# Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Theory

#### 2.1. Stochastic Differential-Algebraic Equations

- $\mathit{x}(\xb7)$ is adapted to the filtration ${({\mathcal{F}}_{t})}_{t\ge {t}_{0}}$,
- ${\int}_{{t}_{0}}^{{t}_{f}}\left|{f}_{0}^{\ell}({x}_{s})\right|ds<\infty $ almost sure (a.s.), for all $\ell =1,\dots ,n$,
- ${\int}_{{t}_{0}}^{{t}_{f}}{\left|{f}_{j}^{\ell}({x}_{s})\right|}^{2}d{w}_{s}^{j}<\infty $ a.s., for all $j=1,\dots ,m$, and $\ell =1,\dots ,n$, and
- (2) holds for every $t\in \mathbb{I}$ a.s.

#### 2.2. Random Dynamical Systems Generated by SDEs

#### 2.3. Lyapunov Exponents of Ergodic RDSs

## 3. $\mathit{QR}$ Methods for Computing LEs

#### 3.1. Discrete $QR$ Method

#### 3.2. Continuous $QR$ Method

#### 3.3. Computational Considerations

#### 3.4. Numerical Examples

#### 3.4.1. Example 1

#### 3.4.2. Example 2

## 4. Application of LEs to Power Systems Stability Analysis

#### 4.1. Modeling Power Systems through SDAEs

#### 4.2. Modeling Stochastic Perturbations

#### 4.3. Test Cases

#### 4.3.1. Case 1: SMIB with Stochastic Load

#### 4.3.2. Case 2: SMIB with Regulator Perturbed by Noise

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AVR | automatic voltage regulator |

CIGRE | Conseil International des Grands Réseaux Électriques |

DAE | differential-algebraic equation |

IEEE | Institute of Electrical and Electronics Engineers |

LE | Lyapunov exponent |

LLE | largest Lyapunov exponent |

MDS | metric dynamical system |

MET | Multiplicative Ergodic Theorem |

ODE | ordinary differential equation |

OU | Ornstein–Uhlenbeck |

PSS | power system stabilizer |

RDSs | random dynamical system |

SDAE | stochastic differential-algebraic equation |

SDE | stochastic differential equation |

SMIB | single-machine infinite-bus |

SSSA | small-signal stability assessment |

## Appendix A

**Table A1.**Numerical results of the calculated LE for SDAE system (29) computed via Discrete $QR$-EM method.

T | h | $\mathbb{E}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathit{\sigma}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathbb{V}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | Rel. Error [%] | CPU-Time [s] |
---|---|---|---|---|---|---|

6000 | $1\times {10}^{-1}$ | $-1.51906$ | $0.00476$ | $2.266\times {10}^{-5}$ | $13.48994$ | $0.9827$ |

6000 | $1\times {10}^{-2}$ | $-1.35231$ | $0.00296$ | $8.734\times {10}^{-6}$ | $1.03179$ | $12.6422$ |

6000 | $1\times {10}^{-3}$ | $-1.33874$ | $0.00234$ | $5.483\times {10}^{-6}$ | $0.01807$ | $125.6875$ |

6000 | $1\times {10}^{-4}$ | $-1.33903$ | $0.00225$ | $5.042\times {10}^{-6}$ | $0.03958$ | $4367.2676$ |

$\mathrm{12,000}$ | $1\times {10}^{-1}$ | $-1.51870$ | $0.00334$ | $1.116\times {10}^{-5}$ | $13.46268$ | $2.0336$ |

$\mathrm{12,000}$ | $1\times {10}^{-2}$ | $-1.35217$ | $0.00184$ | $3.392\times {10}^{-6}$ | $1.02160$ | $24.9517$ |

$\mathrm{12,000}$ | $1\times {10}^{-3}$ | $-1.33920$ | $0.00161$ | $2.579\times {10}^{-6}$ | $0.05206$ | $252.4366$ |

$\mathrm{12,000}$ | $1\times {10}^{-4}$ | $-1.33902$ | $0.00134$ | $1.795\times {10}^{-6}$ | $0.03848$ | $8815.9035$ |

$\mathrm{20,000}$ | $1\times {10}^{-1}$ | $-1.51780$ | $0.00262$ | $6.858\times {10}^{-6}$ | $13.39551$ | $3.3806$ |

