Reduced order modeling for transient simulation of power systems using trajectory piecewise linear approximation
 Muhammad Haris Malik^{1}Email authorView ORCID ID profile,
 Domenico Borzacchiello^{1},
 Francisco Chinesta^{1} and
 Pedro Diez^{2}
https://doi.org/10.1186/s4032301600846
© The Author(s) 2016
Received: 18 July 2016
Accepted: 15 November 2016
Published: 5 December 2016
Abstract
This paper concerns the application of reduced order modeling techniques to power grid simulation. Swing dynamics is a complex nonlinear phenomenon due to which model order reduction of these problems is intricate. A multi point linearization based model reduction technique trajectory piecewise linearization (TPWL) method is adopted to address the problem of approximating the nonlinear term in swing models. The method combines proper orthogonal decomposition with TPWL in order to build a suitable reduced order model that can accurately predict the swing dynamics. The method consists of two stages, an offline stage where model reduction and selection of linearization points is performed and an online stage where the reduced order multipoint linear simulation is performed. An improvement of the strategy for point selection is also proposed. The TPWL method for a swing dynamics model shows that the method provides accurate reduced order models for nonlinear transient problems.
Keywords
Model order reduction Power grid Swing dynamics POD TPWLBackground
Transient stability of the power grids is a keen area of research because of its implications in the power system planning, operation and control. Energy based methods for the transient analysis of power grids were developed originally by Mangnusson [1] and Aylett [2]. The substantial size of power grids makes the transient analysis computationally costly to simulate and therefore a need of model order reduction arises. Reduced order models need to be computationally costeffective while retaining considerable accuracy of the full model in large network grids simulations.
A number of approximation schemes are available for model reduction and selection of an appropriate scheme depends upon the problem to be solved so that a suitable reduced order model is achieved [3]. Some of the earliest methods in the domain of model order reduction are Truncated Balance Realization proposed by Moore in 1981, Hankelnorm reduction published in 1984 by Glover [4], proper orthogonal decomposition (POD) [5], assymptotic waveform evaluation, PRIMA [4] and a more recent proper generalized decomposition (PGD) [6].
The techniques defined by Bai et al. [3] includes Krylovsubspace techniques, Lanczos based methods such as MPVL algorithm and SyMPVL among others for the reduced order modeling in the electromagnetic applications. The study by Parillo et al. [7] presents the use of POD to reduce the hybrid, nonlinear model of a power network. The authors used the “Swing Equations” which are the differentialalgebraic equations of second order to simulate the cascading failures in power systems. They employed model order reduction in parts of the system that remains unaffected by the failures. POD based model order reduction has found applications in diverse fields and is the preferred method in electrical engineering applications such as in the study by Montier et al. [8]. In their study, POD is applied in combination with discrete empirical interpolation method.
Kashyap et al. [9] have used model order reduction for the purpose of state estimation of phasor measurement units (PMUs). The authors have proposed an algorithm based on reduceddimension matrices which operate separately on PMU measurements and on conventional measurements. The proposed scheme is applicable to distributed implementation and is reported to be numerically stable. The algorithm was applied on a IEEE 14bus system and compared with existing schemes in the literature and demonstrated good accuracy. WilleHaussmann et al. [10] used symbolic reduction approach to model lower order grid segments. The study shows reduction by a factor of 2 for a typical grid.
The main hurdle in the effective model order reduction of the power grids is the strong nonlinearity appearing in swing dynamics models. Trajectory PiecewiseLinear method (TPWL) is a welldefined method for the model order reduction of nonlinear time varying applications [11]. This method proposes a suitable strategy for treatment of nonlinearities which presents the real bottleneck of model order reduction. This method has been applied on several nonlinear problems especially to electronics engineering applications [12–19]. A similar method to the one adopted in this paper is found in the work of Bugard et al. [20] and Panzer et al. [21] who have proposed a parametric model order reduction. The main idea presented in these works is to reduce several local models and then produce a parametric reduced order model using a suitable interpolation strategy. Compared to these methods, TPWL has one global reduced basis and uses interpolation of locally linearized models just to represent the nonlinear term in the reduced variables space. More than one training trajectories can be added together to form the single global reduced basis similar to the concept of POD.
