#
The Application of Ant Colony Algorithms to Improving the Operation of Traction Rectifier Transformers^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. The General Issue: Applying the Swarm Intelligence Algorithm to Optimization Problems

## 2. Tram Traction Substations in Poland: Voltage Rectification System

## 3. Voltage and Current Waveforms in Transformer-Rectifier Set

_{b2c2}voltage waveform, which is present and operates at the output terminals (+, -) (DC traction line terminals—see Figure 3). The shape of this waveform corresponds to a top fragment of sine waveform (this particular drawing refers to purely sinusoidal supply voltage, hence all pulses are symmetrical: duration of each pulse is 30°, amplitudes of successive pulses are equal). A more realistic case is when voltages are not equal in magnitude, even though the phase shifts remain same as before (let it be noted, that phase shift, or angular displacement of one voltage against the other, depends on spatial distribution of windings in transformer). The outcome is that pulsations in DC (output) voltage are increased (magnitude of voltages corresponding to rectified voltages of one winding is greater) and time durations are different (see Figure 4b). This may occur when numbers of turns in star-connected winding and delta-connected winding are such that induced phase-to-phase voltages are not exactly the same. Ideally, numbers of turns should meet the following conditions:

_{1}and N

_{2}relate to delta- and star-connected windings, respectively.

_{5%}= U

_{5}/U

_{1}= 0.06, u

_{7}

_{%}= U

_{7}/U

_{1}= 0.05. When THD coefficient of the supply voltage is calculated, then it is equal to 7.81%; this is less than obligatory limit of 8% enforced by appropriate Polish regulations [19]. An example of system operation, when phase shifts of 5th and 7th harmonic are equal to, respectively, φ

_{5}= 180° and φ

_{7}= 180°, is shown in Figure 6a,b. The equal loading of both windings does no longer occur for unbalance coefficient equal to 1 (unbalance coefficient is defined as the ratio of star to delta winding phase-to-phase voltage). Only these voltages differ slightly, i.e., with unbalance coefficient different from unity; the two windings are evenly loaded (Figure 6b).

## 4. Reduction of AC Component in DC Voltage

_{5}= 0.06 U

_{1}and U

_{7}= 0.05 U

_{1}). The phase shifts for higher harmonics may range from 0° to 180°. In accordance with relationships discussed in the previous section, it is obvious that loading of star- and delta-connected windings will depend on the phase shifts—see Figure 7.

_{DC}) is defined as follows:

_{DCmax}and u

_{DCmin}are maximum and minimum values of the pulsating rectified voltage (cf. Figure 4b: the peaks of phase-to-phase voltages marked in black /these correspond to first secondary winding are equal to u

_{DCmax}, while the intersection points between voltages of first and second winding—known as points of natural commutation—correspond to u

_{DCmin}.

## 5. Optimization of Voltage Transformation—Application of Ant Colony Algorithm

#### 5.1. Reduction of AC Component in DC Rectified Voltage

_{ph}is maximum value of appropriate harmonic, index “1” relates to fundamental, and index “5” and “7” to 5th and 7th harmonic, respectively. Symbols a1, b1, or c1 relate to appropriate phase of the winding.

_{a}, k

_{b}, k

_{c}are termed gain coefficients, and they relate to correction (subtraction or addition) of operating number of turns at the second secondary winding (adjustment of taps by on-load tap changer). This is also called unbalance coefficient.

_{ph}(t) is voltage u

_{a1}or u

_{b1}or u

_{c1}and so on and u

_{phmin}is minimum voltage value; time interval considered is 1/6 of time period corresponding to line frequency 50 Hz; this has been explained before (cf. Figure 9).

_{a}, k

_{b}, k

_{c}diverge from 1 by ±15% (k < 1—decreased number of turns, k > 1—increased number of turns in the winding). For this particular problem, we have tried to adopt slightly different ACO parameters (in the calculations it is possible to change number of ants, number of iterations and neighbourhood parameters, see [18]). We have found that while the results do not change much, the computation time is substantially longer.

#### 5.2. Load Balancing in Transformer

_{1}…t

_{12}. If t

_{i}denotes ith commuting time and $\overline{t}$ is the average time of all commuting times, then we may state the objective function simply as follows:

