Power System Transient Stability Analysis Using Sum Of Squares Programming

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1 Power System Transient Stability Analysis Using Sum Of Squares Programming M. Tacchi, B. Marinescu, M. Anghel, S. Kundu, S. Benahmed and C. Cardozo Ecole Centrale Nantes-LS2N-CNRS, France Los Alamos National Laboratory, USA Pacific Northwest National Laboratory, USA RTE-R&D, France Abstract Transient stability is an important issue in power systems but difficult to quantify analytically. Most of the approaches use a simplified model of the generators, that often reduces to the swing equation. In this work, a more detailed model which includes voltage dynamics and both voltage and frequency regulators is considered to get more realistic results. The proposed analysis framework uses an algebraic reformulation technique that recasts the system s dynamics into a set of polynomial differential algebraic equations in conjunction with a sum of squares method to search for a Lyapunov function and to estimate the region of attraction of the stable operating point. The results are checked against a numerical evaluation of the region of attraction. Index Terms Lyapunov theory, nonlinear systems, power systems analysis, region of attraction, sum of squares. I. INTRODUCTION Transient stability is one of the most important power systems analysis problems. From a physical viewpoint, transient stability can be defined as the ability of a power system to maintain its synchronism when subjected to large and transient disturbances [1]. Widespread assessment tools are generally based on indirect methods which rely on the numerical integration of nonlinear differential equations describing system dynamics. However, this approach is not suited to synthesize controllers with a direct quantification of stability margin since it does not provide an analytical characterization of stability [3]. Alternatively, direct methods are based on the estimation of the stability domain of the equilibrium point; they ensure that all the trajectories initiated in this domain converge to the equilibrium point [3]. The main drawback of direct methods is that they rely on the identification of Lyapunov functions which are hard to determine. Indeed, there is not a systematic method for the construction of Lyapunov functions. Moreover, it has been shown that quadratic (i.e., energy type Lyapunov functions do not exist for grids with losses [1]. Recent advances in semidefinite programming relaxation and the use of the Sum Of Squares (SOS decomposition to check non-negativity have opened the way for efficient algorithmic analysis of systems with polynomial vector fields [2]. This approach was used in [3] for transient stability analysis of a power system. However, the generators were modeled only by their swing equations. As the voltage and frequency regulators have an important impact on stability analysis, in the present paper this approach is extended by considering a generator with voltage dynamics and both voltage and frequency regulations. This also provides a way to take into account high penetration of power electronics elements in the grid due to the integration of renewable energies and HVDC transmission lines, thus having an important impact on the transient stability of the system. SOS method gives an analytical solution for the construction of Lypunov functions in order to estimate the Region Of Attraction (ROA for a locally asymptotically stable equilibrium point of the system. The paper is organized as follows: the problem is formulated in Section II on a Single Machine Infinite Bus (SMIB system. SOS formalism is briefly recalled in Section III. The change of variables needed to reach a SOS form (i.e., recasting the model of the system with trigonometric non linearities into a set of polynomial differential algebraic equations is given in Section IV. The recasting procedure is proved to be a Lie-Bäcklund transformation, which means that the transformed system has equivalent trajectories and stability properties [5]. Next, in Section V we relax Lyapunov's conditions for stability and model constraint equations to suitable SOS conditions using theorems from real algebraic geometry in order to formulate the problem as an optimization one. Hence, a Lyapunov function for the asymptotically stable equilibrium point is constructed using the expanding interior algorithm developed in [2], [3]. An estimate of the ROA is given by a level set of the Lyapunov function. We finally test the ROA estimation error in Section VI by numerically computing the real one in all state directions via full nonlinear simulations. The codes are implemented in MATLAB using SOSTOOLS [6], [7]