TPWL method has been implemented in nonlinear control of integrated circuits and MEMS [14]. Xie and Theodoropoulos [22] have used the capability of TPWL of reducing large scale nonlinear dynamic models and demonstrated it through the stabilisation of the oscillatory behavior tubular reactors as the case study.
Trajectory piecewiselinear methods are not limited to just power electronics and control systems applications, indeed there are vast areas of research where the application of TPWL based model order reduction will be beneficial [23, 24]. The study by He et al. [23] involves the implementation of TPWL macromodeling for subsurface flow simulations. In another study by Cardoso and Durlofsky [24] the work on model order reduction using TPWL methods for subsurface flow simulations is presented.
In the current study, we implemented TPWL method to accurately obtain a reduced order model for the nonlinear transient dynamics of power grids, mathematically modeled by swing dynamics. The swing dynamics model is highly nonlinear and it is very difficult to have accurate results with linearized reduced order models and the nonlinear POD is inefficient with respect to time consumption. Therefore, the adoption of TPWL method in the current study is suitable for the model order reduction of power grids.
The article has been divided into three sections, where “Swing dynamics equations” section discusses about the mathematical model of the power grid. “Model order reduction” section deals with the model reduction techniques of POD and introduces the method of TPWL adapted for the swing model. “Numerical experiments” section provides a couple of test cases as an example and the robustness of the application of TPWL method.
Swing dynamics equations
Problem statement
Numerical integration of the high fidelity model
Swing equations are numerically integrated in the commercial software MATLAB. A builtin function ‘ode15s’ has been used in the evaluation of the time dependent ODE represented in Eq. (4), because of the potential stiffness of the problem this is the most effective ODE solver available in MATLAB. It is based on the numerical differentiation formulas (NDF) and optionally use backward differentiation formulas (BDF) and is a multistep solver. A detailed description of this method and its incorporation in MATLAB environment is provided in the article by Shampine and Reichelt [26].
Model order reduction
Proper orthogonal decomposition
The matrix \([\tilde{U}]\) can be calculated using several techniques. In essence, POD is similar to the Karhunen–Loeve decomposition (KLD) and it is often referred to as KLD, principal component analysis (PCA) or the singular value decomposition (SVD) [27]. In the current study, SVD interpretation has been used to obtain a reduced order model.
Here, we will briefly describe the SVD reduction procedure which is available as a builtin function in MATLAB.
For detailed insight into the method and the variations in the above mentioned procedures of KLD, PCA and SVD, the author refers to the studies by Liang et al. [27]. Additionally one can also refer to Kerschen et al. [28] and Berkooz et al. [29].
As an example of the problem with increased computational cost, we performed a simulation with POD based reduced model, the same simulation is presented later in detail. The full simulation without the model reduction required around 202 s, with POD and the nonlinear function as defined in Eq. (15) the simulation took about 290 s.
Trajectory piecewise linear method
An approach based on TPWL method has been adopted in the current study to reap the benefits of reduced order modeling while also maintaining a good approximation of the nonlinear function.
Evaluation of \(\{\tilde{p}(z)\}\) requires \(N^2 \times q^2\) operations which results in similar time consumption as high fidelity model. The objective of introducing TPWL method is to construct a locally affine mapping \(\{\tilde{L}_p(z)\}\) from \(\mathbb {R}^q\) to \(\mathbb {R}^q\) at some time steps s where \(s \ll n\) which involves less operations and such that \(\{\tilde{L}_p(z)\} \approx \{\tilde{p}(z)\}\).
Trajectory piecewise linear method is a method combining the model order reduction and the linearization of the nonlinear functions. The system in the current study given by Eq. (1) has strong nonlinear characteristic and as described in earlier sections, nonlinear reduced order model does not reduce the time consumption. The TPWL method provides a combination of linearized models obtained at selected snapshots.
Steps of TPWL Simulation

