components library for simulation and analysis of aircraft electrical power systems using modelica

A components library for simulation and analysis of aircraftelectrical power systems using Modelica

Martin R. Kuhn1, Antonio Griffo 2, Jiabin Wang 2, and Johann Bals1

1 German Aerospace Center (DLR), Munchner Strae 20, 82234 Wessling, Germany, Phone:+49 (8153) 28-2461, Fax: +49 (8153) 28-1441, Email: [email protected], URL:

http://www.robotic.de2Department of Electronic and Electrical Engineering , The University of Sheffield , MappinStreet, S1 4EX, Sheffield UK , Phone: +44 (0)114 2225817 , Email: [email protected],

[email protected]

June 12, 2009

AcknowledgmentsThis research is being conducted in the frame of the MOET project (More-Open Electrical Technologies),a FP6 European Integrated Project. See www.moetproject.eu for details.

Keywords<<Aerospace>>, <<Modelling>>, <<Simulation>>

AbstractA library of components for simulation and analysis of large vehicular electrical power systems usingModelica language is presented. Components are described using different levels of model complexity,catering for both detailed high fidelity transient switching dynamics and averaged value descriptionswhich, being time invariant, are a computationally efficient and useful tool for design, stability andsensitivity analyses. The merits of employing the Modelica based modelling tool are discussed, and itsutilities and effectiveness are demonstrated through a test system consisting of a three-phase, variablefrequency synchronous generator which feeds high voltage DC loads via an auto-transformer rectifierunit.

IntroductionFuture ”more electric” terrestrial, marine and aerospace vehicular electrical power systems will be basedon the interconnection of a wide range of components, resulting in a significantly more complex elec-trical distribution system with multiple distributed loads most of which are supplied and controlled bypower electronic converters [1]. Due to the destabilizing effect of tightly regulated loads which result inequivalent negative impedance behavior, these electrical networks are susceptible to instabilities [2],[3].In order to investigate potential instability causes and develop corrective actions detailed investigationsby means of numerical simulations are required. Time-domain simulation using detailed nonlinear, timevarying power system models (which will be referred to as behavioral models) can be employed foraccurate transient performance evaluation, stability assessment and power quality investigation. Sev-eral commercial and non-commercial software tools are available for detailed behavioral simulations[4],[5],[6],[7]. Despite the highest level of model accuracy, the main drawback of behavioral modelingis the need for vast computational resources and time. Since switching transients from power electronicdevices do not normally have a significant influence on systems stability, state-space averaging tech-niques have been developed to derive equivalent non-linear time-invariant system models (which will bereferred to as functional models) [8]. Due to the unavailability of validated averaged models suitable for

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stability assessment and the lack of commercial libraries for simulation and analysis of functional mod-els, a dedicated library has been developed using the Modelica language [9]. It is demonstrated that theproposed library, coupled with the commercial software Dymola [10] can be effectively employed as atool for modeling and simulation of complex power systems, both at a functional and behavioral level. Itis also demonstrated how important simulation steps such as system’s initialization, linearization aroundan operating point, formulation of the Jacobian matrix and parametric analysis can be automatically car-ried out in order to provide a reliable and computationally efficient tool for systems analysis. The paperdescribes in detail the use and the advantages of Modelica language for power systems modeling, aswell as provides details on the developed library of components. Detailed descriptions of the two-levelmodeling of the key power systems components will be given in subsequent sections, as well as thoroughsimulations of a test system containing the modeled ATRU fed by a three-phase synchronous generator,with transients initiated by changing in load conditions. Numerical simulations are reported to clearlyshow the capabilities of the proposed framework in terms of behavioral and functional systems model-ing. The potentiality of the proposed modelling framework in stability analysis will also be describedand numerical results of small-signal analysis of a test system, validated with time-domain simulationswill also be presented.