2 which is a free, third-party MATLAB toolbox that solves SOS problems. The analysis is carried out in the case of a single machine-infinite bus system. II. PROBLEM FORMULATION We consider a synchronous machine G connected to an infinite bus N through two lines in parallel (see Figure 1. The infinite bus imposes a nominal voltage of amplitude V s = 1pu and frequency ω s = 1pu. The synchronous machine has two rotating axes d (direct and q (quadratic, which support the currents i d and i q, generating a voltage (v d, v q. The synchronous machine is characterized by a resistance r and equivalent reactances x d, x d and x q. It receives a mechanical power P m which makes it rotate at frequency ω, generating e.m.f e q with voltage field E fd. Let δ be the phase between the machine and the infinite bus and H the machine s mechanic inertia constant. G 2( S V G (v d, v q V (V s, ω s Figure 1: Synchronous machine connected to an infinite bus. The equations describing the dynamics of this system are given in [8] (p.105. They can be written as follows T d de q 0 2H dω dδ N = e q (x d x di d + E fd (1a = P m (v d i d + v q i q + ri 2 d + ri 2 q (1b = ω ω s (1c i q = (X + x d V s sin δ (R + r(v s cos δ e q (R + r 2 + (X + x d (X + x (1d q i d = X + x q R + r i q 1 R + r V s sin δ (1e v d = x q i q ri d (1f v q = Ri q + Xi d + V s cos(δ (1g where T d 0 is a characteristic time. The machine is governed by two regulators whose equations are the following de fd T a = E fd + K a (V ref V t, (2 where T a, K a are the parameters of the voltage regulator (AVR, V ref is the reference, and V t = vd 2 + v2 q (3a T g dp m = P m + P ref + K g (ω ref ω (3b where T g, K g are the parameters of the turbine regulator (governor and P ref, ω ref are reference values - see Apendix A. We next introduce a temporary short-circuit at node S (see Figure 1, according to the following protocol: At a time t c a short-circuit occurs and we switch from the nominal system (Figure 1 to a new short-circuited system (Figure 2, and thus we are no longer at an equilibrium point. During the short-circuit the system s equations are the same as (1, but with (R, X replaced by 2 3 (R, X and V s replaced by 1 3 V s. The regulation equations (2, (3a and (3b are not affected by the short-circuit. The system leaves the equilibrium point of the nominal model and follows the short-circuit equations for the duration t, until the short-circuit is eliminated. At a time t cl = t c + t, we switch back to the nominal topology and equations: the problem is to know whether the system reaches an equilibrium point or not, i.e. whether it is still in the ROA of one equilibrium points of the nominal model or not. G 2( S V G (v d, v q V (V s, ω s Figure 2: The short-circuited system In order to decide if, after the short-circuit is cleared at time t cl, the trajectory of (1 will return to its stable equilibrium point (SEP, we use SOS programming tools and the Lyapunov function method in order to find an analytical approximation of its region of attraction. III. THE SOS APPROACH SOS are real polynomials that can be written as sums of squares of polynomials: s = k j=1 N p 2 j where p 1,..., p k R n. Here, R n is the set of real polynomials in n variables and we denote by Σ n the set of SOS in n variables. The SOS approach is detailed in [2]. It is based on the Positivstellensatz (P-satz, [9] theorem - see Appendix B - that provides certificates for checking when a semi-algebraic set described by polynomial inequalities, equalities, and nonequalities is empty. The idea is to estimate the ROA of the system using a set whose complement can be described as in the P-satz theorem. Indeed, according to a theorem by Lyapunov [4], for a given vector field f : R n R n, x = 0 is a locally stable equilibrium point of the equation ẋ = f(x, iff there exists a continuously differentiable function V (x : D R, defined on an open set D R n containing the equilibrium point x = 0, such that

