDC transmission test model
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In this test system, there are basically three converter stations. One is rectifier (VSC – I) and two of them are inverters (VSC – II and VSC – III). The DC transmission cable between VSC – I and VSC – II is 200 km long. The DC transmission cable between VSC – I and VSC – III is 100 km long. This test system is simulated under healthy and faulty conditions to analyze it’s working as shown in Fig. 4. Figure 4(a) depicts the current of rectifier station under normal conditions. Values of current are graphically displayed against time. Current is reaching steady state value of 1.463 Amperes in less than 0.2 s. In the similar fashion, current attains a steady state value of 2.48 Amperes and 1.074 Amperes in minimum time at converter station II (inverter) and converter station III (inverter) as shown in Fig. 4 (b) and Fig. 4 (c) respectively. This depicts the achievement of stability of HVDC system in minimum possible time. All the converter stations are working under normal conditions. Figure 4 (d) depicts the variation in DC current measured at converter station I (rectifier) under DC fault on a line of 200 km long. Variations are quite visible under fault condition. DC current increases and is found to be of 1.629 Amperes. In the same way, DC current is measured at converter station II and is found to be of 2.334 Amperes as shown in Fig. 4 (e). As the fault occurs at 200 km (very closed to converter station II) from rectifier station, so no significant change in current is observed at converter station II. It is observed that a current decreases to 0.09625 Amperes at VSC – III which is a big variation as shown in Fig. 4 (f). This decrease in DC current is an indication that a neighboring line is under fault condition. When a fault occurs on DC transmission cable connected between VSC – I and VSC – III, increase in DC current is observed at converter station I as shown in Fig. 4 (g). Figure 4 (h) shows that no significant change is observed at converter station III because DC fault is very closed to it. However, decrease in DC current is found at converter station II which is an indication that the neighboring line is under fault as depicted in Fig. 4 (i). Table 3 summarizes the change in value of DC current with respect to different locations of fault and with respect to its measurement points.

Fig. 4.
figure 4figure 4

This figure shows the following conditions. (a) Rectifier station current under normal conditions (b) Inverter station (VSC – I) current under normal conditions (c) Inverter station (VSC – II) current under normal conditions (d) Rectifier station current under DC fault conditions at VSC - II (e) Inverter station (VSC- II) current under DC fault condition at VSC - II (f) Inverter station (VSC – III) current under DC fault condition at VSC - II (g) Rectifier station current under DC fault condition at VSC - III (h) Inverter station (VSC – II) current under DC fault condition at VSC - III (i) Inverter station (VSC – III) current under DC fault condition at VSC – III.

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Table 3. Nature of DC current under different sates of system
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Figure 5 depicts the variations in error vector with respect to change in time steps. It is found that error decreases with the increase in time steps. This error vector is obtained from the contrastive divergence learning of restricted Boltzmann machine (RBM). After training of RBM, this machine is tested under different conditions in a sample of 1000 time steps. Sampling patterns are different under different conditions which help in estimation of fault in MT- HVDC transmission system as shown in Fig. 6. In the normal condition, highest variation in sampling pattern is observed at a time step just before 600 as shown in Fig. 6 (a). In the fault condition of transmission line of 200 km long, highest variation is observed at a time step near to 200 as depicted in Fig. 6 (b). In Fig. 6 (c), highest variation is observed at time step closed to 150. These variations aid in determining the fault in three terminal HVDC test model. These variations are actually the gradient of difference of divergence utilized in restricted Boltzmann machine. These results help in determining the fault conditions established near converter stations.

Fig. 5.
figure 5

Error vector with respect to time steps obtained while training RBM with contrastive divergence (CD) algorithm.

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Fig. 6.
figure 6

This figure depicts the following conditions: (a) Sampling pattern of normal condition of test model in 1000 time steps. (b) Sampling pattern of faulty conditions at inverter station (VSC – II) in test model in 1000 time steps. (c) Sampling pattern of faulty conditions at inverter station (VSC – III) in test model in 1000 time steps.

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7 Conclusion

Restricted Boltzmann machine is an emerging technique to estimate fault in VSC – HVDC transmission system. In this research, contrastive divergence based learning is developed for restricted Boltzmann machine. The gradient of the difference of divergence is evaluated under different states of HVDC transmission model. Based on the gradient of difference of divergences, healthy and faulty state of HVDC system are found effectively. Moreover, change in the value of gradient with respect to fault distance helps in determining the fault location in multi terminal HVDC transmission system. In future, this idea can be extended to classification with respect to different type of faults.