Nonlinear Observer Design for Power System Monitoring

2nd IFAC Workshop on Convergence of Information Technologies and Control Methods with Power Systems May 22-24, 2013. Cluj-Napoca, Romania

Nonlinear Observer Design for Power System Monitoring Didier Georges ∗ ∗

Grenoble INP / Univ Grenoble 1 / Univ Grenoble 3 / CNRS, GIPSA-lab, F-38402 Saint Martin d’H`eres, (e-mail: [email protected]).

Abstract: In this paper, a methodology for designing a nonlinear state observer for power systems is proposed. The algorithm is shown to consist in a sequence of approximate LQ optimal observers whose solution can be explicitely computed thanks to Kalman Filter Riccati equations. A decentralized version of the approach based on augmented Lagrangian relaxation is also discussed. The application to a 5-bus power system is finally presented and demontrates the effectiveness of the approach. ∗ ∗

This work was supported by the french project ANR WINPOWER.

Keywords: Power system monitoring, nonlinear optimal filtering, decentralized state estimation. NOMENCLATURE PMU PDC EKF δi θi ωi Pim PGi QGi 0 Eqi ω0 Di Hi 0 Td0i xdi x0di Kai Tai Ef di Bij Gij PLi QLi Vi Ci

observer whose formulation exactly conresponds to the extension of the Kalman filter to the nonlinear case. Unlike using a Kalman Filter (as in Huang (2007) for instance) or the Extended-Kalman Filter (as in Georges (2012)), the here-proposed approach provides a numerical solution to the exact nonlinear optimal filter.

Phasor Measurement Unit. Phasor Data Concentrator. Extended Kalman Filtering. Subscript i denotes the ith generator. Angle of the ith generator in radian. Bus i phase angle, in radian. Relative speed in rad/s. Mechanical input power, in p.u.. Active power delivered, in p.u.. Reactive power, in p.u.. Transient EMF in quadrature axis , in p.u.. Synchronous machine spedd, in rad/s. Per unit damping constant. Inertia constant in second. Direct axis transient short circuit time constant, in s. Direct axis reactance, in p.u.. Direct axis transient reactance, in p.u.. Exciter gain, in p.u.. Exciter time constant in p.u.. Exciter voltage in p.u.. Susceptance of admittance matrix element i, j. Conductance of admittance matrix element i, j. Load active power at bus i, in p.u.. Load reactive power at bus i, in p.u.. Voltage at bus i, in p.u.. Set of bus indices connected to bus i, including i.

The paper is now organized as follows: Section 2 is dedicated to some backgrounds on wide-area power system modelling. In section 3, a nonlinear optimal finite-horizon state observer for power system monitoring is proposed. Section 4 describes how this design can be used in a receding horizon approach. In section 5, a decentralized version of the approach is discussed. Section 6 is devoted to the application of the methodology to a 5-bus power system. Finally some conclusions and perspectives are provided. 2. SOME BACKGROUNDS ON POWER SYSTEM MODELING FOR WIDE-AREA MONITORING In this paper, a multi-machine power system, with N generators connected to a grid made of M buses, where M ≥ N , is considered. Using a one-axis model of each generator 1 , the model of the ith generator equiped with a dynamical exciter can be expressed as follows (see Ilic (2000) for example): Mechanical Dynamics of Generator i:

1. INTRODUCTION The monitoring of power systems with the goal of providing some reliable information on the health of the system, especially in the context of large transients, remains a challenging issue. The PMU’s introduced by Phadke (1983) provide accurate voltage and current phasor measurements, since they are synchronized from the common global positioning system (GPS) radio clock. These devices are therefore well suited for the goal of dynamic state estimation. In this paper, we propose a nonlinear state 978-3-902823-32-8/2013 © IFAC

δ˙i = ωi , ω0 Di ωi + (P m − PGi ). ω˙ i = − 2Hi 2Hi i

(1) (2)

Electrical Dynamics of Generator i: 1

For sake of simplicity and without any restriction, we do not consider a two-axis or a more detailed model of the generator, including for instance flux, valve and turbine dynamics.

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IFAC ICPS'13 May 22-24, 2013. Cluj-Napoca, Romania

xdi 0 (xdi − x0di ) 1 0 − Vi cos(δi − θi ), E˙ qi = 0 (Ef di − 0 Eqi Td0i xdi x0di (3) 1 Kai E˙ f di = − (Ef di − Ef d0i ) + (Vref i − Vi ). (4) Tai Tai

measured variables will be expressed as a nonlinear function of the state. When some sensors (PMU for example) are used to measure phase angle and voltage magnitudes, the related component of y will be simply one of the state components. Again, if some current measurements are performed, the related output will be a nonlinear function of the states.

