INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH

David Martinez and Ramon Zamora Vol.9, No.2, June, 2019

Electrical Implementations of an Empirical

Electrolyser Model for Improved Matlab/Simulink
Simulations

David Martinez and Ramon Zamora*

Electrical and Electronic Engineering Department, Auckland University of Technology, Auckland 1010, New Zealand
(david.martinez@outlook.co.nz, ramon.zamora@aut.ac.nz)

*Corresponding Author; Dr. Ramon Zamora, Electrical and Electronic Engineering Department, Auckland University of
Technology, Auckland 1010, New Zealand, Tel: +64-9-921-9999, ramon.zamora@aut.ac.nz

Received: 05.04.2019 Accepted:18.05.2019

Abstract- Empirical electrical electrolyser models have proven to be effective in simplifying the analysis of multiple systems
involving electrolyser. However, the conventional representation of current as an input of the electrolyser model present
challenges for circuits involving power converters specifically in Matlab/Simulink implementations. This work presents an
alternative representation of one of such models, using Lambert W function to use voltage as the independent variable. The
function is implemented by using lookup tables to overcome slow calculations and algebraic loop errors. After validating the
accuracy of the new representation against referred publications, a basic circuit using a non-isolated DC/DC boost converter is
implemented to compare the performance of the conventional electrolyser model and the new representation. The simulation
results show that the proposed representation not only works successfully in such systems, but also improves the simulation
time up to 13% while fixing the inherent limitations of the conventional electrolyser model.
Keywords Electrolyser, Lambert W, Matlab/Simulink, DC/DC converter, Power converter.

1. Introduction different parameters of the model or to implement the overall

systems of equations that describe the system [8-10]. Due to
Hydrogen generation from water through electrolysers is their simplicity, empirical models are best suited for large
a clean and sustainable alternative for energy requirements systems analysis and specifically the one developed by
[1-5]. Designing, evaluating and improving renewable Ulleberg has shown great level of accuracy and flexibility in
systems that utilize this technology requires modelling and representing different types of electrolysers [8, 11-14].
simulation techniques to accomplish the desirable goal of
A Matlab / simscape / powersystems / Specialized
each application.
technology model was presented by this author in [15] where
The study of renewable energy systems integrated with the standard description of voltage given as a function of
hydrogen technology started back in 1980 primarily with current was extended with boundary conditions to include
alkaline electrolysers and solar panels. Most of publications transient responses on the electrolysers. However, if the
have focused on connecting directly the electrolysers to the model is used with power converters, MATLAB solvers
energy source as an alternative to simplify the overall require the handling of multiple algebraic loops which
system. This trend peaked on 2014 and may have started generates inconsistencies in the overall response of the
falling due to the inherent sacrifice in efficiency [2]. To electrolyser.
improve the overall efficiency, power converters are usually
Previous studies have avoided the algebraic loops by
deployed either to extract the maximum power available of
removing the switching elements on the resulting electrical
the source or to regulate the electrical signals delivered to the
circuit or by using Simulink signals instead of electrical
load [6-7].
connections [16-17]. However, considering all the elements
In terms of the electrolyser itself, several water of the system in the electrical domain is essential for
electrolyser models have been proposed to either estimate the assessing the electrical behaviour of the resulting circuit.
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019
To obtain a reliable electrical response of the electrolyser For many applications, Eq. (1) provides an accurate
model, this work will present a variation of the model representation of the electrical response of the electrolyser
presented in [15], where the current is defined as function of providing reliable simulation outcomes [11]. In some cases,
voltage using lambert W prodlog function. The contributions however, adjustments are needed to overcome the following
of the proposed model are as follows: issues:
Ø Resolve the existing limitations of the selected Ø If current is equal to zero, Eq. (1) reduces the
empirical model. electrolyser to a dc voltage source which can supply
Ø Reduce the complexity of the model implementation energy to an external circuit.
in the MATLAB/Simulink environment. Ø When electrolyser current is low, the term inside the
Ø Allow smooth simulation results without removing logarithm can be negative producing complex
switching elements or electrical components. values.
Ø Validate the accuracy of the electrolyser model Ø In Eq. (1), any small current can generate a voltage.
against known data. However, the chemical reaction only occurs after
the external applied voltage is greater than the
Initially the original model is presented highlighting its
reversible voltage. Below this limit no current
disadvantages. Next the new representation of the model is
should be flowing through the electrolyser.
introduced and validated using data from two previous
Ø Implementing electrical models in Matlab/Simulink,
publications. Finally, a MATLAB/Simulink implementation
such as the one represented in Eq. (1), results in the
is presented with a basic switching application to show the
creation of an algebraic loop that poses difficulties
improvements with the proposed representation.
to the software solvers. Resolving these loops can
cause distortions in the expected response of the
2. Electrolyser Electrical Model model.
The chemical process of water electrolysis separates a A proposed way to solve these issues was published in
water molecule into its two constituent elements by applying [15] where a set of boundary conditions are included to Eq.
energy in the form of electricity. This electrochemical (1) to guarantee valid values regardless of the behaviour of
process is temperature and pressure dependant and changes the external circuit. The implementation in Matlab/Simulink,
substantially depending on the type of ionic transport media shown in Fig. 1, includes a diode that restricts current to flow
or electrolyte. This electrolyte classifies an electrolyser in only when the applied voltage is above the limit value of
either alkaline, proton exchange membrane (PEM) or solid NcVrev. The algebraic loop was resolved with a memory
oxide (SOE). The full chemical description can be found in block that works as a sample unit delay.
[8,18-19].
The design of the electrolytic cell, including electrodes
materials, gap space, form, etc, change the response of the
electrolyser even within the same type of electrolysers.
Therefore, a model that can cover all the different variables
providing a reliable response is a topic of constant research
[8].
Among the many alternatives to electrolyser modelling,
a simple solution is to utilize polarization curves of a real
device and fit a given curve in terms of a small set of
parameters. The one proposed in [20] takes six parameters to
describe voltage as a function of current and temperature in
the form of a logarithmic curve given by Eq. (1):
Fig. 1. Electrolyser simscape power systems block [15].