3 V (0 = 0 and V (x > 0, x D {0}, and whose derivative along the system s trajectory is negative, i.e. V (x 0, 1 x D. The function V is called a Lyapunov function and any level set Ω = {x Ω V (x } such that Ω D is a positively invariant subset of the system s ROA. The SOS approach consists in finding the largest possible Ω D by optimizing over polynomial V and semi-algebraic D, which is possible as soon as f R n. Two algorithms allowing to perform this research are discussed in [2]: the expanding D algorithm and the expanding interior algorithm. Since the latter is more efficient than the former, we only implemented the expanding interior algorithm. The aim of the SOS program is to use a positive definite p Σ n and an expansion parameter > 0 in order to define a variable sized region, P := {x R n p(x }, included in the level set Ω 1 := {x R n V (x 1} of a yet unknown Lyapunov function V, which we choose to be positive on R n {0}. The optimization problem is to expand as long as we can find a V such that P Ω 1. The level set Ω 1 corresponding to the largest is our best approximation of the ROA. In order for Ω 1 to satisfy the second condition of Lyapunov s theorem, we must have {x R n V (x 1} {0} {x R n V (x < 0}. This optimization problem can be formulated using set emptiness constraints as, V R n,v (0=0 (4a s.t. {x R n V (x 0, x 0} = (4b {x R n p(x, V (x 1, V (x 1} = (4c {x R n V (x 1, V (x 0, x 0} = (4d which can be reformulated as, s.t. V R n,v (0=0 {x R n V (x 0, l 1 (x 0} = {x R n p(x 0, V (x 1 0, V (x 1 0} = {x R n 1 V (x 0, V (x 0, l 2 (x 0} = where l 1, l 2 Σ n are positive definite ensuring the non polynomial constraint x 0. By applying the P-satz theorem, this optimization problem can be now formulated as the SOS programming problem V Rn,V (0=0,k 1,k 2,k 3 Z + s 1,...,s 10 Σn s.t. s 1 s 2 V + l 2k1 1 = 0 s 3 + s 4 ( p + s 5 (V 1 + s 6 ( p(v 1 +(V 1 2k2 = 0 s 7 + s 8 (1 V + s 9 V + s10 (1 V V + l 2k3 2 = 0 In order to limit the size of the SOS problem, we select k 1 = k 2 = k 3 = 1 and we simplify the first constraint by selecting s 2 = l 1 and factoring out l 1 from s 1. Since the second constraint contains quadratic terms in the coefficients 1 With the well known convention V = ( V x T f(x of V, we select s 3 = s 4 = 0, and factor out (V 1 from all the terms. Finally, we select s 10 = 0 in the third constraint in order to eliminate the quadratic terms in V and factor out l 2. Thus, after renumbering the remaining SOS polynomials, we reduce the SOS problem to V R n,v (0=0, s 1,s 2,s 3 Σ n (6a s.t. V l 1 = s 4 Σ n (6b (( ps 1 + (V 1 = s 5 Σ n (6c ((1 V s 2 + V s 3 + l 2 = s 6 Σ n. (6d Since Σ n and the set of positive semi-definite matrices are isomorphic, this is reduced to an LMI problem which can be solved within an iterative algorithm (e.g., [10], [11] (see Figure 3. YES choose (0 > 0 and V (0 Lyapunov function on Ω 1 = {x R n V (0 (x 1} such that P (0 := {p(x (0 } Ω 1 search for s (i+1 6,8,9 Σ n and a imal (i+1 > (i satisfying (6 for V (i+1 = V (i search for s (i+2 6 Σ n, V (i+2 R n and a imal (i+2 > (i+1 satisfying (6 for s (i+2 8,9 = s (i+1 8,9 test (i+2 (i+1 < tol The resulting Ω 1 = {x R n V (i (x 1} is an estimate of 0 s R.O.A. linear search for NO Figure 3: The expanding interior algorithm Some features of SOSTOOLS include the setting of polynomial optimization problems and the search for a polynomial Lyapunov function after expressing the SOS problem as a LMI feasibility problem. In [3], SOSTOOLS is used to implement the expanding interior algorithm for a dynamic model without regulation. IV. RECASTING THE POWER SYSTEMS MODEL The system (1, (2, (3a, (3b can be reformulated as an autonomous ODE: ẋ = F (x (7