Step 1: Simulation of the high fidelity model used as the training trajectories (See details in “Selection of training trajectories” section).

Step 2: Generation of reduced basis using POD.

Step 3: Linearization of the nonlinear function and construction of the set of linearization points S (See details in “Selection of linearization points” section). During the construction of set S, weights have to computed for the combination of linear functions which is detailed in the “Weighting function” section.

Step 4: Reduction of the linearized system and the storage of \(\{\delta \}^j, \{\tilde{p}\}^j, [\tilde{J}]^j\) and S, where j is the element in the set of linearization points S.

Step 1: Load the stored reduced basis.

Step 2: Calculate weighting functions.

Step 3: Combine the linearized systems in a convex combination.

Step 4: Solve the reduced linearized system.
Selection of training trajectories
Training trajectories form an integral part of the TPWL method which theoretically, should be able to cover all the domain of the nonlinear function. The selection of training trajectories, therefore, requires careful selection of initial conditions upon which the trajectories of the swing model depend. One may consider it is inefficient to compute so many nonlinear functions to cover the whole domain. However, in practice there are only a few possible conditions a system can achieve in real time applications. Training trajectories provide the points at which the nonlinear system has to be linearized (see “Selection of linearization points” section). It is to be stressed that the TPWL method can interpolate between the training trajectories but not to extrapolate. Therefore, it is necessary to include all the trajectories in the training set that are considered to be visited by the nonlinear function [30].
Selection of linearization points
A simple nonlinear trigonometric function and its linear approximation based on the concept of fixed distance in the context of its norm is plotted in Fig. 2a while a new method developed in the current study is shown in Fig. 2b . As it is observable from Fig. 2a and b , the number of linearization points are comparable as 5 in the first case to 7 in the current case has reduced the approximation error by about 40%. Generally, the increase in number of linearization points is of no significant loss in computation time as the selection is done during the offline phase while in the online phase the computation of linear functions is very quick.
The modified version of the selection of linearization points is more time consuming then the original method proposed by Albunni [30]. However, selection of linearization points is performed during the offline phase where time is not a constraint, the proposed method in the current study has higher accuracy with the problem discussed here.
A very important note that the nonlinear function \(\{\tilde{p}\}^j\) and the Jacobian matrix \([\tilde{J}]^j\) are stored in the reduced basis.
Weighting function
TPWL method combines the linearized model in a convex combination approximating the original nonlinear system. In a convex combination all the coefficients are greater or equal to zero with the sum of all the coefficients equal to one. If there are ‘s’ linearized models and ‘q’ is the order of the linearized system, then the computation of these weights is in the order of O(sq) (for detailed study on the weighting function refer to thesis of Rewienski [11]). The calculation of weights is carried out during both the online and offline phases in the current study.
Numerical integration of the reduced model
With the above information, we now have a system in reduced basis which fully exploits the benefits of reduced order modeling. As it can be observed from Eq. (24), the number of operations now depend on s rather than \(N^2\).
Numerical experiments

The power grid is lossless

The generators are small and the ratio between the length of transmission line joining generators to the infinite bus and the length of transmission line joining two consecutive generators is much bigger. Hence, the interaction between a generator and infinite bus is much smaller than the interaction between two neighboring generators

Transmission lines joining two consecutive generators is shorter than the line joining the generators with the infinite bus

Transmission lines between the infinite bus and all the generators are of same length

Transmission lines connecting the generators are of same length
Grid data
Symbol  Description  Value 