For simple circuits of few interconnected elements, this

implementation is enough to obtain valid results. However,
when the complexity of the system increases the expected
outcome of the electrolyser model is heavily distorted. As a
(1) novel alternative, Eq. (1) can be changed to be a function of
voltage instead of current, which allows the possibility to
define the electrolyser operation in terms of the external
where: T is the temperature of the cell, A is the area applied voltage.
of the electrodes, Vrev is the reversible voltage of the cell, Nc
is the number of cells in series and r1, r2, t1, t2, t3, and s are 3. New model representation
the fitting parameters also known as overvoltage parameters.
At standard conditions (1 bar, 25 oC), Vrev is approximated 3.1. Voltage as the domain of the function
to 1.229 V and can be considered constant for low
temperature ranges up to 100 0C [20].

1061
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019
Eq. (1) can be solved with respect to current using the
prodlog or W Lambert function [21-22]. The steps shown
below can be applied to any other model using logarithm
functions. (10)

First, the equation is rearranged, and some terms are Returning b, c’ and d’ terms back to the original form:
renamed to simplify the visualization of the steps:

(2)

(3)
Where,

(11)
, ,
To complete the electrolyser model, the domain of the
Taking power of ten to both sides of Eq.(3), rearranging function is defined for specific ranges of operation. For real
and renaming: values of voltage, the W function has two solutions or

(4) branches in the interval as shown in Fig. 2. For

this application, the principal branch satisfies
the model.
(5)

(6)

Where,

and

Substituting on Eq.(6):

(7) Fig. 2. Two main branches of the lambert W function [23].

Furthermore, to avoid the issues highlighted in the

(8) previous section, the voltage range is limited from 0 to
when there is no chemical reaction thus no current;
and greater to when the reaction is active and current
Using Lambert function definition passes through the electrolyser. The final electrolyser model
with respect to voltage and temperature can be written as in
on Eq.(8): Eq.(12).
The hydrogen production model remains unchanged and
is rewritten from [15] in Eq.(13) where is the mass rate
of hydrogen gas in kg per second, f1 and f2 are parameters
(9) related to Faraday efficiency, z is the number of electrons
transferred in the reaction (2 electrons for water electrolysis),
F is Faraday constant, c is a conversion constant equal to
Back substitution or 0.08988 kg/m3 ; and is the volume of an ideal gas at
standard conditions 0.0224136 m3/mol.