Modelica for dynamical systems modelingModelling and simulation of electrical, mechanical, electro-mechanical and other physical systems usu-ally consists of writing a system of algebraic-differential equations in the form:

x = f(x,y,u) (1)

0 = g(x,y) (2)

which describe the evolution in time of state-space variables x of the system with inputs u. Algebraicrelationships describe the instantaneous memoryless dynamics of variables y which are implicitly definedas functions of the state variables x by (2) which usually result from the mathematical description of theinterconnection of different subsystems.Small signal stability analysis requires the construction of a state-space model in the differential form:

x = f(x,h(x),u) (3)

where (2) has been explicitly solved to yield y = h(x), and the subsequent linearization of (3) arounda steady-state equilibrium point corresponding to a specified operating condition x0 determined by theinputs u0, to yield:

Δx = A ·Δx+B ·u (4)

with the equilibrium point obtained as the solution of:

0 = f(x0,u0). (5)

Jacobian matrix A and B are obtained as:

A =∂f∂x |x0,u0

,B =∂f∂u |x0,u0

(6)

Solution of (2) for y and calculation of steady-state conditions require algebraic manipulations which areoften extremely difficult to be analytically performed for all but very simple systems, and therefore areusually tackled by iterative numerical calculations at simulation stage which usually slows down compu-tation by introducing fictious algebraic loops. Furthermore analytical calculation of linearized equationscan also be difficult and some simulation software such as Simulink in these cases performs linearizationby resorting to numerical perturbation of the system’s equations about the specified equilibrium point.

The Modelica language based simulation program Dymola has been selected for the building of the pro-posed library and simulation environment thanks to its capability to symbolically pre-process a Modelicamodel and to transform implicit equation systems to state space form, where possible, allowing for effi-cient simulation. The transformed equations are provided as C-Code that is compiled and embedded inDymola’s own simulation engine, as stand-alone programme or in other simulation environments (e.g.in Simulink or Matlab). Parameter variations, initialization, pre and post-processing of data can be au-tomatically carried out. Those operations can be performed directly in Dymola and also controlled via aMatlab interface which is part of the distribution. Most of the calculations are carried out in analyticalsymbolic form, therefore simplifying design and avoiding loss of precision and increase in computationburden which would result from iterative numerical procedure.


Figure 1: Schematic of a simplified electric power system

Modelica is an object oriented multi-domain modeling language for component-oriented modeling ofcomplex systems. In contrast to data flow-oriented languages with directed inputs and outputs, suchas the widely known Matlab and its associated Simulink tool, Modelica employs an equation basedmodeling, which results in a faster modeling process and a significantly increased reusability of themodels. The language is developed under open source license since 1996 by the non-profit ModelicaAssociation, based at Linkoping University, Sweden. Many free Modelica libraries for different physicaldomains modeling are available. The Modelica standard library contains a large collection of compo-nents to model among others analogue and digital electronics, electrical machines, 1-dim. translationaland rotational mechanical systems, as well as input/output control blocks. Modelica has also been usedfor simulation of electric power systems and power electronics components [11],[12]. Free and com-mercial simulation environments acting as Modelica front-end are available with useful functionalitiesto graphically construct and simulate Modelica models and perform post-processing operations. Besidesthe potentialities of its computational algorithms the Dymola simulation environment has also usefulcapabilities in modeling and handling large libraries of components.

Power system modelingA simplified schematic for a representative power system for ’more electric’ aircraft is shown in Fig. 1.A three-phase synchronous generator provides power to the AC bus whose voltage is regutated by anautomatic voltage regulator (AVR) that provides the field excitation to the generator. A direct symmetriceighteen-pulse autotransformer and rectifier unit (ATRU) is used to rectify AC voltage and power the DCloads. AC and DC capacitors may also be present for power quality and stability improvement.Signal-flow simulation programs such as Simulink requires the user to explicitly describe the intercon-nections between different components in a system. For an electrical system these relationships may begiven by voltage/current balance equations. Each component is usually represented with a block havinga predefined set of inputs and outputs. Causality dictates that the outputs at each time instant are a func-tion of the inputs at that instant and the internal states. Figure 2 shows the block diagram that describesa signal-flow modelling of the power system of Fig. 1, while Fig. 3 shows the block diagram for thedescription of the full order synchronous generator model. Thick arrows in Fig. 2 are used to denoteAC values which can be given by either three-phase abc or two-phase dq quantities, depending on themodelling and the level of detail the user is interested in.