$\mathrm{20,000}$ | $1\times {10}^{-2}$ | $-1.35236$ | $0.00139$ | $1.944\times {10}^{-6}$ | $1.03533$ | $41.2975$ |

$\mathrm{20,000}$ | $1\times {10}^{-3}$ | $-1.33936$ | $0.00133$ | $1.781\times {10}^{-6}$ | $0.06437$ | $416.3228$ |

$\mathrm{20,000}$ | $1\times {10}^{-4}$ | $-1.33902$ | $0.00119$ | $1.415\times {10}^{-6}$ | $0.03922$ | $\mathrm{13,870.1475}$ |

**Table A2.**Numerical results of the calculated LE for SDAE system (29) computed via Discrete $QR$-Milstein method.

T | h | $\mathbb{E}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathit{\sigma}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathbb{V}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | Rel. Error [%] | CPU-Time [s] |
---|---|---|---|---|---|---|

6000 | $1\times {10}^{-1}$ | $-1.47000$ | $0.00356$ | $1.267\times {10}^{-5}$ | $9.82471$ | $1.3947$ |

6000 | $1\times {10}^{-2}$ | $-1.34911$ | $0.00205$ | $4.217\times {10}^{-6}$ | $0.79239$ | $14.2033$ |

6000 | $1\times {10}^{-3}$ | $-1.33883$ | $0.00185$ | $3.415\times {10}^{-6}$ | $0.02480$ | $139.4657$ |

6000 | $1\times {10}^{-4}$ | $-1.33901$ | $0.00224$ | $5.039\times {10}^{-6}$ | $0.03812$ | $4786.6657$ |

$\mathrm{12,000}$ | $1\times {10}^{-1}$ | $-1.46914$ | $0.00249$ | $6.202\times {10}^{-6}$ | $9.75996$ | $2.8596$ |

$\mathrm{12,000}$ | $1\times {10}^{-2}$ | $-1.34925$ | $0.00186$ | $3.466\times {10}^{-6}$ | $0.80302$ | $27.8426$ |

$\mathrm{12,000}$ | $1\times {10}^{-3}$ | $-1.33889$ | $0.00176$ | $3.093\times {10}^{-6}$ | $0.02924$ | $280.9451$ |

$\mathrm{12,000}$ | $1\times {10}^{-4}$ | $-1.33900$ | $0.00134$ | $1.794\times {10}^{-6}$ | $0.03700$ | $9629.9437$ |

$\mathrm{20,000}$ | $1\times {10}^{-1}$ | $-1.46973$ | $0.00159$ | $2.517\times {10}^{-6}$ | $9.80448$ | $4.6544$ |

$\mathrm{20,000}$ | $1\times {10}^{-2}$ | $-1.34924$ | $0.00141$ | $1.982\times {10}^{-6}$ | $0.80274$ | $46.9282$ |

$\mathrm{20,000}$ | $1\times {10}^{-3}$ | $-1.33915$ | $0.00130$ | $1.6991\times {10}^{-6}$ | $0.04854$ | $465.2272$ |

$\mathrm{20,000}$ | $1\times {10}^{-4}$ | $-1.33900$ | $0.00119$ | $1.416\times {10}^{-6}$ | $0.03770$ | $\mathrm{15,121.3377}$ |

**Table A3.**Numerical results of the calculated LE for SDAE system (29) computed via Continuous $QR$-EM method.

T | h | $\mathbb{E}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathit{\sigma}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathbb{V}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | Rel. Error [%] | CPU-Time [s] |
---|---|---|---|---|---|---|

6000 | $1\times {10}^{-1}$ | $-1.35864$ | $0.00326$ | $1.061\times {10}^{-5}$ | $1.50459$ | $1.3841$ |

6000 | $1\times {10}^{-2}$ | $-1.34005$ | $0.00278$ | $7.743\times {10}^{-6}$ | $0.11616$ | $13.6211$ |

6000 | $1\times {10}^{-3}$ | $-1.33822$ | $0.00277$ | $7.651\times {10}^{-6}$ | $0.02128$ | $135.0360$ |

6000 | $1\times {10}^{-4}$ | $-1.33892$ | $0.00224$ | $5.025\times {10}^{-6}$ | $0.03152$ | $4642.6202$ |

$\mathrm{12,000}$ | $1\times {10}^{-1}$ | $-1.35932$ | $0.00226$ | $5.091\times {10}^{-6}$ | $1.55512$ | $2.7334$ |