1062
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019

(12)

(13)

Table 1. Electrolyser fitting parameters

Electro- Model parameters Goodness of fit from curve fitting tool

lyser
r1 r2 s t1 t2 t3 SSE R-sqr A-R-sqr RMSE

Alkaline 1.969e-4 -5.754e-7 0.1862 0.008887 -21.46 2362 3.402e-4 0.9989 0.9987 0.002781

PEM 2.379e-5 -1.13e-8 0.05518 0.7831 548.3 4335 0.00836 0.9989 0.9987 0.01866

3.2. Model construction and validation of the new Table 2. Electrolyser fitting parameters
representation
Alkaline PEM
As previously mentioned, the process starts by finding
the polarization curves of the devices to be modelled. Two NRMSE 40 0C 0.935666 0.977537
electrolysers from published studies [20,24] are used as
reference. Ten points are used to estimate the set of NRMSE 50 0C 0.955586 -
parameters for Eq. (1) using Matlab curve fitting tool. The
list of final parameters is presented in Table 1 and a visual NRMSE 60 0C 0.951929 0.970202
comparison between the reference and the model is shown in
Figure 3. NRMSE 70 0C 0.980023 -
To assess the new representation, the data points of the NRMSE 80 0C 0.975527 0.971501
reference voltage for the two electrolysers are used as input
of Eq. (12) and the results are compared to the reference Data Points 10 10
current using the normalized root mean square error
(NRMSE). This measure of differences is described by Eq.
(14) and is useful when the comparison involves different
scales. A result closer to 1 indicates a good fit. The results
are presented in Table 2 and a visual comparison is shown in 3.3. Matlab implementation
Figure 4.
The approach presented in [15] can be followed to
implement Eq. (12) in Matlab/simscape/power
systems/Specialized technology toolbox. However, the built
in Matlab Lambert W function requires considerable
computation time and is not practical for simulations that
include multiple elements.
(14)

1063
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019

Fig. 3. Electrolysers electrical response, model vs reference

Fig. 4. New representation of the model vs reference.

To overcome the slow computational problem, a look up response of the new representation and allows a much
table is implemented where voltage and temperature are simpler implementation as shown in Fig. 6.
input variables and current is a matrix output generated by
Eq. (12). The domain validation is now achieved with
different switch selectors as shown in Fig. 5 and the diode is
no longer required.

Fig. 6. Implementation of hydrogen production in

Matlab/Simulink
To validate the implementation, the polarization curves
of the electrolysers are generated using a controlled voltage
source as shown in Fig. 7, and a numerical comparison
between the reference and the model is calculated using the
same technique as in the previous section. The results are
shown in Fig. 8 and Table 3.
The integration method chosen is continuous mode to
Fig. 5. Implementation of the New representation in rely on the accuracy of the Simulink variable-step solver
Matlab/Simulink algorithms [25]. The Matlab recommendation for nonlinear
models is followed by choosing Simulink solver ode23t mod.
The activation condition of the hydrogen production Stiff/Trapezoidal with default values [26]. A maximum step
model in Eq. (13), is now implicit in the electrical current size of 1 x10-06 is chosen only for the circuit shown in Fig. 7

1064
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019
to get voltage values approximately close to the reference (D) required to achieve this conversion is found to be 0.7143
values and to obtain a meaningful comparison. using equation (15) [27]. The values of inductance and
capacitance are set to provide good filtering with a smooth
Table 2. Electrolyser fitting parameters response for a switching frequency of 10 kHz.
Alkaline PEM
(15)
NRMSE 40 0C 0.935664 0.977537

