Viable Computation of the Largest LyapunovCharacteristic Exponent for Power Systems

Brendan Hayes and Federico Milano, Fellow, IEEESchool of Electrical and Electronic Engineering, University College Dublin, Ireland

{brendan.hayes, federico.milano}@ucd.ie

Abstract—Stochastic Differential Algebraic Equations (SDAEs)are used to model power systems. However, there is no universallyaccepted method to properly evaluate the stability of such models.The theoretical and numerical aspects of the computation ofthe largest Lyapunov Characteristic Exponent (LCE) for powersystems with the inclusion of stochastic processes is discussedas a method to provide a measure of stability. A semi-implicitformulation of power systems is employed in order to exploitparallelism, sparsity and to have low memory requirements.Three case studies are considered, two based on the IEEE 14-bus system as well as a 1,479-bus model of the all island Irishtransmission grid.

Index Terms—Lyapunov characteristic exponents (LCEs), sta-bility, limit sets, stochastic processes, differential-algebraic equa-tions (DAEs), deterministic chaos, stochastic chaos.

I. INTRODUCTION

In recent years, power systems have undergone major struc-tural changes due to the desire to promote the use of clean andsecure renewable energy. Wind energy is the fastest growingrenewable energy source for electricity worldwide which hasled to an increase in the volatility and uncertainty present inpower systems. As a result, this has incurred drastic changesto how power systems are modeled and simulated. Volatilitycan be modeled as a set of stochastic processes that, if coupledwith dynamics models, lead to describe power systems as aset of SDAEs. There is no universally accepted method toproperly evaluate the stability of such models. Hence, thefocus of this research is to addresses this issue by proposinga computationally viable method to calculate the largest LCEwhich can be utilized as a measure of the stability of stochasticpower system models.

Modeling power systems as a set of Stochastic DifferentialEquations (SDEs) was first proposed back in the nineties [1].However, it is only in recent years that methods to study theimpact of stochastic fluctuations on the dynamic response ofpower systems have started to be developed [2]–[7]. In thesepapers, the common approach to study the stability of powersystems is a Monte Carlo method based on strong solutionsof the SDAEs. This requires a high number of time domainsimulations, b, which is computationally expensive. The MonteCarlo method is suited to small systems up to a few tens

Brendan Hayes and Federico Milano are supported by Science FoundationIreland (SFI) under Grant No. SFI/15/IA/3074. F. Milano is also a beneficiaryof the EC Marie Skłodowska-Curie CIG No. PCIG14-GA-2013-630811.

of buses but is not suitable to large systems which yieldhigh dimension SDEs. This paper proposes the application ofLyapunov theory to study the stability of SDE-based powersystem models which reduces the computational burden from bsimulations to 1. The proposed method provides a quantitativesolution to outline if a power system is chaotic. It can be usedby electrical engineers as an aid to design robust transmissionsystems as the renewable energy sector grows into the futureand is based on the concept of stochastic chaos.

Lyapunov theory was first proposed in [8] and has beenwidely used in the numerical analysis of nonlinear dynamicssince the work of Seydel in [9]. Lyapunov theory is a tool thatcan be used to capture the sensitive dependence of a systemto initial conditions. The LCE provides a measure of how twoorbits that start close together converge (or diverge) as timeprogresses. Roughly speaking, LCEs relate to the trajectoriesof dynamical systems as eigenvalues relate to equilibriumpoints. Where eigenvalues indicate the stability of equilibriumpoints, LCEs indicate the stability of trajectories.

The main difficulty that limits the widespread adoption ofLCEs to study chaotic motions and stochastic processes forreal world large transmission networks is that such a tool iscomputationally expensive. In [10] and [11], various methodsto compute LCEs are proposed but these methods are limitedto linearized models and only small benchmark systems upto a few tens of buses are considered. The calculation ofLCEs of real-world nonlinear power system models requiresthe numerical integration of the variational equation associatedwith the original dynamic system. Unfortunately, the sizeof the variational equation increases with the square of thenumber of state variables of the system. While methods existfor its computation, from a practical point of view, none areapplicable to high dimensional systems such as real-worldpower systems which could have several thousand state andalgebraic variables. Herein lies the novelty of this research, todevelop a computationally efficient method to viably computeLCEs for power system models of arbitrary size.