Electrical Equations at Generator bus i, i = 1, ..., N : 0 Eqi Vi PGi = 0 sin(δi − θi ), xdi

(5)

0 QGi = (Eqi Vi cos(δi − θi ) − Vi2 )/x0di , (6) X Vi Vk (Gik sin θik − Bik cos θik ) + PLi , (7) PGi = k∈Ci

X

QGi =

3. NONLINEAR FINITE-HORIZON OPTIMAL STATE OBSERVER DESIGN The main goal is now to design the equivalent to the continuous finite-horizon Kalman filter (Gibbs (2011) for instance) for algebro-differential system (11)-(13), by solving the following optimization problem:

Vi Vk (Gik cos θik + Bik sin θik ) + QLi , (8) min

k∈Ci

v(t),w(t),t∈I

where θik = θi − θk .

s.t. PLj +

Vj Vk (Gjk sin θjk − Bjk cos θik ) = 0,

(9)

k∈Cj

QLj +

X

ZT

{kym (t) − H(ˆ x(t), zˆ(t))k2R−1 + kv(t)k2Q−1 1

0

Electrical Equations at the Non Generator Bus j, j = 1, ..., M − N : X

1 2

Vj Vk (Gjk cos θjk + Bjk sin θjk ) = 0. (10)

k∈Cj

The overall dynamical model of the power system may be expressed as the following algebraic-differential system: x˙ = F (x, z, u),

(11)

0 = G(x, z, l),

(12)

where x denotes the vector of the N generator state vari0 ables (δi , ωi , Eqi , Ef di ), i = 1, ..., N and possibly the state variables of turbines, valves and additional controllers such as PSS or FACTS, z is the 2M vector of the voltage magnitude and phase angle at the M buses. u is the vector of reference control inputs, i.e. the mechanical power and the reference inputs of each controller. l is the vector of load currents at each bus. w can also include interconnection variables if the studied system is part of a larger power system. F denotes the vector field of the differential part of the state-space representation, while G is a nonlinear function with the same dimensions as the ones of vector z. In this section, l(t) is supposed to be known at each time t. If Gkz (t) has full rank, ∀t ∈ I (this assumption is usual in the case of power systems in normal operations), it is well known that nonlinear singular system (11)-(12) has linearized systems with no infinite dynamic modes (index 1 singular systems). By using the implicit function theorem z(t) can be locally expressed as a function φ(x(t), l(t)).

This problem is a nonlinear deterministic formulation (see Bornard (1995) for the LQ case) of the continuous finite-horizon Kalman filter design problem which seeks the minimum variance state estimate of overall states x ˆ and zˆ, where Q1 and Q2 are interpreted as some covariance matrice of zero mean gaussian noises affecting state equations (11) and (12) through matrices E and K respectively, and R is the covariance of a zero mean gaussian noise affecting the measurement output y. M is the covariance matrix of random initial state x(0). With this deterministic interpretation, final states x ˆ(T ) and zˆ(T ) are seeked as solutions to optimization problem (14), on the basis of known output measurements ym (t), inputs u(t) and l(t), ∀t ∈ I 2 . Optimal nonlinear output tracking problem (14) is closely related to the classical optimal trajectory tracking problem, except that the cost penalizes the initial states rather than the terminal states. For that reason, it appears to be very convenient to reverse time by considering the change (t → T − t), and thus turning the optimal output tracking problem into the following nonlinear tracking control problem:

min v ˜(t),w(t),t∈I ˜

1 2

ZT

{k˜ ym (t) − H(˜ x(t), z˜(t))k2R−1 + k˜ v (t)k2Q−1 1

0

s.t. (13)

where H(x, z) defines the measured variables as a function of the states x and z. When some traditional sensors are used to measure active and reactive powers, the related

(14)

0 = G(ˆ x(t), zˆ(t), l(t)) + Kw(t), where I = [0, T ], Q1 , Q2 , R, and M , are some symmetric positive-definite matrices of adequate dimensions, and E and H are also matrices of appropriate dimensions.

In addition to the state-representation (11)-(12), we need to define the measurement vector y: y = H(x, z)

1 +kw(t)k2Q−1 }dt + kˆ x(0)k2M −1 2 2 x ˆ˙ (t) = F (ˆ x(t), zˆ(t), u(t)) + Ev(t),

2

1 2 +kw(t)k ˜ }dt + k˜ x(T )k2M −1 Q−1 2 2 x ˜˙ (t) = −F (˜ x(t), z˜(t), u(t)) − E˜ v (t), 0 = G(˜ x(t), z˜(t), ˜l(t)) + K w(t), ˜

(15)

The a priori elimination of stable variable z (apart from algebraic equation G(x(t), z(t), l(t)) = 0 as a function of z(t) and l(t)) would not lead to the optimal nonlinear state observer as formulated above.