$\mathrm{12,000}$ | $1\times {10}^{-2}$ | $-1.33999$ | $0.00186$ | $3.459\times {10}^{-6}$ | $0.11134$ | $26.9335$ |

$\mathrm{12,000}$ | $1\times {10}^{-3}$ | $-1.33813$ | $0.00159$ | $2.535\times {10}^{-6}$ | $0.02786$ | $272.8842$ |

$\mathrm{12,000}$ | $1\times {10}^{-4}$ | $-1.33891$ | $0.00134$ | $1.794\times {10}^{-6}$ | $0.03052$ | $9217.7371$ |

$\mathrm{20,000}$ | $1\times {10}^{-1}$ | $-1.35888$ | $0.00196$ | $3.835\times {10}^{-6}$ | $1.52252$ | $4.4753$ |

$\mathrm{20,000}$ | $1\times {10}^{-2}$ | $-1.34010$ | $0.00096$ | $9.306\times {10}^{-7}$ | $0.11965$ | $45.0326$ |

$\mathrm{20,000}$ | $1\times {10}^{-3}$ | $-1.33807$ | $0.00148$ | $2.187\times {10}^{-6}$ | $0.03224$ | $465.5119$ |

$\mathrm{20,000}$ | $1\times {10}^{-4}$ | $-1.33892$ | $0.00119$ | $1.417\times {10}^{-6}$ | $0.03121$ | $\mathrm{14,583.8282}$ |

**Table A4.**Numerical results of the calculated LE for SDAE system (29) computed via Continuous $QR$-Milstein method.

T | h | $\mathbb{E}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathit{\sigma}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | $\mathbb{V}\left[{\mathit{\lambda}}_{\mathit{T}}\right]$ | Rel. Error [%] | CPU-Time [s] |
---|---|---|---|---|---|---|

6000 | $1\times {10}^{-1}$ | $-1.33950$ | $0.00287$ | $8.228\times {10}^{-6}$ | $0.07468$ | $1.5009$ |

6000 | $1\times {10}^{-2}$ | $-1.33767$ | $0.00253$ | $6.386\times {10}^{-6}$ | $0.06235$ | $14.8586$ |

6000 | $1\times {10}^{-3}$ | $-1.33810$ | $0.00232$ | $5.402\times {10}^{-6}$ | $0.02998$ | $147.5276$ |

6000 | $1\times {10}^{-4}$ | $-1.33890$ | $0.00225$ | $5.052\times {10}^{-6}$ | $0.02988$ | $5082.6248$ |

$\mathrm{12,000}$ | $1\times {10}^{-1}$ | $-1.33931$ | $0.00259$ | $6.692\times {10}^{-6}$ | $0.06075$ | $3.0427$ |

$\mathrm{12,000}$ | $1\times {10}^{-2}$ | $-1.33864$ | $0.00121$ | $1.460\times {10}^{-6}$ | $0.01053$ | $29.3666$ |

$\mathrm{12,000}$ | $1\times {10}^{-3}$ | $-1.33769$ | $0.00153$ | $2.329\times {10}^{-6}$ | $0.06087$ | $299.2588$ |

$\mathrm{12,000}$ | $1\times {10}^{-4}$ | $-1.33889$ | $0.00134$ | $1.802\times {10}^{-6}$ | $0.02878$ | $\mathrm{10,035.9100}$ |

$\mathrm{20,000}$ | $1\times {10}^{-1}$ | $-1.33990$ | $0.00182$ | $3.296\times {10}^{-6}$ | $0.10453$ | $4.9331$ |

$\mathrm{20,000}$ | $1\times {10}^{-2}$ | $-1.33828$ | $0.00152$ | $2.310\times {10}^{-6}$ | $0.01654$ | $49.5662$ |

$\mathrm{20,000}$ | $1\times {10}^{-3}$ | $-1.33853$ | $0.00140$ | $1.960\times {10}^{-6}$ | $0.00258$ | $505.4304$ |

$\mathrm{20,000}$ | $1\times {10}^{-4}$ | $-1.33889$ | $0.00119$ | $1.416\times {10}^{-6}$ | $0.02948$ | $\mathrm{15,917.1861}$ |

**Table A5.**Numerical results of the calculated LEs for the Chua’s system (32) computed via the four $QR$-based methods for $T=6000$.