4 where x = ( δ, ω, e T q, E fd, P m R 5 is the state vector and the vector field F : R 5 R 5 is defined by: F 1 (x = x 2 ω s F 2 (x = 1 2H (x 5 (v d (xi d (x + v q (xi q (x+ ri d (x 2 + ri q (x 2 F 3 (x = 1 T ( x d0 3 (x d x d i (8 d(x + x 4 F 4 (x = 1 T a ( x 4 + K a (V ref V t (x F 5 (x = 1 T g ( x 5 + P ref + K g (ω ref x 2 with i q (x, i d (x, v q (x and v d (x introduced in (1d, (1e, (1g and (1f. Here one can see that F is not a polynomial vector field, as requested in the SOS approach. However, the following variables allowed us to recast the system as a polynomial ODE: z 1 = sin(δ δ eq z 2 = 1 cos(δ δ eq z 3 = ω ω s z 4 = e q e eq q z 5 = E fd E eq fd z 6 = P m P ref z 7 = V t V eq z 8 = 1 V t 1 V eq (9a (9b (9c (9d (9e (9f (9g (9h where x eq is the value of the variable x at the considered equilibrium point. One can then compute the dynamics of the recasted system, with an equlibrium point z = 0, as ż = H(z, (10 with H : R 8 R 8 defined by H 1 (z = (1 z 2 z 3 H 2 (z = z 1 z 3 H 3 (y = 1 2H (z 6 + P ref (v d (zi d (z+ v q (zi q (z + ri d (z 2 + ri q (z 2 ( H 4 (z = 1 T (z d0 4 + e eq q (x d x d i d(z + z 5 + E eq fd ( H 5 (z = 1 T a (z 5 + E eq fd + K a(v ref (z 7 + V eq H 6 (z = 1 T g ( z 6 K g z 3 H 7 (z = ( z (v d (z zi v d (z + V eq H 8 (z = ( z V eq 3 i {1,2,4} v q (z zi v q (z H i (z (v d (z zi v d (z+ i {1,2,4} v q (z zi v q (z H i (z (11 The size of the recasted system has a direct impact on the computation burden of the ROA estimation. Let us notice that, if (V ref V t (x in F 4 (x is replaced by (Vref 2 V t(x 2, i.e., if F 4 (x = 1 ( x4 + K a (Vref 2 V t (x 2 (12 T a is used in (8, we no longer need to introduce z 7 and z 8 in the recasted system which consists in this case of only the first 6 equations of (11, i.e. ż = H (z, with H : R 6 R 6 a polynomial vector field. From a physical point of view, instead of comparing the magnitude (or modulus of the voltage (phasor over the synchronous machine to a reference, we are comparing its squared magnitude to the square of the reference. As the model is written in per-unit variables, voltages are around 1 and this approximation has little impact (this has also been checked in simulation. However, for the new equations to model the same system as the former, it is necessary to make sure that the change of variable is a Lie-Bäcklund transformation (see Appendix C, i.e. a (local one-to-one, invertible, correspondence between the trajectories of the two systems. In the present case z = Φ(x, with Φ : R 5 R 6, is not a Lie-Bäcklund isomorphism: indeed, F and H are Φ-related, but Φ obviously does not have a smooth inverse Ψ : R 6 R 5. In fact, it is intuitive that one has to add some algebraic constraints on z so as for Φ to be a licit change of variables: Here, the algebraic constraint is G(z = 0. (13 G(z = z z 2 2 2z 2, (14 reflecting the trigonometric identity z1+(1 z = 1. One can verify that Φ : R 5 {z R 6 G(z = 0} is an endogenous transformation of the system [12]. It is then possible to apply the SOS approach to the differential algebraic equation, { ż = H (z (15 G(z = 0. V. SOS ESTIMATION OF THE ROA In the expanding interior algorithm, the addition of the equality constraints G(z = 0 only influences the definition of the domain and of Ω 1 = {z R 6 G(z = 0 and V (z 1}, (16 P = {z R 6 p(z ; G(z = 0}. (17 This leads, according to the P-satz, to the following modification of the expanding interior problem (6 V l 1 q 1 G = s 4 Σ 6 (( ps 1 + (V 1 q 2 G = s 5 Σ 6 ((1 V s 2 + V s 3 + l 2 q 3 G = s 6 Σ 6 (18a (18b (18c where q 1,2,3 R 6 are free polynomial variables (q i G is therefore the definition of an element of the ideal I(G. Using an expanding domain P = {z R 6 p(z := z 2 }, the algorithm in Figure 3 builds an increasing sequence of

5 approximations to the ROA which, when the iterations stop for = , returns the following Lyapunov function, V (z = z z 1 z z 1 z z 1 z z 1 z z 1 z z z 2 z z 2 z z 2 z z 2 z z 2 3 ( z 3 z z 3 z z 3 z z z 4 z z 4 z z z 5 z z 2 6, whose Ω 1 level set provides the largest estimation of the ROA. Two-dimensional projections of the resulting ROA in original coordinates are plotted in Figs. 4 and 5 in red lines. While these estimations are quite large, a comparison to the exact (numerically estimated ROA, thick blue lines in Figure 5, shows that further improvements are possible by increasing the degrees of the polynomials in the SOS program. VI. ROA ESTIMATION VIA NUMERICAL TIME-DOMAIN