_{7}we have four columns; each column pair represents difference in phase currents. Before optimization, discrepancy is great; after optimization, phase currents are more or less equal. This means that transformer’s secondary windings, one delta-connected and one star-connected, are loaded almost equally; the same is true of two rectifiers connected to these windings. The entire load of the system is thus evenly distributed between two circuits.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kennedy, J. Particle swarm optimization. In Encyclopedia of Machine Learning; Springer: Berlin/Heidelberg, Germany, 2011; pp. 760–766. [Google Scholar]
- Dorigo, M. Optimization, Learning and Natural Algorithms (in Italian). Ph.D. Thesis, Politecnico di Milano, Milano, Italy, 1992. [Google Scholar]
- Toksari, M.D. Ant colony optimization for finding the global minimum. Appl. Math. Comput.
**2006**, 176, 308–316. [Google Scholar] [CrossRef] - Połap, D.; Woźniak, M.; Napoli, C.; Tramontana, E.; Damaševičius, R. Is the colony of ants able to recognize graphic objects? In International Conference on Information and Software Technologies; Dregvaite, G., Damasevicius, R., Eds.; Springer: Cham, Switzerland, 2015; Volume 538, pp. 376–387. [Google Scholar]
- Woźniak, M.; Połap, D. On some aspects of genetic and evolutionary methods for optimization purposes. Int. J. Electron. Telecommun.
**2015**, 61, 7–16. [Google Scholar] [CrossRef] - Karaboga, D.; Gorkemli, B.; Ozturk, C.; Karaboga, N. A comprehensive survey: Artificial bee colony (ABC) algorithm and applications. Artif. Intell. Rev.
**2014**, 42, 21–57. [Google Scholar] [CrossRef] - Lee, T.-Y. Operating schedule of battery energy storage system in a time-of-use rate industrial user with wind turbine generators: A multipass iteration particle swarm optimization approach. IEEE Trans. Energy Convers.
**2007**, 22, 774–782. [Google Scholar] [CrossRef] - Lee, C.-S.; Ayala, H.V.H.; Dos Santos Coelho, L. Capacitor placement of distribution systems using particle swarm optimization approaches. Int. J. Electr. Power Energy Syst.
**2015**, 64, 839–851. [Google Scholar] [CrossRef] - Kumar, D.; Samantaray, S.R.; Kamwa, I.; Sahoo, N.C. Reliability-constrained based optimal placement and sizing of multiple distributed generators in power distribution network using cat swarm optimization. Electr. Power Compon. Syst.
**2014**, 42, 149–164. [Google Scholar] [CrossRef] - Di Fazio, A.R.; Russo, M.; De Santis, M. Zoning evaluation for voltage optimization in distribution networks with distributed energy resources. Energies
**2019**, 12, 390. [Google Scholar] [CrossRef] - Mohammadi, H.R.; Akhavan, A. Parameter estimation of three-phase induction motor using hybrid of genetic algorithm and particle swarm optimization. J. Eng.
**2014**, 2014. [Google Scholar] [CrossRef] - Wu, Q.; Cole, C.; McSweeney, T. Applications of particle swarm optimization in the railway domain. Int. J. Rail Transp.
**2016**, 4, 167–190. [Google Scholar] [CrossRef] - Zhu, H.P.; Luo, L.F.; Li, Y.; Rehtanz, C. A hybrid active power compensation device for current balance of electrical railway system. In Proceedings of the 2010 International Conference on Power System Technology (POWERCON), Hangzhou, China, 24–28 October 2010; pp. 1–6. [Google Scholar]
- McPherson, G.; Laramore, R.D. An Introduction to Electrical Machines and Transformers, 2nd ed.; Wiley: New York, NY, USA, 1990. [Google Scholar]
- Polish Committee of Standardization. Transformers. General Requirements; Polish Committee of Standardization: Warsaw, Poland, 2011; p. 75. [Google Scholar]
- Sikora, A.; Kulesz, B.; Zielonka, A. Application of swarm algorithm to solving voltage unbalance problem in DC tram traction supply system. In Proceedings of the 2018 Innovative Materials and Technologies in Electrical Engineering (i-MITEL), Sulecin, Poland, 18–20 April 2018; pp. 1–6. [Google Scholar] [CrossRef]
- Sikora, A.; Kulesz, B.; Grzenik, R. 12-pulse and 24-pulse ac/dc voltage transformation circuits. Sci. J. Sil. Univ. Technol. Elektr.
**2015**, 3, 29–64. [Google Scholar] - Di Manno, M.; Varilone, P.; Verde, P.; De Santis, M.; Di Perna, C.; Salemme, M. User friendly smart distributed measurement system for monitoring and assessing the electrical power quality. In Proceedings of the 2015 AEIT International Annual Conference, Naples, Italy, 14–16 October 2015; pp. 1–5, ISBN 978-8-8872-3728-3. [Google Scholar] [CrossRef]
- Ministry of Economy. Ruling on detailed conditions of power engineering system operation. J. Laws
**2007**, 93, 623. [Google Scholar] - Sikora, A.; Kulesz, B.; Zielonka, A. Minimization of power pulsations in traction supply—Application of ant colony algorithm. In International Conference on Information and Software Technologies; Damaševičius, R., Vasiljevienė, G., Eds.; Springer: Cham, Switzerland, 2018; Volume 920, pp. 399–411. [Google Scholar] [CrossRef]

**Figure 1.**Tram traction substation—simplified functional diagram: a—transformer used for loads other than traction network, b—rectifier transformer, c—rectifiers, and d—return cables.

**Figure 2.**Six-phase transformer: (

**a**) winding configuration; (

**b**) vector diagram of phase-to-phase voltages.

**Figure 3.**Configuration of the energy transformation circuit: six phase transformer and two diode rectifiers connected in parallel; I

_{a2}marks a phase current in one secondary winding and I

_{r2}marks total output load current of the same winding (other currents are not marked here for clarity’s sake).