Method | h | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | $\mathcal{S}$ |
---|---|---|---|---|---|

$1\times {10}^{-3}$ | $0.23994$ | $-0.06890$ | $-3.22041$ | $-3.04936$ | |

D-EM | $5\times {10}^{-4}$ | $0.24207$ | $-0.07063$ | $-3.23055$ | $-3.05911$ |

$1\times {10}^{-4}$ | $0.24824$ | $-0.07093$ | $-3.20410$ | $-3.05678$ | |

$1\times {10}^{-3}$ | $0.24166$ | $-0.07304$ | $-3.23649$ | $-3.06788$ | |

D-Mil | $5\times {10}^{-4}$ | $0.24486$ | $-0.06837$ | $-3.24114$ | $-3.06465$ |

$1\times {10}^{-4}$ | $0.24738$ | $-0.06726$ | $-3.24892$ | $-3.05880$ | |

$1\times {10}^{-3}$ | $0.22970$ | $-0.06103$ | $-3.22547$ | $-3.05680$ | |

C-EM | $5\times {10}^{-4}$ | $0.22962$ | $-0.06231$ | $-3.23562$ | $-3.06831$ |

$1\times {10}^{-4}$ | $0.23400$ | $-0.06837$ | $-3.24523$ | $-3.06960$ | |

$1\times {10}^{-3}$ | $0.23514$ | $-0.06963$ | $-3.23238$ | $-3.05686$ | |

C-Mil | $5\times {10}^{-4}$ | $0.23169$ | $-0.06858$ | $-3.21973$ | $-3.05662$ |

$1\times {10}^{-4}$ | $0.23287$ | $-0.07468$ | $-3.21613$ | $-3.05794$ |

**Table A6.**Numerical results of the approximated LLE of SMIB system (40) corresponding to the study-case 1, computed via the four $QR$-based techniques.

$\mathit{\rho}$ | D-EM | D-Mil | C-EM | C-Mil | $\mathit{\rho}$ | D-EM | D-Mil | C-EM | C-Mil |
---|---|---|---|---|---|---|---|---|---|

$0.00$ | $-0.02849$ | $-0.02849$ | $-0.02864$ | $-0.02864$ | $1.05$ | $-0.00287$ | $-0.00267$ | $-0.00268$ | $-0.00114$ |

$0.05$ | $-0.02848$ | $-0.02847$ | $-0.02863$ | $-0.02863$ | $1.10$ | $-0.00075$ | $-0.00159$ | $-0.00158$ | $-0.00278$ |

$0.10$ | $-0.02843$ | $-0.02845$ | $-0.02860$ | $-0.02863$ | $1.15$ | $-0.00436$ | $-0.00259$ | $-0.00261$ | $-0.00136$ |

$0.15$ | $-0.02845$ | $-0.02841$ | $-0.02857$ | $-0.02859$ | $1.20$ | $0.00184$ | $0.00093$ | $0.00091$ | $-0.00191$ |

$0.20$ | $-0.02832$ | $-0.02852$ | $-0.02867$ | $-0.02863$ | $1.25$ | $0.00175$ | $-0.00149$ | $-0.00154$ | $0.00217$ |

$0.25$ | $-0.02815$ | $-0.02836$ | $-0.02852$ | $-0.02833$ | $1.30$ | $0.00149$ | $0.00029$ | $0.00026$ | $0.00059$ |

$0.30$ | $-0.02837$ | $-0.02811$ | $-0.02828$ | $-0.02837$ | $1.35$ | $0.00038$ | $0.00044$ | $0.00040$ | $0.00409$ |

$0.35$ | $-0.02827$ | $-0.02795$ | $-0.02811$ | $-0.02811$ | $1.40$ | $0.00870$ | $0.00284$ | $0.00279$ | $0.00311$ |

$0.40$ | $-0.02734$ | $-0.02778$ | $-0.02797$ | $-0.02762$ | $1.45$ | $0.00338$ | $0.00072$ | $0.00075$ | $0.00314$ |

$0.45$ | $-0.02722$ | $-0.02758$ | $-0.02775$ | $-0.02773$ | $1.50$ | $0.00409$ | $0.00570$ | $0.00564$ | $0.00607$ |

$0.50$ | $-0.02676$ | $-0.02658$ | $-0.02674$ | $-0.02675$ | $1.55$ | $0.00644$ | $0.00806$ | $0.00802$ | $0.01024$ |