SIMULATIONS We aim to validate the estimate of the ROA found by the SOS approach, by testing, in simulation, the limits of stability of the system in all the state-space directions. For this, the system is systematically initialized at a starting point far from the considered equilibrium point, and we check by simulation if it goes back to equilibrium or not. Since the system has 5 state variables which means a huge number of combinations and because δ and ω are the most important state variables for transient stability analysis, we decide to make the test in a projection of the state space on the planes (δ, ω, (e q, E fd, and (P m, ω. Figure 5 shows that the estimated ROA (red lines is inside the real one (thick blue line, computed numerically by simulation and this validates the previous results. The arrows in the plots show that the real ROA is larger in the direction of the arrows. The same figure shows that the trajectories of the system initialized in several points inside the numerically computed ROA converge asymptotically to the considered equilibrium point. VII. CONCLUSIONS SOS approach and tools have been successfully used to quantify transient stability of a SMIB system for which the generator has been modeled in more detail as in previous studies. Indeed, voltage dynamics and voltage and frequency regulations were taken into account in this formalism. First, this provides more accuracy in estimation of the stability margin in terms of the ROA. Indeed, the estimated ROA is large enough compared to the exact ROA computed by simulation. Next, the Lyapunov approach is well suited for the control synthesis and this quantification can be further exploited to build/tune regulators in order to imize ROA. As a matter of fact, in the SOS optimization one can next include the regulators parameters as decision variables. Future work will focus on (a (b (c Figure 4: 2D projections of the estimated ROA (red line and expanding domain P (blue line, and 3D views of the LF in the coordinate pairs: (a (e q, E fd, (b (P m, ω, and (c (δ, ω. estimation with the full model, i.e., without the approximation (12 and with more detailed models for the generator and regulators, inclusion of the non linearities of the machine related to saturations of the actuating variables, application to larger grids, extension to the tuning of regulators parameters.

6 (a (b T d 0 = 9.67 x d = 2.38 x d = x q = 1.21 H = 3 r = ω s = ω ref = 1 R = 0.01 X = V s = 1 T a = 1 K a = 70 V ref = 1 T g = 0.4 K g = 0.5 P ref = 0.7 The considered equilibrium point of 8, approximated with (12, is: E eq fd = ; e eq q = ; δ eq = ω eq = ω ref = 1 ; Pm eq = P ref = 0.7 APPENDIX B BASIC ALGEBRAIC GEOMETRY In this section we introduce the basic algebraic definitions that are necessary in order to present one of the most important theorems in real algebraic geometry. Definition 1. Given {g 1,..., g t } R n, the Multiplicative Monoid generated by g j s is M(g 1,..., g t = {g 1 k 1 g 2 k 2... g t k t k 1,..., k t Z + } (20 which is the set of all finite products of g j s including the empty product, defined to be 1. It is denoted as M(g 1,..., g t. Definition 2. Given {f 1,..., f s } R n, the Cone generated by f j s is { C(f 1,..., f s := s 0 + } s i b i s i Σ n, b i M(f 1,..., f s (21 Definition 3. Given {h 1,..., h u } R n, the Ideal generated by h k s is { } I(h 1,..., h u := hk p k p k R n (22 With these definitions we can now state the following fundamental theorem. Theorem 1 (Positivstellensatz. Given polynomials {f 1,..., f s }, {g 1,..., g t }, and {h 1,..., h u } in R n, the following are equivalent: (c Figure 5: 2D projections of the real ROA (thick blue line, the SOS estimate (thin red line, and expanding domain P (thin blue line on the coordinate subspaces: (a (e q, E fd. (b (P m, ω, and (c (δ, ω. Trajectories of the system initialized in several points (stars inside the numerically computed ROA converge asymptotically to the considered equilibrium point. APPENDIX A MODEL PARAMETERS These are the parameters of the test system: 1 The set x f 1 (x 0,..., f s (x 0 Rn g 1 (x 0,..., g t (x 0 h 1 (x = 0,..., h u (x = 0 (23 is empty. 2 There exist polynomials f C(f 1,..., f s, g M(g 1,..., g t, and h I(h 1,..., h u such that f + g 2 + h = 0. (24 The LMI based tests for SOS polynomials can be used to prove that the set emptiness condition from the Positivstellensatz (P -satz holds, by finding specific f, g and h such that f + g 2 + h = 0. These f, g and h are known as P- satz certificates since they certify that the equality holds. The search for bounded degree P-satz certificates can be done using semidefinite programming (SDP. If the degree bound is chosen large enough the SDP will be feasible and give the refutation certificates. It is important to notice that the