\(m_i\)  Mass of the generators  1 (p.u.) 
\(d_i\)  Damping of the generators  0.25 (p.u.) 
\(p^m\)  Power generated by the generators  0.95 (p.u.) 
b  Susceptance between generator and slack node  1 (p.u.) 
\(b_{int}\)  Susceptance between consecutive generators  100 (p.u.) 
N  Number of generators  1000 
The values used in the current study are adapted from the study of Susuki et al. [33] with the addition of damping to ensure the steady state stability of the power grid. Also, the number of generators in the study of Susuki et al. are only 20 and the focus of their study is to demonstrate the coherent swing instabilities. Although different from the study by Susuki et al., the grid loop as described in their study presented a good opportunity to showcase the ability of TPWL method for the model order reduction and fast simulation of electrical power grids.
Training trajectories and reduced order models
In the current study, there are three different scenarios of initial conditions that must be taken into account for the training trajectories, these are listed in Table 3. The dependence of the trajectories on the initial conditions is evident since the trajectory will be different in each case.
 1.
Initial conditions at the equilibrium point for all the generators bar one.
 2.
All the generators start from a nonequilibrium point and one generator out of synchronicity.
 3.
All the generatros start from the same nonequilibrium state and are synchoronous.
Training trajectories data
Symbol  Description  Value (s) 

T  Total time of simulation  50 
\(\Delta \)T  Time step  0.005 
Initial conditions
Training trajectory  \(\varvec{\delta }_{\varvec{2}}\)  \(\varvec{\delta }_\mathbf{i } \quad \varvec{\forall } \mathbf i \varvec{\ne } \mathbf 2 \) 

Trajectory 1  1.45  1.25 
Trajectory 2  1.1  1 
Trajectory 3  0.8  0.8 
Time consumption data
Training trajectory  Modes ‘\(\mathbf q \)’  Lin Pts ‘\(\mathbf s \)’  Time (s)  

Full model  POD  Lin Pts Sel  
1  312 (\(\eta \, <\, 10^{4}\))  26 (\(\epsilon _s=0.01\))  202  33  276 
312 (\(\eta \,<\, 10^{4}\))  47 (\(\epsilon _s=0.005\))  632  
203 (\(\eta \,<\, 10^{2}\))  47 (\(\epsilon _s=0.005\))  580  
2  312 (\(\eta \,<\, 10^{4}\))  16 (\(\epsilon _s=0.005\))  191  27  155 
199 (\(\eta \,<\, 10^{2}\))  16 (\(\epsilon _s=0.005\))  150  
199 (\(\eta \,<\, 10^{2}\))  49 (\(\epsilon _s=0.001\))  850  
3  130 (\(\eta \,<\, 10^{4}\))  8 (\(\epsilon _s=0.005\))  60  30  105 
62 (\(\eta \,<\, 2\times 10^{2}\))  8 (\(\epsilon _s=0.005\))  100  
62 (\(\eta \,<\, 2\times 10^{2}\))  17 (\(\epsilon _s=0.001\))  200 
The modes and the linearization points from the training trajectories were saved and then used with cases which are slightly different from the training cases. The results we obtained are encouraging for this kind of model order reduction for the nonlinear functions.
First test case: single node perturbation in nonequilibrium conditions
Initial conditions for test cases
Test cases  \(\varvec{\delta }_{\varvec{2}}\)  \(\varvec{\delta }_\mathbf{i } \quad \varvec{\forall } \mathbf i \varvec{\ne } \mathbf 2 \) 

Test case 1  1.12  1 
Test case 2  1.15  1.15 
Test cases of TPWL simulations
Test case  Modes ‘\(\mathbf q \)’  Lin Pts ‘\(\mathbf s \)’  Time (s)  Error in \(\varvec{\delta }\)  