$0.55$ | $-0.02702$ | $-0.02575$ | $-0.02590$ | $-0.02537$ | $1.60$ | $0.01014$ | $0.00642$ | $0.00638$ | $0.00553$ |

$0.60$ | $-0.02508$ | $-0.02606$ | $-0.02620$ | $-0.02591$ | $1.65$ | $0.00797$ | $0.01089$ | $0.01086$ | $0.00915$ |

$0.65$ | $-0.01250$ | $-0.00779$ | $-0.00781$ | $-0.02359$ | $1.70$ | $0.00724$ | $0.00896$ | $0.00892$ | $0.00826$ |

$0.70$ | $-0.01016$ | $-0.00549$ | $-0.00550$ | $-0.00353$ | $1.75$ | $0.00828$ | $0.00808$ | $0.00805$ | $0.00815$ |

$0.75$ | $-0.00412$ | $-0.01408$ | $-0.01417$ | $-0.00428$ | $1.80$ | $0.01366$ | $0.00658$ | $0.00654$ | $0.01361$ |

$0.80$ | $-0.00503$ | $-0.00544$ | $-0.00546$ | $-0.00454$ | $1.85$ | $0.00776$ | $0.00977$ | $0.00974$ | $0.01083$ |

$0.85$ | $-0.00505$ | $-0.00656$ | $-0.00658$ | $-0.00505$ | $1.90$ | $0.01068$ | $0.01346$ | $0.01341$ | $0.01192$ |

$0.90$ | $-0.00372$ | $-0.00471$ | $-0.00475$ | $-0.00350$ | $1.95$ | $0.01537$ | $0.01313$ | $0.01305$ | $0.01072$ |

$0.95$ | $-0.00503$ | $-0.00452$ | $-0.00454$ | $-0.00406$ | $2.00$ | $0.01248$ | $0.01225$ | $0.01219$ | $0.01031$ |

$1.00$ | $-0.00353$ | $0.00024$ | $0.00023$ | $-0.00314$ |

**Table A7.**Numerical results of the approximated LLE of SMIB system (41) corresponding to the study-case 2, computed via C-EM method.

$\mathit{\rho}$ | LLE | $\mathit{\rho}$ | LLE | $\mathit{\rho}$ | LLE | $\mathit{\rho}$ | LLE | $\mathit{\rho}$ | LLE |
---|---|---|---|---|---|---|---|---|---|

$0.00$ | $-0.74586$ | $0.60$ | $-0.76208$ | $1.20$ | $-0.88585$ | $1.80$ | $-0.64744$ | $2.40$ | $-0.17751$ |

$0.05$ | $-0.74593$ | $0.65$ | $-0.76663$ | $1.25$ | $-0.89951$ | $1.85$ | $-0.64093$ | $2.45$ | $-0.09607$ |

$0.10$ | $-0.74572$ | $0.70$ | $-0.77893$ | $1.30$ | $-0.88447$ | $1.90$ | $-0.60895$ | $2.50$ | $-0.07736$ |

$0.15$ | $-0.74632$ | $0.75$ | $-0.78894$ | $1.35$ | $-0.89645$ | $1.95$ | $-0.56286$ | $2.55$ | $0.01194$ |

$0.20$ | $-0.74647$ | $0.80$ | $-0.78972$ | $1.40$ | $-0.87056$ | $2.00$ | $-0.53564$ | $2.60$ | $-0.00056$ |

$0.25$ | $-0.74694$ | $0.85$ | $-0.81366$ | $1.45$ | $-0.86469$ | $2.05$ | $-0.51171$ | $2.65$ | $0.04541$ |

$0.30$ | $-0.74786$ | $0.90$ | $-0.83571$ | $1.50$ | $-0.84152$ | $2.10$ | $-0.45690$ | $2.70$ | $0.16176$ |

$0.35$ | $-0.74901$ | $0.95$ | $-0.85522$ | $1.55$ | $-0.81798$ | $2.15$ | $-0.41292$ | $2.75$ | $0.20227$ |

$0.40$ | $-0.75015$ | $1.00$ | $-0.85872$ | $1.60$ | $-0.80238$ | $2.20$ | $-0.28496$ | $2.80$ | $0.27489$ |

$0.45$ | $-0.75187$ | $1.05$ | $-0.86683$ | $1.65$ | $-0.77636$ | $2.25$ | $-0.35850$ | $2.85$ | $0.28201$ |