7 P -satz offers no guidance on how to select the degrees of the polynomials involved in the definition of the monoid M, cone C, and ideal I. By putting an upper bound on these degrees and checking whether (24 holds, one can create a series of tests for the emptiness of (23. Each of these tests requires the construction of some sum of squares and polynomial multipliers, resulting in a sum of squares program that can be solved using SOSTOOLS. APPENDIX C LIE-BÄCKLUND TRANSFORMATIONS Consider a (finite-dimensional autonomous system ξ = F (ξ (25 with the vector field F defined on the smooth manifold M. Then we can introduce the notion of Lie-Bäcklund transformation: Definition 4. [5] Let (M, F, (N, H be two systems, Φ : M C N, p M and q := Φ(p N. Then, if ξ is a trajectory of (M, F in a neighbourhood of p, then ζ := Φ ξ stays in a neighbourhood of q and we have ζ(t = Φ(ξ(t F (ξ(t (26 REFERENCES [1] H. D. Chiang, Direct Methods for Stability Analysis of Electric Power Systems, Wiley [2] Z. W. Jarvis-Wloszek, Lyapunov based analysis and controller synthesis for polynomial systems using sum-of-squares optimization, Ph.D. dissertation, University of California, Berkeley, CA, [3] M. Anghel, F. Milano, and A. Papachristodoulou, Algorithmic construction of Lyapunov functions for power system stability analysis Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 60, no. 9, pp , Sept [4] H. K. Khalil, Nonlinear Systems. New Jersey: Prentice Hall, [5] M. Fliess, J. Levine, P. Martin, and P. Rouchon, A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems, IEEE Transactions on Automatic Control,44(5: , may [6] S. Prajna, A. Papachristodoulou, and P.A. Parrilo, SOS- TOOLS: Sum of squares optimization toolbox for Matlab, [7] S. Prajna, A. Papachristodoulou, and P.A. Parrilo, Introducing sostools: A general purpose sum of squares programming solver, in Proceedings of the Conference on Decision and Control, pp , [8] P.W. Sauer and M.A. Pai, Power System Dynamics and Stability. Prentice Hall, [9] J. Bochnak, M. Coste, and M.-F. Roy. Geometrie Algebrique Reelle. Springer, [10] J.F. Sturm, Using sedumi 1.02, a matlab toolbox for optimizaton over symmetric cones, Optimization Methods and Software, 1112:pp , [11] Y. Nesterov, and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, Studies in Applied and Numerical Mathematics, SIAM, [12] M. Tacchi, Stability analysis and synthesis of hybrid closed-loops for modern power systems: Sums Of Squares for Lyapunov Stability, End-ofstudy Engineer Internship Report, February which holds even in infinite dimension: everything depends only on a finite number of coordinates. Φ is an endogenous transformation iff, for any ξ in a neighbourhood of p Φ(ξ F (ξ = H(Φ(ξ (27 (we say that F and H are Φ-related at (p, q; then ζ = H(ζ and Φ has a smooth inverse Ψ (then, H and F are automatically Ψ-related. Φ is a Lie-Bäcklund isomorphism iff we locally have T Φ(span(F = span(h and Φ has a smooth inverse Ψ such that T Ψ(span(H = span(f 2. In other words, for ξ in a neighbourhood of p and ζ in a neighbourhood of q, we should have Φ(ξ (span(f (ξ = span(h(φ(ξ (28a Ψ(ζ (span(h(ζ = span(f (Ψ(ζ. (28b From this definition, it is obvious that an endogenous transformation (which is a particular case of Lie-Bäcklund isomorphism preserves the trajectories (and so the stability properties of a system, which is exactly what we need for our transformations not to modify the ROA of the considered equilibrium point. 2 A distribution D :=span(v 1,..., V k being intuitively defined on a manifold M by D(x =span(v 1 (x,..., V k (x T M, where V 1,..., V k are vector fields on M.