NRMSE 50 0C 0.955585 - The two electrolysers are approximately set up to the same
range of operation by adjusting the number of cells. At 35
NRMSE 60 0C 0.951927 0.970202 volts and temperature of 23 0C, the currents of a 21-cells-
alkaline and a 23-cells-PEM electrolyser are found to be
NRMSE 70 0C 0.980025 - 13.8797 A and 14.6294 A, respectively. These values of
voltage and current relate to an equivalent resistance of
NRMSE 80 0C 0.975527 0.971501 approx. 2.5 ohms.
The simulations total run time is 1 second and the
Data Points 10 10 switching is enabled after 0.3 seconds. The algebraic loop
introduced by the electrical implementation is forced to be
solved iteratively by Simulink algorithms to obtain high
accuracy [28].
When the system uses the electrolyser conventional
model, Simulink is unable to solve the loop for specific
scenarios and therefore the following approaches are
followed to obtain valid solutions:
Ø Introducing two common loop breaking techniques
with either a unit delay block or a first order transfer
function.
Ø Implementing a lookup table for the conventional
Fig. 7. Circuit to obtain polarization curves in model.
Matlab/Simulink The unit delay approach was implemented in [15] with a
memory block and is equivalent to a unit delay block. The
4. Simulation of a switching application first order transfer function uses the same form of a low pass
filter and the time constant works as the cut-off frequency.
To assess the benefits of the new representation, a With a switching frequency of 10 kHz, a time constant lower
system implementing a DC/DC boost converter feeding an than 1.6 x10-5 is needed to at least cover the switching
electrolyser is developed as shown in Fig. 9. The input and dynamics without harmonics. Unfortunately, the simulation
output voltage values of the boost converter are arbitrarily time increases considerable as the time constant is reduced
defined to be 10 and 35 Volts, respectively. The duty cycle

Fig. 8. New representation using simulink, model vs reference.

1065
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019

Fig. 9. Boost converter and electrolyser system.

and only a time constant of 1 x 10-5 is assessed. The alkaline electrolyser is introduced next producing a
degraded ripple content due to its non-linearity against small
A lookup table is constructed similarly as described in
variations of voltage. For this scenario, Simulink can solve
the previous section by creating a voltage output matrix
the loop for all models, showing equivalent results between
using Eq. (1). The electrical implementation remains
them as shown in Fig. 11 and Table 3.
unchanged but instead of constructing Eq. (1) with Simulink
blocks, a lookup table is introduced as shown in Fig. 10.
Table 3. Results with filter capacitor of 400 µF

Eq. Resistance Conventional New Rep.

DC Ripple DC Ripple DC Ripple

% % %

Iin 48.6 A 1.22 41.27 1.61 41.29 1.61

A A

Vout 34.65 6.65 34.67 5.6 34.67 5.6

V V V

Fig. 10. Lookup table implementation for the Iout 13.85V 6.65 13.08 95.29 13.08 95.33
conventional model A A

5. Results and discussions

To reduce the ripple content on the output signals the
The system is first tested using the equivalent resistance filter capacitor is increased to 8000 µF. Under this scenario,
to assess the converter conversion and observe the overall Simulink cannot solve the algebraic loop in the conventional
results. With a filter capacitor of 400 µF, the linear load model and the techniques described in the previous section
produces a ripple content of 6.65% in both output signals. are assessed to verify the best results.

1066
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019

Fig. 11. Simulation results for the system using 400 µF

Fig. 12. Loop solving techniques results for the conventional model using 800 µF

Initially the unit delay is tested followed by the lookup

1067
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019

Fig. 13. Simulation results for two electrolyser types using different models
table implementation producing the results in Fig. 12. The
controlled voltage source and the capacitor voltage of the Table 5. Simulation results for PEM type
conventional model dictates the input current which needs to
vary to satisfy the circuit state equations. For a unit delay, the
model never receives the correct value of current and ends up Conventional New Representation
oscillating with a divergent amplitude. For a first order (142.086 s) (125.919 s)
transfer function, the current oscillates to guarantee a correct
voltage transition between the capacitor and the controlled
voltage source. For the lookup table, the complexity of the DC Ripple % DC Ripple %
model is reduced which allows Simulink to solve correctly
the system of equations.
Vout 34.8 V 0.29 34.8 V 0.29
Finally, a comparison is performed between the new
representation and the conventional model using the lookup
table implementation for both models. Additional to the
circuit metrics, total simulation time is added to the Iout 12.53 A 8.22 12.54 A 8.23
performance metrics to assess the best alternative.
Simulations are run for the two electrolyser types, PEM and
Alkaline using both models. The results presented in Fig. 13 6. Conclusion
and Table 4-5, show that the circuits metrics are equivalent
with small discrepancies in the current signals. Simulation A new representation of the electrolyser empirical model was
time is longer for the conventional model requiring approx. presented to extend its application to larger electrical systems
13 % extra time than the new representation. with several components. Lambert W function is used to
change the form of the model to be voltage dependant,
Table 4. Simulation results for alkaline type allowing a simpler implementation that results in an
improvement of overall simulation time.