Full model  TPWL  Abs  Rel  
1  199  49  240  18  8.75 \( \times \, 10^{4}\)  6.75 \( \times \, 10^{4}\) 
2  62  17  60  4  6.62 \( \times \, 10^{4}\)  5.27 \( \times \, 10^{4}\) 
The results are very promising and the TPWL simulation is considerably faster as the full simulation in a similar case takes about 240 s while the TPWL simulation took about 18 s with good accuracy. The errors are listed in the Table 6. The comparison of the average \(\delta \) between the original and reduced order models are presented in Fig. 4.
Second test case: synchronous nonequilibrium
This is a case similar to the third training trajectory where we had all the nodes starting from a synchronous nonequilibrium position, in this case we gave the initial conditions of 1.15 rather than 0.8 in the training case \(\delta _i = 1.15 \ \forall i \). The initial conditions used for this test case are presented in Table 5.
Convergence analysis
It can be conclusively said that the major impact on the accuracy of the reduced order linearized model depends on having more linearized point. This is quite intuitive, since adding more points around which the linearization of the nonlinear function is performed it will be able to capture the nonlinear behavior more effectively and hence reducing the error.
Confidence interval
Initial conditions used in the build up for confidence interval
Trajectory for confidence interval  \(\varvec{\delta }_{\varvec{2}}\)  \(\varvec{\delta }_\mathbf{i } \quad \varvec{\forall } \mathbf i \varvec{\ne } \mathbf 2 \) 

Trajectory 1  1.5  1 
Confidence interval for parameters with training trajectory of Table 7
Parameter  Upper limit  Lower limit 