The remainder of the paper is organized as follows, inSection II relevant background information of Lyapunov the-ory and dynamical systems is presented. In Section III, theproposed method is explained along with the mathematics thatunderpin it. The method discussed in this section is appliedto three case studies in Section IV; one deterministic caseand two stochastic cases. The computationally efficiency ofthe proposed approach is illustrated through the calculation ofthe LCEs of the 1,479-bus model of the all-island Irish grid

including stochastic wind speed fluctuations. Finally, SectionV draws conclusions and outlines possible areas of future workin the field.

II. LYAPUNOV THEORY

This section provides some basic background and defini-tions of dynamical systems that are relevant for chaotic mo-tions and describes the rationale and practical implementationof LCEs.

Lyapunov exponents are a quantitative method that capturesthe sensitive dependence of systems to initial conditions. Theyprovide a measure of how two orbits that start close togetherconverge or diverge as time progresses. Diverging orbits aretermed chaotic orbits and, in this work, the definition of chaosprovided by [12] will be used; a dynamical system is chaoticif it has a positive Lyapunov exponent.

Figure 1: Divergence of two orbits starting from nearby initial points.

Let us consider a set of nonlinear Ordinary DifferentialEquations (ODEs) in the form:

x = h(x) , (1)

where x, x ∈ Rn is the state vector at time t and h is smooth.Consider two nearby points x0 and x0+u0 in the phase spaceM, as illustrated in Fig. 1, where u0 is a small perturbationof the initial point x0. After time t, their images under theflow will be φ(x0, t) and φ(x0 + u0, t) and the perturbationu(t) is defined as:

u(t) = φ(x0 + u0, t)− φ(x0, t) = φx(x0, t) · u0 , (2)

where φx(x0, t) is the gradient of φ with respect to x0. Theaverage exponential rate of convergence (or divergence) of thetwo trajectories is defined by:

limt→∞

eλ(x0,u0)t = limt→∞

‖u(t)‖‖u0‖

= limt→∞

‖φx(x0, t) · u0‖ ,

where ‖u‖ is the norm of the vector u. The number λ istermed the LCE and is indicative of the stability of the system.

λ(x0,u0) = limt→∞

1

tln‖u(t)‖‖u0‖

(3)

Under weak smoothness conditions on the dynamic system, thelimit (3) exists and is finite for almost all points x0 ∈M andfor almost all tangent vectors u0 and it is equal to the largestLCE, λ1 [13]. If the largest LCE is positive, the nearby orbitsexponentially diverge. These are termed chaotic orbits.

For a continuous-time dynamical system (1) and an initialpoint x0, the tangent vector u(t) defined in (2) evolves in timesatisfying the so called variational equation [14]:

Φ(x0, t) = J(φ(x0, t)) ·Φ(x0, t), Φ(x0, 0) = In , (4)

where Φ(x0, t) is the derivative with respect to x0 of φ(t)at x0 i.e. Φ(x0, t) = φx(x0, t) and is termed the transitionmatrix. Equation (4) is a linear time-variant differential equa-tion whose coefficients depend on the evolution of the originalsystem (1). Hence, in general, (4) can be solved only togetherwith (1), as follows:[

x

Φ

]=

[h(x)J(x)

],

[x(t0)Φ(t0)

]=

[x0

In

](5)

where J is the Jacobian matrix of h(x). The application oftraditional integration techniques to the variational equationtypically leads to all solutions converging to the largest LCE.In this work, the largest LCE is sufficient to define theasymptotic behavior of a dynamical system. The largest LCEλ1 can then be estimated by applying (3):

λ1 ≈1

tln ‖Φ(x0, t) · u0‖ , (6)

where u0 is a randomly generated vector of order n. Thereare three situations to consider for λ1:• λ1 < 0: The flow is attracted to a stable fixed point or

stable periodic orbit.• λ1 = 0: The orbit is a neutral fixed point.• λ1 > 0: The flow is unstable and chaotic.