Conventional New Representation A Matlab/Simulink implementation using lookup tables is

(146.316 s) (126.628 s) presented as a simplification approach to speed up the
simulation and to allow recursive algebraic loop by the
DC Ripple % DC Ripple % solver for greater accuracy. When running simulations for a
system composed of a DC/DC boost converter and an
Vout 34.8 V 0.29 34.8 V 0.29 electrolyser, the lookup table implementation shows an
accurate result against the common loop handling techniques
of a unit delay and a first order transfer function. In terms of
Iout 12.54 A 5.95 12.51 A 5.36

1068
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019
model comparison, the new representation improves the total Journal of Hydrogen Energy, Vol. 39, pp. 13063-13078,
simulation time by 13%. 2014
[12] I. Mendaza, B Bak-Jensen and Z. Chen “Alkaline
The analysis presented here can also be applied to other Electrolyzer and V2G System DIgSILENT Models for
electrolyser models that may face similar challenges when Demand Response Analysis in Future Distribution
interacting with multiple elements Networks”, 2013 IEEE Grenoble Conference,
November 2013
References [13] A. Dukic “Autonomous hydrogen production system”,
International Journal of Hydrogen Energy, Vol. 40, pp.
[1] F. Zhang, P. Zhao, M. Niu and J. Maddy, “The survey 7465-7474, 2015
of key technologies in hydrogen energy storage”, [14] D. Ali, R. Gazey and D. Aklil “Developing a thermally
International Journal of Hydrogen Energy, Vol. 41, pp. compensated electrolyser model coupled with
14535-14552, 2016. pressurised hydrogen storage for modelling the energy
[2] T. Ogawa, M. Takeuchi, Y. Kajikawa, “Analysis of efficiency of hydrogen energy storage systems and
Trends and Emerging Technologies in Water identifying their operation performance issues”,
Electrolysis Research Based on a Computational Renewable and Sustainable Energy Reviews, Vol. 66,
Method: A Comparison with Fuel Cell Research”, pp. 27-37, 2016
Sustainability, Vol 10(2), pp. 478, 2018 [15] D. Martinez, R. Zamora, “MATLAB Simscape Model
[3] O. Bilgin, “Evaluation of hydrogen energy production of An Alkaline Electrolyser and Its Simulation with A
of mining waste waters and pools”, 4th International Directly Coupled PV Module for Auckland Solar
Conference on Renewable Energy Research and Irradiance Profile”, International Journal of Renewable
Applications, Palermo, pp.557-561, November 2015. Energy Research, Vol. 8, pp. 552-560, 2018
[4] S. Kaya , B. Öztürk, H. Aykaç, “Hydrogen Production [16] R. Garcia-Valverde, C. Miguel, R. Martınez-Bejar, A.
from Renewable Source: Biogas”, International Urbina, “Optimized photovoltaic generator-water
Conference on Renewable Energy Research and electrolyser coupling through a controlled DC-DC
Applications, Madrid, pp.633-637, October 2013. converter”, International Journal of Hydrogen Energy,
[5] K. Koiwa, R. Takahashi, J. Tamura, “A Study of Vol. 33, pp. 5352-5362, 2008
Hydrogen Production in Stand-alone Wind Farm”, 2012 [17] S. Dahbi, R. Aboutni, A. Aziz, N. Benazzi, M.
International Conference on Renewable Energy Elhafyani, K. Kassmi, “Optimised hydrogen production
Research and Applications, Nagasaki, pp.1-6, by a photovoltaic-electrolysis system DC/DC converter
November 2012. and water flow controller”, International Journal of
[6] D. Yamashita, H. Nakao, Y. Yonezawa, Y. Nakashima, Hydrogen Energy, Vol. 41, pp. 20858-20866, 2016