$0.50$ | $-0.75370$ | $1.10$ | $-0.88458$ | $1.70$ | $-0.72155$ | $2.30$ | $-0.25838$ | $2.90$ | $0.33127$ |

$0.55$ | $-0.75980$ | $1.15$ | $-0.89293$ | $1.75$ | $-0.73500$ | $2.35$ | $-0.13307$ | $2.95$ | $0.38053$ |

## References

- Biegler, L.; Campbell, S.; Mehrmann, V. Control and Optimization with Differential-Algebraic Constraints; Advances in Design and Control; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2012. [Google Scholar]
- Brenan, K.; Campbell, S.; Petzold, L. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations; Classics in Applied Mathematics; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1996. [Google Scholar]
- Kunkel, P.; Mehrmann, V. Differential-Algebraic Equations: Analysis and Numerical Solution; EMS Textbooks in Mathematics; European Mathematical Society: Zurich, Switzerland, 2006. [Google Scholar]
- Kloeden, P.; Platen, E. Numerical Solution of Stochastic Differential Equations; Stochastic Modelling and Applied Probability; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Mao, X. Stochastic Differential Equations and Applications; Elsevier Science: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Oksendal, B. Stochastic Differential Equations: An Introduction with Applications; Universitext; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Cong, N.D.; The, N.T. Stochastic Differential-algebraic Equations of Index 1. Vietnam. J. Math.
**2010**, 38, 117–131. [Google Scholar] - Schein, O.; Denk, G. Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits. J. Comput. Appl. Math.
**1998**, 100, 77–92. [Google Scholar] [CrossRef] [Green Version] - Winkler, R. Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math.
**2003**, 163, 435–463. [Google Scholar] [CrossRef] [Green Version] - Lyapunov, A. General Problem of the Stability Of Motion; Control Theory and Applications Series; Taylor & Francis: Abingdon, UK, 1992. [Google Scholar]
- Oseledec, V.I. A multiplicative ergodic theorem. Ljapunov characteristic number for dynamical systems. Trans. Moscow Math. Soc.
**1968**, 19, 197–231. [Google Scholar] - Arnold, L. Random Dynamical Systems; Monographs in Mathematics; Springer: Berlin/Heidelberg, Germany, 1998; 2003. [Google Scholar]
- Cong, N.D.; The, N.T. Lyapunov spectrum of nonautonomous linear stochastic differential algebraic equations of index-1. Stochastics Dyn.
**2012**, 12, 1–16. [Google Scholar] [CrossRef] - Küpper, D.; Kværnø, A.; Rößler, A. A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise. Bit Numer. Math.
**2012**, 52, 437–455. [Google Scholar] [CrossRef] - Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.M. Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica
**1980**, 15, 9–20. [Google Scholar] [CrossRef] - Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.M. Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 2: Numerical application. Meccanica
**1980**, 15, 21–30. [Google Scholar] [CrossRef] - Dieci, L.; Vleck, E.S.V. Lyapunov Spectral Intervals: Theory and Computation. Siam J. Numer. Anal.
**2003**, 40, 516–542. [Google Scholar] [CrossRef] - Dieci, L.; Van Vleck, E.S. Lyapunov and Sacker–Sell Spectral Intervals. J. Dyn. Differ. Equ.
**2007**, 19, 265–293. [Google Scholar] [CrossRef] - Linh, V.H.; Mehrmann, V. Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations. J. Dyn. Differ. Equ.
**2009**, 21, 153–194. [Google Scholar] [CrossRef] - Linh, V.H.; Mehrmann, V.; Van Vleck, E.S. QR methods and error analysis for computing Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations. Adv. Comput. Math.
**2011**, 35, 281–322. [Google Scholar] [CrossRef] - Carbonell, F.; Biscay, R.; Jimenez, J.C. QR-Based Methods for Computing Lyapunov Exponents of Stochastic Differential Equations. Int. J. Numer. Anal. Model. Ser. B
**2010**, 1, 147–171. [Google Scholar] - Lamour, R.; März, R.; Tischendorf, C. Differential-Algebraic Equations: A Projector Based Analysis; Differential-Algebraic Equations Forum; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Caraballo, T.; Han, X. Applied Nonautonomous and Random Dynamical Systems: Applied Dynamical Systems; Springer Briefs in Mathematics; Springer International Publishing: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Arnold, L. Lyapunov Exponents of Nonlinear Stochastic Systems. In Nonlinear Stochastic Dynamic Engineering Systems; Ziegler, F., Schuëller, G.I., Eds.; Springer: Berlin, Heidelberg, Germany, 1988; pp. 181–201. [Google Scholar]
- Pikovsky, A.; Politi, A. Lyapunov Exponents: A Tool to Explore Complex Dynamics; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Dieci, L.; Russell, R.; Van Vleck, E. On the Computation of Lyapunov Exponents for Continuous Dynamical Systems. Siam J. Numer. Anal.