Negative Lyapunov exponents are characteristic of dissipativeor non-conservative systems. Note, bounded infinite trajec-tories that do not converge towards a fixed point are char-acterized by at least one zero LCE which corresponds toa perturbation of the phase point along its own trajectory.1

This certainly happens for SDAEs with bounded trajectories.Hence, the largest LCE satisfies the condition λ1 ≤ 0 for non-chaotic orbits.

III. LCES OF DIFFERENTIAL ALGEBRAIC EQUATIONS

The determination of the largest LCE is relatively straight-forward for ODEs. However, power system models for tran-sient stability analysis cannot be formulated as ODEs. Typi-cally, they are modeled as a set of explicit nonlinear Differen-tial Algebraic Equations (DAEs) [15]. However, in [16], theauthor proposes a novel semi-implicit model of power systems,that is more general and less computationally demanding thanthe explicit model. The semi-implicit model is as follows:

[T 0R 0

] [x0

]=

[f(x,y)g(x,y)

], (7)

where x, x ∈ X ⊂ Rn are the state variables; y, y ∈ Y ⊂ Rmare the algebraic variables; f : X×Y 7→ Rn is the vector field;and g : X × Y 7→ Rm are the algebraic constraints.

1Power system models also show an additional vanishing LCE due to theconstant motion of the synchronous speed reference which is used to definethe rate of change of rotor angles of synchronous machines.

In this paper, for simplicity but without loss of generality,we assume that T and R are time-invariant and very sparse.The time-variant case can be deduced from the developmentsbelow. Apart from numerical properties, which are extensivelydiscussed in [16], the main advantage of the formulationin (7) is that T might not be full rank which allows theimposition of an infinitely fast dynamic response to somestate variables. This approach greatly reduces the number ofoperations to compute equations and elements of the Jacobianmatrix of the DAE, increases the sparsity of the Jacobianmatrix and allows effortless switching between state variablesand algebraic variables. The proposed method can be extendedto other topologies.

The key difficulty with assessing the LCE of power systemsis determining the transition matrix Φ(x0, t). Consider a DAEof the form:

Γz(t) = Ψ(z(t)) (8)

where

Γ =

[T 0R 0

], z =

[x0

]and Ψ(z) =

[f(x,y)g(x,y)

].

The generic solution to (8) is:

Γz(t) =

t∫0

Ψ(z(τ), τ)dτ + Γz(0)

Expand Ψ(z(τ), τ) using the Taylor series expansion aboutthe trajectory χ(t;χ0):

Ψ(z(t), t) = Ψ(χ(t), t)+∂Ψ

∂z

∣∣∣∣z=χ(t)

(z(t)−χ(t))+R2(z)

where R2(z) contains higher order terms that are nonlinear.Only considering first order terms:

γ(t) =

t∫0

Ψ(χ(τ), τ)dτ +

t∫0

J(τ ;χ)(γ(τ)− χ(τ))dτ

where J(τ ;χ) is the Jacobian matrix of Ψ. Letting ζ(t) =γ(t)− χ(t):

Γζ(t) =

t∫0

J(τ ;χ)ζ(t)dτ + Γζ(0) (9)

Note that ζ(t) is the solution of the differential equation:

Γζ = J(t;χ)ζ (10)

The solution to (10) is:

ζ(t) = Φ(t, 0;χ)ζ(0) (11)

where Φ(t, 0;χ) is the state transition matrix for (11).If an implicit integration scheme is utilized, which is highly

recommended due to the stiffness of power system models(see, for example, [17] and [18]), the Jacobian matricesrequired to compute Φ are already available because these areneeded to integrate the DAE. Moreover, since the variationalequation is linear, one can implement the explicit expression

to integrate such an equation. For example, using the implicitbackward Euler method with a time step of ∆t, (10) becomes:

Γζ(∆t) = ζ(0) + ∆tJ(χ(∆t))ζ(∆t)

ζ(∆t) = [Γ−∆tJ(χ(∆t))]−1ζ(0) (12)

and

ζ(k∆t) =

k∏i=1

[Γ−∆tJ(χ((k − i+ 1)∆t))]−1ζ(0) (13)

Thus, the state transition matrix in (11) is therefore given by:

Φ(k∆t, 0;χ0) =

k∏i=1

[Γ−∆tJ(χ((k − i+ 1)∆t))]−1 (14)

In the next section, we will demonstrate the computationalviability of the proposed approach to determine the transitionmatrix and the largest LCE for power systems.

IV. CASE STUDIES

This section presents three case studies. Subsection IV-Afocuses on the IEEE 14-bus system, which is known toshow a deterministic chaotic behavior for sufficiently highloading levels [19]. The IEEE 14-bus system with the inclusionof noise in the load power consumption is considered inSubsection IV-B to define the behavior of LCEs for non-deterministic chaotic motions. Subsection IV-C considers theall-island Irish transmission system. This is a large real-worldnetwork where the computational burden of the variationalequation would typically restrict the calculation of LCEs. Thecomputational viability of the proposed method is outlinedhere.

All simulations are obtained using Dome, a Python-basedpower system software tool [20]. The Dome version utilizedin this case study is based on Python 3.4.3; ATLAS 3.10.2 fordense vector and matrix operations; CVXOPT 1.1.8 for sparsematrix operations; and KLU 1.3.6 – included in SUITESPARSE4.5.1 – for sparse matrix factorization. All simulations wereexecuted on a 64-bit Ubuntu 14.04 operating system runningon a 8 core 3.60 GHz Intel Xeon with 12 GB of RAM.

A. IEEE 14-bus System - Deterministic Chaos

The IEEE benchmark network consists of 2 synchronousmachines and 3 synchronous compensators, 2 two-winding and1 three-winding transformers, 15 transmission lines and 11loads. The system also includes primary voltage regulators(AVRs) and a PSS connected at machine 1. All data of theIEEE 14-bus system as well as a detailed discussion of itstransient behavior can be found in [15].

The system includes 63 state variables, of which only 52are associated to non-null time constants, and 84 algebraicvariables. Null-time constants are associated to synchronousmachine stator fluxes, ψsd and ψsq , and the variable e′d whosetime constant, T ′q0, is null for the machine connected at bus1. Hence in (7) n = 63 and m = 84.

In [19], the authors show that increasing the loading levelof the IEEE 14-bus system without PSS leads to a Hopf bi-furcation followed by a series of period-doubling bifurcations

900 901 902 903 904 905

Time − [s]

0.9998

1.0002

ω2−

[pu]

(a)

0.98 1.02

ω1 − [pu]

0.99

1.008

ω2−

[pu]

(b)

Figure 2: (a) Synchronous generator 2 speed (b) Poincare section for ω1 andω2 for Kω = 0.5.

that eventually leads to the appearance of chaos. It is alsoshown that chaos can be eliminated by including the PSS if aproper PSS model with an accurate tuning of the parameters isimplemented. However, if the PSS is not properly tuned, it ispossible that the PSS may trigger a series of Hopf bifurcationsleading to chaos.

Consider the case when Kω = 0.5, the PSS is not properlytuned and the system is oscillating, Fig. 2 (a). It is clear thatthe output is a limit cycle with a period of T ≈ 0.653 (s).One conventional method to assess the behavior of oscillatingsystems is a Poincare section; two state variables are sampledat a frequency f and plotted against one another. Figure 2 (b)shows the Poincare section where ω1 and ω2, rotor speeds 1and 2, are sampled at f = 1/0.653. Since the system oscillateswith the same period as the sampling frequency a single pointis visible on the Poincare section which is indicative of a stablelimit-cycle i.e. the system is not chaotic.