\(p^m\)  0.97 (p.u.)  – 
\({d_i}\)  –  0.15 (p.u.) 
\(\delta _2\)  2.25  1 
\(b_{int}\) (p.u.)  105  95 
As we have described earlier that in this method several training trajectories can be included in the offline phase to make the reduced basis. This implies that for values over the limits in the Table 8, we can add another training trajectory so that the error remains acceptable.
Conclusions
TPWL proves to be a very robust method for model order reduction of models containing nonlinear functions. It has been proved as a fast, reliable and accurate MOR technique as observed from the results presented by the test cases in “Numerical experiments” section. The method as described is separated into offline and online phases, where in the offline phase the selection of linearization points is carried out. That is the only time consuming part of the method and as it is performed only once during offline phase the time penalty on the overall procedure is not severe.
For the confidence interval, some other values of disturbances in \(\{\delta \}\), and different values of \(p^m, {d_i}, b_{int}\) were trialed and the results as shown in “Confidence interval” section proves the robustness of the method considering the large variations possible for simulation with reduced model. Note that, the confidence interval’s simulation were performed with only one training trajectory and if more trajectories are included in the reduced basis the interval where the method can be applied increases and the results improves.
The method is very well adapted to the problem discussed in this study and more application for example, for the differentialalgebraic equations (DAEs) of the network grids containing both generators (PV nodes) and the loads (PQ nodes), can further consolidate the current method as a well established model reduction method.
Declarations
Author's contributions
All the authors contributed in the development and adaptation of the method for the current study. Malik performed the coding in MATLAB and drafted the manuscript. Borzacchielo, Diez and Chinesta supervised the study and advised on the draft and the corrections in the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Magnusson PC. The transientenergy method of calculating stability. Trans Am Inst Electr Eng. 1947;66(1):747–55. doi:https://doi.org/10.1109/TAIEE.1947.5059502.View ArticleGoogle Scholar
 Aylett PD. The energyintegral criterion of transient stability limits of power systems. Proc IEE Part C Monogr. 1958;105(8):527–36. doi:https://doi.org/10.1049/pic.1958.0070.View ArticleGoogle Scholar
 Z. Bai PD, Freund R. Reducedorder modeling. In: Schilders EJWT, Maten W, editors. Handbook of numerical analysis Vol XIII, numerical methods in electromagnetics vol 13, 2nd edn. North Holland : Elsevier; 2005. p. 825–95.Google Scholar
 Schilders W. Introduction to model order reduction. In: Schilders WHA, van der Vorst HA, Rommes J, editors. Model order reduction: theory, research aspects and applications. Mathematics in industry, vol 13. Berlin: Springer; 2008. p. 3–32. doi:https://doi.org/10.1007/9783540788416.
 Pinnau R. Model reduction via proper orthogonal decomposition. In: Schilders, WHA, van der Vorst HA, Rommes J, editors. Model order reduction: theory, research aspects and applications. Mathematics in industry, vol 13. Berlin: Springer; 2008. p. 95–109. doi:https://doi.org/10.1007/9783540788416.
 Chinesta F, Leygue A, Bordeu F, Aguado JV, Cueto E, Gonzalez D, Alfaro I, Ammar A, Huerta A. Pgdbased computational vademecum for efficient design, optimization and control. Arch Comput Methods Eng. 2013;20(1):31–59. doi:https://doi.org/10.1007/s118310139080x.MathSciNetView ArticleGoogle Scholar
 Parrilo PA, Lall S, Paganini F, Verghese GC, Lesieutre BC, Marsden JE. Model reduction for analysis of cascading failures in power systems. In: American control conference, 1999. Proceedings of the 1999, vol 6. 1999. p. 4208–42126. doi:https://doi.org/10.1109/ACC.1999.786351.
 Montier L, Henneron T, Clénet S, Goursaud B. Transient simulation of an electrical rotating machine achieved through model order reduction. Adv Model Simul Eng Sci. 2016;3(1):1–17. doi:https://doi.org/10.1186/s403230160062z.View ArticleGoogle Scholar
 Kashyap N, Werner S, Riihonen T, Huang YF. Reducedorder synchrophasorassisted state estimation for smart grids. In: Smart grid communications (SmartGridComm), 2012 IEEE third international conference. 2012. p. 605–10 . doi:https://doi.org/10.1109/SmartGridComm.2012.6486052.
 WilleHaussmann B, Link J, Wittwer C. Simulation study of a smart grid approach: Model reduction, reactive power control. In: Innovative smart grid technologies conference Europe (ISGT Europe), 2010 IEEE PES. 2010. p. 1–7. doi:https://doi.org/10.1109/ISGTEUROPE.2010.5638971.
 Rewieński MJ. A trajectory piecewiselinear approach to model order reduction of nonlinear dynamical systems. PhD thesis, Massachusetts Institute of Technology. 2003.Google Scholar
 Liu Y, Yuan W, Chang H. A global maximum error controllerbased method for linearization point selection in trajectory piecewiselinear model order reduction. IEEE Trans Comp Aided Des Integr Circuits Syst. 2014;33(7):1100–4. doi:https://doi.org/10.1109/TCAD.2014.2307000.View ArticleGoogle Scholar