**1997**, 34, 402–423. [Google Scholar] [CrossRef] [Green Version] - Grorud, A.; Talay, D. Approximation of Lyapunov exponents of stochastic differential systems on compact manifolds. In Analysis and Optimization of Systems; Bensoussan, A., Lions, J.L., Eds.; Springer: Berlin/Heidelberg, Germany, 1990; pp. 704–713. [Google Scholar]
- Talay, D. Second-order discretization schemes of stochastic differential systems for the computation of the invariant law. Stoch. Stoch. Rep.
**1990**, 29, 13–36. [Google Scholar] [CrossRef] - Dieci, L.; Russell, R.D.; Vleck, E.S.V. Unitary Integrators and Applications to Continuous Orthonormalization Techniques. Siam J. Numer. Anal.
**1994**, 31, 261–281. [Google Scholar] [CrossRef] [Green Version] - Ryagin, M.Y.; Ryashko, L.B. The analysis of the stochastically forced periodic attractors for Chua’s circuit. Int. J. Bifurc. Chaos
**2004**, 14, 3981–3987. [Google Scholar] [CrossRef] - Hirsch, M.; Smale, S.; Devaney, R. Differential Equations, Dynamical Systems, and an Introduction to Chaos; Academic Press: Cambridge, MA, USA; Elsevier Science: Amsterdam, The Netherlands, 2013. [Google Scholar]
- Arnold, L. IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. In Proceedings of the IUTAM Symposium (Solid Mechanics and Its Applications), Madras, Chennai, India, 4–8 January 1999; Springer: Dordrecht, The Netherlands, 2012; pp. 15–28. [Google Scholar]
- Kundur, P. Power System Stability and Control, 1st ed.; McGraw–Hill: Palo Alto, CA, USA, 1994. [Google Scholar]
- Kundur, P.; Paserba, J.; Ajjarapu, V.; Andersson, G.; Bose, A.; Cañizares, C.; Hatziargyriou, N.; Hill, D.; Stankovic, A.; Taylor, C.; et al. Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. IEEE Trans. Power Syst.
**2004**, 19, 1387–1401. [Google Scholar] - Sauer, P.; Pai, M.; Chow, J. Power System Dynamics and Stability: With Synchrophasor Measurement and Power System Toolbox; Wiley-IEEE, Wiley: Hoboken, NJ, USA, 2017. [Google Scholar]
- Machowski, J.; Bialek, J.; Bumby, J. Power System Dynamics: Stability and Control; Wiley: Hoboken, NJ, USA, 2011. [Google Scholar]
- Verdejo, H.; Vargas, L.; Kliemann, W. Stability of linear stochastic systems via Lyapunov exponents and applications to power systems. Appl. Math. Comput.
**2012**, 218, 11021–11032. [Google Scholar] [CrossRef] - Verdejo, H.; Escudero, W.; Kliemann, W.; Awerkin, A.; Becker, C.; Vargas, L. Impact of wind power generation on a large scale power system using stochastic linear stability. Appl. Math. Model.
**2016**, 40, 7977–7987. [Google Scholar] [CrossRef] - Hayes, B.; Milano, F. Viable Computation of the Largest Lyapunov Characteristic Exponent for Power Systems. In Proceedings of the 2018 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), Sarajevo, Bosnia and Herzegovina, 21–25 October 2018; pp. 1–6. [Google Scholar]
- Wadduwage, D.P.; Wu, C.Q.; Annakkage, U. Power system transient stability analysis via the concept of Lyapunov Exponents. Electr. Power Syst. Res.
**2013**, 104, 183–192. [Google Scholar] [CrossRef] - Milano, F.; Zarate-Minano, R. A Systematic Method to Model Power Systems as Stochastic Differential Algebraic Equations. IEEE Trans. Power Syst.
**2013**, 28, 4537–4544. [Google Scholar] [CrossRef] - Geurts, B.J.; Holm, D.D.; Luesink, E. Lyapunov Exponents of Two Stochastic Lorenz 63 Systems. J. Stat. Phys.
**2019**, 18, 1–23. [Google Scholar] [CrossRef] [Green Version] - González-Zumba, A. Wind Power Grid Integration: A Brief Study About the Current Scenario and Stochastic Dynamic Modeling. Master’s Thesis, Universitat Politècnica de València, Valencia, Spain, 2017. [Google Scholar]
- Allen, E. Modeling with Itô Stochastic Differential Equations; Mathematical Modelling: Theory and Applications; Springer: Dordrecht, The Netherlands, 2007. [Google Scholar]
- Pota, H.R. The Essentials of Power System Dynamics and Control, 1st ed.; Springer: Singapore, 2018. [Google Scholar]