**Figure 4.**Transformer voltages: secondary side phase-to-phase voltages are marked with in black and grey; each colour corresponds to one winding: (

**a**) star and delta winding voltages (phase-to-phase) are exactly the same and (

**b**) star and delta winding voltages (phase-to-phase) are slightly different (U

_{D}= 0.95U

_{Y}, where U

_{D,}U

_{Y}—RMS-values of phase-to-phase delta and star winding voltages, respectively); conducting times of corresponding phases (c1 and c2) in both secondary windings are also shown. Supply voltage is sinusoidal.

**Figure 5.**The effect of secondary winding unbalance: (

**a**) divergence of phase currents in secondary windings, (

**b**) total diode conduction angles in both windings vs. unbalance coefficient; data is shown for corresponding phases (c1 and c2—see Figure 2 and Figure 3 for phase currents). Supply voltage is sinusoidal. Currents are given in reference to total DC load current of the entire circuit.

**Figure 6.**Operation in case of balanced windings and distorted supply voltage (RMS values U

_{5}= 0.06 U

_{1}, U

_{7}= 0.05 U

_{1}; phase shifts are φ

_{5}= 180° and φ

_{7}= 180°); (

**a**) secondary voltages (phase-to-phase), black and grey lines denote star and delta-connected windings, respectively; conducting times of corresponding phases (c1 and c2) in both secondary windings are also shown; (

**b**) total conduction angles in both windings vs. unbalance coefficient.

**Figure 7.**Phase currents of both secondary windings; supply voltage is distorted, RMS values U

_{5}= 0.06 U

_{1}, U

_{7}= 0.05 U

_{1}; phase shifts of 5th (φ

_{5}) and 7th harmonic (φ

_{7}) range from 0° to 180°, as marked in the chart. Values of winding currents are referenced to load current.

**Figure 8.**Counteracting the unbalance of phase currents in secondary windings; supply voltage is deformed, RMS-values of 5th and 7th harmonic are U

_{5}= 0.06 U

_{1}and U

_{7}= 0.05 U

_{1}, phase shifts are φ

_{5}= 180° and φ

_{7}= 180°; points A1 and A2 mark currents’ divergence in case of balanced windings; point B marks equal loading of windings, when number of turns of one winding is changed by application of tap changer (unbalance coefficient ≈ 1.025).

**Figure 9.**Compensation principle for reducing voltage ripple in DC voltage: black and red lines mark the voltage of star-and delta-connected windings, respectively; (

**a**) areas S1 and S2 differ; (

**b**) gain coefficients have been introduced, number of turns has been slightly changed, and areas S1 and S2 are similar. Thick dashed line marks voltage ripple minimum value.

**Figure 10.**Single example of tap-changer correction of turn (voltage) unbalance of secondary windings in 6-phase rectifier transformer: black/red lines denote voltages of two secondary windings (after rectification—tops of pulses are shown); supply voltage is distorted as described in the text (THD = 7.81%, φ

_{5}= 5°, φ

_{7}= 110°; (

**a**) before correction, i.e., gain coefficient k

_{a}, k

_{b}, k

_{c}is 1; (

**b**) gain coefficient is corrected in accordance with ACO and is equal to 1.08747. Green line marks minimum ripple voltage (u

_{phmin}), and light and dark green areas correspond to S1 and S2 areas explained in Figure 9.

**Figure 11.**Several pulses of phase-to-phase voltages (black and grey lines denote voltages of both secondary windings—star-connected and delta-connected winding, respectively); natural commutation points are distinguished by arrows and commuting angles/times shown.

**Figure 12.**Optimization results: (

**a**) total conducting time per phase before (solid lines) and after optimization (dashed lines); (

**b**) winding phase current before (brown shades) and after optimization (green shades); supply voltage is distorted (RMS-values of 5th and 7th harmonic are U

_{5}= 0.06 U

_{1}and U

_{7}= 0.05 U

_{1}); phase shift of 5th harmonic is φ

_{5}= 0°; in each 4-column set, columns 1 and 3 refer to one secondary winding and columns 2 and 4 to the other secondary winding.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kulesz, B.; Sikora, A.; Zielonka, A.
The Application of Ant Colony Algorithms to Improving the Operation of Traction Rectifier Transformers. *Computers* **2019**, *8*, 28.
https://doi.org/10.3390/computers8020028

**AMA Style**

Kulesz B, Sikora A, Zielonka A.
The Application of Ant Colony Algorithms to Improving the Operation of Traction Rectifier Transformers. *Computers*. 2019; 8(2):28.
https://doi.org/10.3390/computers8020028

**Chicago/Turabian Style**

Kulesz, Barbara, Andrzej Sikora, and Adam Zielonka.
2019. "The Application of Ant Colony Algorithms to Improving the Operation of Traction Rectifier Transformers" *Computers* 8, no. 2: 28.
https://doi.org/10.3390/computers8020028