 Chen Y, White J. A quadratic method for nonlinear model order reduction. In: Technical proceedings of the 2000 international conference on modeling and simulation of microsystems. 2000. p. 477–80. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.8951.
 Vasilyev D, Rewienski M, White J. A tbrbased trajectory piecewiselinear algorithm for generating accurate loworder models for nonlinear analog circuits and mems. In: Design automation conference, 2003. Proceedings. 2003; p. 490–5. doi:https://doi.org/10.1109/DAC.2003.1219055.
 Qu L, Chapman PL. A trajectory piecewiselinear approach to model order reduction for nonlinear stationary magnetic devices. In: Computers in power electronics, 2004. Proceedings. 2004 IEEE workshop. 2004; p. 15–19. doi:https://doi.org/10.1109/CIPE.2004.1428113.
 Rewienski M, White J. A trajectory piecewiselinear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans Comp Aided Des Integr Circuits Syst. 2003;22(2):155–70. doi:https://doi.org/10.1109/TCAD.2002.806601.View ArticleGoogle Scholar
 Farooq MU, Xia L, Hussin FAB, Malik AS. Automated model generation of analog circuits through modified trajectory piecewise linear approach with Chebyshev Newton interpolating polynomials. In: Intelligent systems modelling simulation (ISMS), 2013 4th international conference. 2013; p. 605–9. doi:https://doi.org/10.1109/ISMS.2013.28.
 Long C, Simonson LJ, Liao W, He L. Floorplanning optimization with trajectory piecewiselinear model for pipelined interconnects. In: Proceedings of the 41st annual design automation conference. 2004. p. 640–5.Google Scholar
 Farooq MU, Xia L. Local approximation improvement of trajectory piecewise linear macromodels through chebyshev interpolating polynomials. In: Design automation conference (ASPDAC), 2013 18th Asia and South Pacific. 2013. p. 767–72. doi:https://doi.org/10.1109/ASPDAC.2013.6509693.
 Burgard S, Farle O, Klis D, DyczijEdlinger R. Orderreduction of fieldslevel models with affine and nonaffine parameters by interpolation of subspaces. IFAC PapersOnLine. 2015;48(1):170–5. doi:https://doi.org/10.1016/j.ifacol.2015.05.111.View ArticleGoogle Scholar
 Panzer H, Mohring J, Eid R, Lohmann B. Parametric model order reduction by matrix interpolation. Automatisierungstechnik. 2010;58(8):475–84. doi:https://doi.org/10.1524/auto.2010.0863.View ArticleGoogle Scholar
 Xie W, Theodoropoulos C. An offline model reductionbased technique for online linear MPC applications for nonlinear largescale distributed systems. In: Pierucci S, Ferraris GB, editors. In: 20th European symposium on computer aided process engineering. Computer aided chemical engineering, vol 28, . Amsterdam: Elsevier; 2010. p. 409–14. doi:https://doi.org/10.1016/S15707946(10)280690. http://www.sciencedirect.com/science/article/pii/S1570794610280690.
 He J, Sætrom J, Durlofsky LJ. Enhanced linearized reducedorder models for subsurface flow simulation. J Comput Phys. 2011;230(23):8313–41. doi:https://doi.org/10.1016/j.jcp.2011.06.007.MathSciNetView ArticleMATHGoogle Scholar
 Cardoso MA, Durlofsky LJ. Linearized reducedorder models for subsurface flow simulation. J Comput Phys. 2010;229(3):681–700. doi:https://doi.org/10.1016/j.jcp.2009.10.004.MathSciNetView ArticleMATHGoogle Scholar
 Bergen AR, Vittal V. Power system analysis. 2nd ed. New Jersey: Prentice Hall; 1999.Google Scholar
 Shampine LF, Reichelt MW. The MATLAB ODE suite. SIAM J Sci Comput. 1997;18:1–22. doi:https://doi.org/10.1214/aoms/1177729959.MathSciNetView ArticleMATHGoogle Scholar
 Liang YC, Lee HP, Lim SP, Lin WZ, Lee KH, Wu CG. Proper orthogonal decomposition and its applications—part I: theory. J Sound Vib. 2002;252(3):527–44. doi:https://doi.org/10.1006/jsvi.2001.4041.MathSciNetView ArticleMATHGoogle Scholar
 Kerschen G, Golinval JC, Vakakis AF, Bergman LA. The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 2005;41(1–3):147–69. doi:https://doi.org/10.1007/s1107100528032.MathSciNetView ArticleMATHGoogle Scholar
 Berkooz G, Holmes P, Lumley JL. The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech. 1993;25(1):539–75. doi:https://doi.org/10.1146/annurev.fl.25.010193.002543.
 Albunni MN. Model order reduction of moving nonlinear electromagnetic devices. PhD thesis, Technical University of Munich. 2010.Google Scholar
 Rewienski M, White J. A trajectory piecewiselinear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. In: Computer aided design, 2001. ICCAD 2001. In: IEEE/ACM international conference on. 2001. p. 252–7. doi:https://doi.org/10.1109/ICCAD.2001.968627.
 Acha E, Agelidis VG, AnayaLara O, Miller TJE. 4  power flows in compensation and control studies. In: Acha E, Agelidis VG, AnayaLara O, Miller TJE, editors. Power electronic control in electrical systems. Newnes Power Engineering Series. Oxford: Newnes; 2002. p. 106–52. doi:https://doi.org/10.1016/B9780750651264/500043. http://www.sciencedirect.com/science/article/pii/B9780750651264500043.
 Susuki Y, Mezić I, Hikihara T. Coherent swing instability of power grids. J Nonlinear Sci. 2011;21(3):403–39. doi:https://doi.org/10.1007/s0033201090875.MathSciNetView ArticleMATHGoogle Scholar