Now consider the case when Kω = 50, the PSS is notproperly tuned and the system is wildly oscillating, Fig. 3 (a).Unlike Fig. 2 (a), it is difficult to visually determine whetherthe output is periodic or aperiodic. However, the Poincaresection reveals that the output is aperiodic as output generatesa fractal pattern on the Poincare map which is indicative ofchaos. In this instance, it cannot be determined if the orbit ischaotic or a quasi-periodic; the motion is associated with afinite number of frequencies that are related to one another byirrational multiples.

While Poincare sections are useful to determine the typeof behavior of deterministic systems operate with, they offer

900 905 910 915 920

Time − [s]

0.988

0.993

ω2−

[pu]

(a)

0.9897 0.9906

ω1 − [pu]

0.989

0.9915

ω2−

[pu]

(b)

Figure 3: (a) Synchronous generator 2 speed (b) Poincare section for ω1 andω2 for Kω = 50.

little insight into the stability of higher order orbits. We willnow consider the largest LCEs for both Kω values.

0 200 400 600 800 1000

Time − [s]

0

0.2

0.4

0.6

0.8

1.0

LCE

Figure 4: Largest LCE of the IEEE 14-bus system with Kω = 0.5 (red) andKω = 50 (black)

Figure 4 shows the largest LCE for the IEEE 14-bus systemfor the two values of Kω . In red, with Kω = 0.5, the systemoperates with a limit cycle with a period of T = 0.6533(s), the largest LCE settles to 0. This is the expected valuefor the largest LCE as the system is not operating in thechaotic region. However, when Kω = 50, the largest LCEsettles to 0.2 i.e. a positive number which is an indicator ofchaos. Unlike the Poincare section, this quantitative measureacts as a binary measure as to whether the system is chaoticor not. When stochastic systems are considered, qualitativemeasures are not applicable as plots will appear random andthus, chaotic. There is no way to qualitatively determine ifthese systems are chaotic.

0.988 1.015

ω1 − [pu]

0.988

1.015ω5−

[pu]

(a)

0.9940 1.006

ω1 − [pu]

0.9930

1.007

ω5−

[pu]

(b)

Figure 5: Flow of the state space of rotor speeds of machines 1 and 5 for (a)scenario 1 and (b) scenario 2.

The CPU simulations time to calculate the largest LCEis 5 (s) with Kω = 50. The total simulation time is 54(s). Therefore, the calculation of the LCE increased the totalsimulation time by approximately 10 %. The simulations useda fixed integration step of ∆t = 0.01 (s). However, this is asmall benchmark system with no stochastic perturbations. Wewill now consider the effect of stochastic perturbations on thecomputational burden of the LCE.

B. IEEE 14-bus System - Stochastic Chaos

This subsection discusses the effect of noise on the dynamicbehavior of the IEEE 14-bus system. Noise is modeled asOrnstein-Uhlenbeck (OU) process as discussed in [6]. Thisis a Wiener process characterized by a normal distribution,exponential autocorrelation and constant standard deviation.The parameters of the OU processes of the loads are thesame as those considered in [7]. The following scenarios areconsidered.• Original system with 5 synchronous machines and two

primary frequency regulators at generators 1 and 2 (seeFig. 5 (a)).

• A wind power plant substituting the synchronous genera-tor at bus 2; only the generator at bus 1 includes primaryfrequency regulation (see Fig. 5 (b)).

Figure 5 illustrates the flow of the state space of rotorspeeds 1 and 5 for both scenarios. Unlike the Poincare sectionsin Section IV-A, these diagrams offer little insight into thestability of the system. Since the system is operating withstochastic loads, the flow in the state space appears to be

random and therefore has the appearance of chaos. Thereis no method to select a sampling frequency and to plot aPoincare section so as to filter out the stochastic perturbations.As a result, conventional qualitative deterministic methods arenot applicable to stochastic situations. In both scenarios, thetrajectories of the machine rotor speeds appear to be similar.There are no major differences. For this reason, the largestLCE is proposed as a method to study the stability of stochasticpower systems as it is a quantitative method and not open tosubjective interpretation. This provides a method to distinguishbetween the two modes of operation

0 100 200 300 400

Time − [s]

0

0.2

0.4

0.6

0.8

1.0

LCE

Figure 6: Largest LCE of the IEEE 14-bus system for (red) scenario 1 and(black) scenario 2.