Figure 2: Block diagram for signal flow power system simulation

Although a signal-flow based modelling is appealing for its simplicity, the need to explicitly defineinterface equations for the interconnection of different components, which can in some cases be givenby complex algebraic or differential/algebraic equations, while taking into account the causality of eachblock can make the modelling and simulation of complex systems with a large number of interconnected


Figure 3: Synchronous generator block diagram

components awkward. Furthermore it is not possible to integrate different levels of modelling detailssuch as functional and behavioural models in the same simulation environment.As an example of the powerful modeling capabilities of Modelica, an extract of the code for the modelingof a synchronous generator is presented. This simple description is compared with the complex modelingthat would result in a traditional signal-flow causal based modeling environment such as Simulink, asreported in the block diagram in Fig. 2 [13].

• Voltage equations

u_qs = Rs*i_qs + we*psi-ds + der(psi_qs);u_ds = Rs*i_ds - we*psi_qs + der(psi_ds);-Rkq*i_kq = der(psi_kq);-Rkd*i_kd = der(psi_kd);u_fd = Rfd*i_fd + der(psi_fd);

• Flux linkage equations

Psi_qs = L_ls*i_qs + K_sat*L_mq*(i_qs+i_kq);psi_ds = L_ls*i_ds + K_sat*L_md*(i_ds+i_fd+i_kd);psi_kq = L_lkq*i_kq + K_sat*L_mq*(i_qs+i_kq);psi_kd = L_lkd*i_kd + K_sat*L_md*(i_ds+i_kd+i_fd);psi_fd = L_lfd*i_fd + K_sat*L_md*(i_ds+i_kd+i_fd);

• Magnetizing currents

i_md = (i_fd + i_ds + i_kd)/k_fd;i_mq = (i_qs + i_kq)/k_fd;

• Saturation equations

i_m = sqrt(i_mdˆ2 + i_mqˆ2);Lm_sat = sqrt(2)*v_sat/(w_e*i_m);K_sat = Lm_sat/Ls_fd;SaturationTable.input = i_m;SaturationTable.output = v_sat;

• Mechanical equations

Te = 3/2*Pp*(psi_ds*i_qs-psi_qs*i_ds);der(w_m) = Te - visc;w_e = der(theta);


• abc ↔ dq transformation

[u_qs; u_ds] = 2/3*[ cos(theta) cos(theta-2/3*pi) cos(theta+2/3*pi);-sin(theta) -sin(theta-2/3*pi) -sin(theta+2/3*pi)]*[u_a; u_b; u_c];

Simbols and variables have an intuitive meaning, details are reported in [14]. The Modelica commandder define the derivative operator. It is clear that there is no need to specifically define an input/outputrelationship for each block.Differently to any causal-based simulators, algebraic loops are automatically solved, in Modelica basedsimulations, at compilation stage by employing powerful symbolic manipulations, therefore no iterativeprocedures are required resulting in faster simulations. Symbolic manipulation also allows Modelicabased simulators to automatically generate linearised equations and Jacobian matrix of the system whichwould have to be derived manually or numerically with signal-flow based modeling.

Multilevel library conceptThe proposed library is based upon a multilevel modelling concept and a base ”netcore” library devel-oped previously by DLR under the European commission funded project VIVACE [15] which addressedthe need for a unitary framework for the integration and management of different suppliers’ models. Themotivation behind embedding multi-level models in one model is that a system architecture only has tobe built once and can be simulated at different levels of accuracy and at different speed depending on thepurpose of the study and results can be compared and validated between different levels of modelling.In the library, each component model is a container for two sub-models which represent different model-ing accuracies of the same system. The so-called ”behavioral” models are close to the hardware level andare suited for detailed simulations as needed for power quality studies, with switching transients of powerelectronics components fully captured. ”Functional” models are non-switching but possibly nonlinearmodels obtained as a result of averaging techniques applied to switching transients. Functional modelscan be used for linearization and for stability investigations applying classical methods of control theoryfor linear time invariant systems. The frequency range in which functional models are representativeof the average dynamics is much lower than the switching frequency of power electronics devices. Thecriteria for validating functional models are [16]:,

• the trajectories have to represent the moving averages along the trajectories of the original system,

• the stability properties of the original and approximate systems have to be identical,

• the models have to be valid for large signals,

• the open-loop models must be usable for closed-loop design.