**Figure 1.**Discrete and continuous $QR$-based approximations of the Lyapunov exponent (LE) corresponding to stochastic differential-algebraic equation (SDAE) (29) via Euler-Maruyama and Milstein integrators, with a stepsize $h=1\times {10}^{-3}$ and $T=250$. The solid circles show mean and the whiskers the $95\%$ confidence intervals of the trajectories.

**Figure 2.**Discrete and continuous $QR$-based approximations of the LE corresponding to SDAE (29) via Euler-Maruyama and Milstein integrators, with a stepsize $h=1\times {10}^{-3}$ and T = 10,000. The black dashed line in the left-hand side subplot shows the analytic value of $\lambda $.

**Figure 3.**Comparison of relative errors for discrete and continuous $QR$-based approximations of the LE corresponding to SDAE (29) via Euler-Maruyama and Milstein integrators, with a range of stepsizes between $h=1\times {10}^{-2},\dots ,1\times {10}^{-3}$; and with T = 1000, …, 12,000.

**Figure 4.**Comparison of the computing-time for discrete and continuous $QR$-based approximations of the LE corresponding to SDAE (29) via Euler-Maruyama and Milstein integrators, with a range of stepsizes between $h=1\times {10}^{-2},\dots ,1\times {10}^{-3}$; and with T = 1000, …, 12,000.

**Figure 6.**Chua’s system phase-portraits in chaotic regime, (

**a**) without stochastic perturbation; (

**b**) with stochastic perturbation.

**Figure 7.**Time evolution of the computed LEs in stochastic Chua’s system (32) using the four $QR$-based methods for a stepsize $h=1\times {10}^{-4}$ and an interval $T=6000$.

**Figure 8.**IEEE/CIGRE Power systems stability classification [34].

**Figure 11.**Largest Lyapunov exponent (LLE) considering different disturbance sizes $\rho $ for the SMIB Test Case 1, tested with four $QR$-based methods.

**Figure 12.**Phase portraits of the system (40) considering different disturbance sizes $\rho $ for the SMIB Test Case 1.

**Figure 13.**SMIB system scheme equipped with automatic voltage regulator (AVR) and power system stabilizer (PSS), corresponding to Test Case 2.

**Figure 14.**Computed LLE for the dimension 7 SMIB system of Test Case 2, considering different disturbance sizes and using the continuous Euler-Maruyama $QR$-based method.

**Figure 15.**Computing-time comparison of LE calculation for the dimension 7 SMIB Test Case 2. Comparison performed for the four $QR$-based methods in a range of step sizes between $h=[1\times {10}^{-2},1\times {10}^{-3}]$ and with $T=[1000,\mathrm{12,000}]$.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

González-Zumba, A.; Fernández-de-Córdoba, P.; Cortés, J.-C.; Mehrmann, V.
Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems. *Mathematics* **2020**, *8*, 1393.
https://doi.org/10.3390/math8091393

**AMA Style**

González-Zumba A, Fernández-de-Córdoba P, Cortés J-C, Mehrmann V.
Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems. *Mathematics*. 2020; 8(9):1393.
https://doi.org/10.3390/math8091393

**Chicago/Turabian Style**

González-Zumba, Andrés, Pedro Fernández-de-Córdoba, Juan-Carlos Cortés, and Volker Mehrmann.
2020. "Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems" *Mathematics* 8, no. 9: 1393.
https://doi.org/10.3390/math8091393