Figure 6 shows the largest LCE for both stochastic sit-uations. With all the synchronous machines connected, thelargest LCE settles to 0 and indicates a stable system. Forthe second scenario, when wind is introduced, the system hasa positive LCE and is unstable. It is to be expected that themore regulation that is present in a system, the more robust asystem is. The evaluation of the LCEs provides a quantitativemeasure of such robustness. Thus, subjective interpretationusing qualitative approaches are not required. This sectiondemonstrates the promising tool of LCEs to define the impactof stochastic processes on the performance of power systemcontrollers.

C. All-Island Irish Transmission System

In this final case study, the all-island Irish transmissionsystem set up at the UCD Electricity Research Center isconsidered. The model includes 1, 479 buses, 1, 851 trans-mission lines and transformers, 245 loads, 22 conventionalsynchronous power plants with AVRs and turbine governors,6 PSSs and 176 wind power plants. The topology and the dataof the transmission system are based on the real-world systemprovided by the Irish TSO, EirGrid. However, dynamic datais estimated and is based on the authors knowledge of thetechnology of power plants. Hence, simulation results, whilerealistic, do not represents actual operating condition of theIrish transmission system.

The main purpose of this section is to highlight the com-putational viability of the proposed method for computing thelargest LCE for a large system. The system includes 2, 112state variables, of which 2, 064 have non-null time constants,and 6, 338 algebraic variables.

0 500 1000 1500

Time − [s]

0

0.2

0.4

0.6

0.8

1.0LCE

Figure 7: Largest LCE for the all-island Irish system.

The system is simulated for 1500 (s) with a time step of 0.01(s). The simulation takes 4 minutes and 35 seconds withoutthe calculation of the variational equation and the largestLCE. Figure 7 shows the largest LCE is stable for the Irishtransmission system. The calculation of the largest LCE takesan additional 5 minutes and 42 seconds. This approximatelydoubles the simulation time for the system.

This section has highlighted the feasibility of using LCEsto quantify chaos in large power systems of arbitrary size. Themain advantages of the proposed approach compared to MonteCarlo based methods are as follows:• Computationally efficient: this work has demonstrated

that the proposed method is computationally viable andefficient for power systems of arbitrary size. Monte Carlobased methods require a high number of simulations tobe performed for statistical accuracy. The calculation ofthe largest LCE requires one.

• Binary result: the largest LCE is a single number whichcan be used to determine whether the system is stable orchaotic.

• Robust: Monte Carlo based methods rely on statisticalaccuracy which does not guarantee that all possibledynamics are captured even if b is high. Lyapunov theorydoes not suffer from the same constraint.

However, it is important to note that the largest LCE islimited in its approach. Monte Carlo based methods can alsobe used as a qualitative tool to assess whether power systemsare chaotic. They have one key advantage compared to thelargest LCE, they can provide other insights into power sys-tems such as the statistical information outlined in references[2]–[7]. The largest LCE gives a measure of the stability of thesystem but requires further simulations to gain more insights.

V. CONCLUSIONS

This paper highlights the key issue stopping the widespreadadoption of the largest LCE as a stability measure for bothdeterministic and stochastic chaotic motions in power systemsas the computational burden associated with the variationalequation for real-world power systems. The size variationalequation is equal to the square of the number of state variablesin the system. Historically, this has been a prohibitive size asthe computational burden is too great to warrant its inclusion.

The main contribution of this paper is to provide a compu-tationally viable method to estimate the largest LCE for power

systems through the use of semi-implicit SDAEs, by proposinga method to efficiently calculate the transition matrix. Thiscontribution was highlighted using three case studies to studythe calculation of LCEs for both deterministic and stochasticpower systems as well as demonstrating its effectiveness forlarge real world power systems through the all-island Irishtransmission system.

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