The presence of both (three-phase) AC and DC subsystems in a typical vehicular power network requiresthe use of a time invariant reference frame. For three-phase AC waveforms in the so called ”abc system”,the DC component of the generalized multi-frequency averaging results in the classical description interms of a synchronously rotating reference frame, which decomposes periodic three-phase values intotheir associate DQ0-axis stationary components [17].The zero component is not included in this library since it is mainly relevant for asymmetric loads nottreated by functional models.A multi-level model for the generator as shown in Figure 4 (a), is built by embedding functional (c)and behavioral (d) models into a single container model (b), which can in turn be connected to othermulti-level models by an appropriate connection plug. A flag acting as a global variable can be set atcompilation stage enabling the translation of either the behavioral or the functional modeling level (objectin upper right corner in (a)). The equation system and simulation code only contains the selected levels.In order to satisfy the need to interconnect models which might be represented either in three-phaseabc system (c) or in DQ reference (d), the Modelica concept of ”expandable connectors” is employed.This data bus concept allows level-dependent models to be interconnected, while enabling only datarelated to the selected level to be present after compilation. It contains either the abc current and voltagevariables or the DQ components plus the generator angular frequency. New multi-level models can beimplemented conveniently by using the infrastructure of the implemented ”netcore” library with baseclasses, partial models, interfaces and models.

Components modellingThe presented components library was developed in context with the EC funded project ”More OpenElectrical Technologies” (MOET). Among others, several generators, (autotransformer-) rectifiers andmotor drives were embedded. The modeling framework is illustrated with respect to two essential


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components of the power distribution network of a proposed aircraft power system: a three-phase syn-chronous generator and an 18-pulse Auto Transformer Rectifier Unit (ATRU). The generator componentmodel is shown in figure 4. Both generators submodels share the same systems equations in the dq sys-tem. The most essential difference between the two levels of the model is the use of the dq componentsin the functional models connector ports while the behavioral model has the additional transformationstep to the abc system in the output. The generator’s output voltage is controlled with an AVR whichis similarly modelled using the two-level concept and employs a PI controller for output regulation(with/without anti-windup limiters).For the voltage feedback, the AC voltage magnitude of the behavioural model is obtained as:

VAC−beh =√

V 2α +V 2

β (7)

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[VαVβ

]=

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0 1 −1

]·⎡⎣ Va

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q.

Figure 5 shows the library structure of the generator. The package ”generator” contains the multi-levelmodels of the generator, here in different versions to include different modelling details. This modelcan be used to build a system and the level can be selected globally or locally as a parameter. Thefunctional and behavioral sub-models and other components are placed in the components folder. Inorder to validate the generator modelling, test systems are also provided in the library.The ATRU model in figure 6 shows the model icon with the two embedded submodels. Both submod-els use the same DC filters on the DC side. Only one of the level dependent models is translated atcompilation time depending upon the level set by the user.On the AC side of each submodel the AC connectors are present. To discriminate between the two dif-ferent modelling levels a yellow square connector carrying dq voltages and currents and angular velocityinformations is used for interconnecting ”functional” components. A blue dot represents a three-phaseconnector carrying information on three-phase voltages and currents for ”behavioural” models. Bothconnectors are connected to the green circle representing the expandable plug which allows the coexis-tence of the two different modelling levels in the same block.Several topologies of ATRU have been analyzed and included in the library. The direct-symmetric topol-ogy without interphase reactors is presented here. The ATRU behavioural model contains a 9 phase


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functional and behaviouralmodel and components

unitary testingenvironmentfor the generator

Figure 5: Library structure of the generator

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autotransformer unit, three six diodes bridge rectifier and some filtering components. The functionalmodels has been derived applying state-space averaging techniques. Detailed analytical derivations andthourough validation of the derived models are reported in [18].

Numerical resultsThe modeling approach has been validated by time domain simulations. A test system representativeof the system in Fig. 1 has been assembled using the components in the library. It consists of thesynchronous generator and its associated AVR connected to the ATRU which supplies resistive loads atthe DC output. The simulation results with stepwise increasing load are reported in Fig. 7. In particularthe DC voltage and the generator’s d and q axis currents are shown. It is demonstrated how the functionalmodels trajectories clearly capture the moving averages along the trajectories of the detailed behaviouralmodel.