Y. Ota, K. Nishioka, M. Sugiyama, “New Solar to [18] K. Zeng, D. Zhang, “Recent progress in alkaline water
Hydrogen Conversion System High Efficiency and electrolysis for hydrogen production and applications”,
Flexibility”, 6th International Conference on Renewable Progress in Energy and Combustion Science, Vol. 36,
Energy Research and Applications, San Diego, pp. 307–326, 2010
November 2017. [19] J. Chi, H. Yu, “Water electrolysis based on renewable
[7] K. Neuhaus, C. Alonso, L. Gladysz, A. Delamarre, K. energy for hydrogen production”, Chinese Journal of
Watanabe, M. Sugiyama, “Solar to Hydrogen Catalysis, Vol. 39, pp. 390–394, 2018
Conversion using Concentrated Multi-junction [20] O. Ulleberg, “Modeling of advanced alkaline
Photovoltaics and Distributed Micro-Converter electrolyzers: a system simulation approach”,
Architecture”, 7th International Conference on International Journal of Hydrogen Energy, Vol. 28, pp.
Renewable Energy Research and Applications, Paris, 21-33, 2003. (Article)
October 2018. [21] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J.
[8] P. Oliviera, C. Bourasseaua, B. Bouamamab, “Low- Jeffrey, D. E. Knuth (1996). "On the Lambert W
temperature electrolysis system modelling: A review”, function", Advances in Computational Mathematics,
Renewable and Sustainable Energy Reviews, Vol 78, Vol. 5 pp. 329–359, 1996
pp. 280-300, 2017. [22] S. Steward, “A new elementary function for our
[9] M. Hammoudi, C. Henao, K. Agbossou, Y. Dube, M.L. curricula?”, Australian Senior Mathematics Journal,
Doumbia, “New multi-physics approach for modelling Vol. 19 Issue 2, p8-26, 2005
and design of alkaline electrolyzers”, International [23] Mathworks Documentation, “Lambert W function”,
Journal of Hydrogen Energy, Vol. 37, pp. 13895-13913, https://www.mathworks.com/help/symbolic/lambertw.h
2012 tml, Last accessed October 2018
[10] A. Ursua, P. Sanchis, “Static-dynamic modelling of the [24] N. Briguglio, G. Brunaccini, S. Siracusano, N.
electrical behaviour of a commercial advanced alkaline Randazzo, G. Dispenza, M. Ferraro, R. Ornelas, A.S.
water electrolyser”, International Journal of Hydrogen Arico, V. Antonucci, “Design and testing of a compact
Energy, Vol. 37, pp. 18598-18614, 2012 PEM electrolyzer system”, International Journal of
[11] E. Amores, J. Rodriguez and C. Carreras, “Influence of Hydrogen Energy, Vol. 38, pp. 11519-11529, 2013
operation parameters in the modelling of alkaline water [25] Mathworks Documentation, “Choosing an Integration
electrolysers for hydrogen production”, International Method”,
https://www.mathworks.com/help/physmod/sps/powers

1069
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH
David Martinez and Ramon Zamora Vol.9, No.2, June, 2019
ys/ug/choosing-an-integration-method.html, [27] N. Mohan, “Power electronics: a first course”, New
Mathworks, Last accessed September 2018 Jersey, USA, John Wiley & Sons, 2012, Chapter 3.
[26] Mathworks Documentation, “Simulating with [28] Mathworks Documentation, “Simulating Discretized
Continuous Integration Algorithms”, Electrical Systems”,
https://www.mathworks.com/help/physmod/sps/powers https://www.mathworks.com/help/physmod/sps/powers
ys/ug/simulating-with-continuous-integration- ys/ug/simulating-discretized-electrical-systems.html,
algorithms.html, Mathworks, Last accessed September Mathworks, Last accessed September 2018
2018

1070