(a) generator, q-axis current (b) generator, d-axis current

(c) DC output voltage

Figure 7: Compared simulation results for generator/ATRU source

Stability analysisThe developed modelling environment can be easily used for network stability studies by analyzing theinfluence of design parameters and operating conditions on system’s small-signal stability. As an ex-ample, small-signal stability of the test system previously described has been analyzed, when a constantpower load (CPL) is connected to the DC output of the ATRU via a 200μF , 20μH LC passive filter.The load power demand PCPL is increased gradually from 110 to 150 kW. For each loading conditionthe system’s equilibrium and linerization are automatically calculated using Dymola algorithms. Theresulting eigenvalue locus is shown in Fig. 8. It is clear that the system is unstable for PCPL ≥ 130kW.In order to confirm this result, time domain simulations using functional models are reported in Fig. 9which shows the transients in DC voltage and current following a step change of 1 kW in load powerdemand at 0.4s. Two conditions are reported: PCPL=128kW and PCPL=130kW. The unstable transientswith growing amplitude oscillations in the latter case confirm the results of small-signal analysis. Morecomplex systems with several loads can easily be analyzed. Detailed results are reported in [14].

ConclusionA library of components for modelling, simulation and stability analysis of complex vehicular powersystems has been described. The library, built using the modelling language Modelica, is able to handledifferent levels of modelling complexity and details, employing in a unified framework both detailedbehavioural models and functional averaged models. The benefits of describing a complex system withModelica in terms of modelling simplicity and capability of automatically performing procedures such asinitialization and linearization, have been highlighted. Numerical results have been reported to demon-strate the modelling approach and its potential for stability analysis of complex vehicular power systems.


Figure 8: Eigenvalues locus as a function of constant power load demand

Figure 9: DC voltage and currents transients for load power PCPL=128kW (a), and PCPL=130kW (b).

References[1] A. Emadi, M. Ehsani, and J.M. Mille. Vehicular Electric Power Systems: Land, Sea, Air, and Space

Vehicles. CRC Press, 2003.

[2] A. Emadi, A. Khaligh, C.H. Rivetta, and G.A. Williamson. Constant power loads and negativeimpedance instability in automotive systems: definition, modeling, stability, and control of power


electronic converters and motor drives. IEEE Trans. on Vehicular Technology, 55(4):1112–1125,July 2006.

[3] L. Han, J. Wang, A. Griffo, and D. Howe. Stability assessment of ac hybrid power systems formore electric aircraft. In Proc. IEEE Vehicle Power and Propulsion Conference VPPC ’08, pages1–6, 3–5 Sept. 2008.

[4] Synopsys Inc. Saber. www.synopsys.com, 2009.

[5] The MathWorks. Simpowersystems. www.mathworks.com/products/simpower, 2009.

[6] Plexim. Plecs. www.plexim.com, 2009.

[7] Alternative Transient Program. Atp. www.emtp.org, 2009.

[8] M. R. Kuhn, M. Otter, and L. Raulin. A multi level approach for aircraft electrical systems design.In Proceedings of the 6th International Modelica Conference, 2008.

[9] Modelica association. Modelica. www.modelica.org, 2009.

[10] Dynasim AB. Dymola. www.dynasim.se, 2009.

[11] M. Larsson. Objectstab-an educational tool for power system stability studies. IEEE Trans. PowerSystems, 19(1):56–63, Feb. 2004.

[12] H. Elmqvist, F.E. Cellier, and M. Otter. Object-oriented modeling of power- electronic circuitsusing dymola. In First Joint Conference of International Simulation Societies, 1994.

[13] Chee-Mun Ong. Dynamic Simulation of Electric Machinery. Prentice Hall, 1998.

[14] A. Griffo and J. Wang. Stability assessment of electrical power systems for ’more electric’ aircraft.In 13th European Conference on Power Electronics and Applications, 2009.

[15] VIVACE Project. Value improvement through a virtual aeronautical collaborative enterprise.www.vivaceproject.com, 2007.

[16] P.T. Krein, J. Bentsman, R.M. Bass, and B.L. Lesieutre. On the use of averaging for the analysis ofpower electronic systems. 5(2):182–190, April 1990.

[17] P.C. Krause, O. Wasynczuk, and S.D. Sudhoff. Analysis of Electric Machinery and Drive Systems.WileyBlackwell, 2002.

[18] A. Griffo and J. Wang. State-space average modelling of synchronous generator fed 18-pulse dioderectifier. In 13th European Conference on Power Electronics and Applications, 2009.


components library for simulation and analysis of aircraft electrical power systems using modelica

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