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1 T x(t) − x ˜k (t)) − F2k p˜(t) + (˜  p˜(T ) = M −1 x ˜(T )

where x ˜(t) = x ˆ(T − t), z˜(t) = zˆ(T − t), u ˜(t) = u(T − t), ˜l(t) = l(T − t), y˜(t) = y(T − t), v˜(t) = v(T − t) and w(t) ˜ = w(T ˆ − t). k

k

∇x˜(0) Hx˜(0) = 0 x ˜˙ (t) = −F1k (t) − F2k (t)˜ x(t)) − E˜ v (t)

If we suppose that a state trajectory (x (t), z (t)) is available, inspired by the so-called ”Auxiliary Problem Principle” originally introduced by Cohen (see Cohen (1984) for instance), problem (15) may be transformed into the following auxiliary ”linearized” problem, under appropriate regularity conditions

+F3k w(t) ˜ T

v˜(t) = Q1 E p˜(t)

(18) (19) (20) (21) (22)

T w(t) ˜ = −Q (t){H3k (t)R−1 (˜ ym (t) k k +H1 (t) − H2 (t)˜ x(t)) + F3k (t)˜ p(t)}, −1

min v ˜(t),w(t),t∈I ˜

1 2

ZT

{k˜ ym (t) − H(˜ xk (t), z˜k (t)))

0

s.t.

−Hxk (t)(˜ x(t) − x ˜k (t)) − Hzk (t)(˜ z (t) − z˜k (t))k2R−1 1 + k˜ x(t) − x ˜k (t)k2  1 2 +k˜ v (t)k2Q−1 + kw(t)k x(T )k2M −1 ˜ }dt + k˜ Q−1 1 2 2 x ˜˙ (t) = −F (˜ xk (t), z˜k (t), u(t))

(23) where H denotes the Hamiltonian associated to problem (17), p˜(t) denotes the adjoint state, and Q(t) = T H3k (t)R−1 H3k (t) + Q−1 2 . Furthermore, if state trajectories x ˜(t) = x ˜k (t) and z˜(t) = z˜k (t), ∀t ∈ I, then x ˜(t) = x ˜k (t) and z˜(t) = z˜k (t), ∀t ∈ I are also solutions satisfying the necessary Lagrange stationnary conditions of problem (15). Proof: It is a direct application of Lagrange’s necessary optimality conditions to problem (16) by using the Hamiltonian functional

−Fxk (t)(˜ x(t) − x ˜k (t)) − Fzk (t)(˜ z (t) − z˜k (t)) −E˜ v (t), G(˜ xk (t), z˜k (t), ˜l(t)) + Gkx (t)(˜ x(t) − x ˜k (t))

1 ym (t) + H1k (t) − H2k (t)˜ x(t) H = k˜ 2 1 2 +H3k (t)w(t)k ˜ k˜ x(t) − x ˜k (t)k2 R−1 + 2 1 1 2 + k˜ v (t)k2Q−1 + kw(t)k ˜ Q−1 1 2 2 2 T k k + p˜ (t)(−F1 (t) − F2 (t)˜ x(t)

+Gkz (t)(˜ z (t) − z˜k (t)) + K w(t) ˜ = 0, ⇔ −1

z˜(t) − z˜k (t) = −Gkz (t)

{G(˜ xk (t), z˜k (t), ˜l(t))

+Gkx (t)(˜ x(t) − x ˜k (t)) + K w(t)}, ˜ y˜(t) = H(˜ xk (t), z˜k (t))) + Hxk (t)(˜ x(t) − x ˜k (t)) +Hzk (t)(˜ z (t) − z˜k (t))

(16)

where notations .kx and .kz stand for the Jacobian matrices w.r.t. x and z and evaluated at (˜ xk (t), z˜k (t)) respectively. This problem can be expressed in a more compact form by eliminating z and using new matrices whose expression are not given here explicitely but are straightforward:

min v ˜(t),w(t),t∈I ˜

1 2

ZT

−E˜ v (t) + F3k (t)w(t)). ˜ (24) (20) means that x ˜(0), which is not fixed, has to be optimal. It can be easily shown by setting x ˜(t) = x ˜k (t) and z˜(t) = k z˜ (t), ∀t ∈ I in conditions (18)-(23), that x ˜(t) = x ˜k (t) and k z˜(t) = z˜ (t), ∀t ∈ I are solutions satisfying the necessary Lagrange stationnary conditions of problem (15). The (straightforward) calculations are left to the reader. This result induces the following fixed-point algorithm:

{k˜ ym (t) + H1k (t) − H2k (t)˜ x(t)

(1) At k = 0, start with some initial (xk (t), z k (t)), ∀t ∈ I. (2) At iteration k, solve auxiliary problem (17). Let (xk+1 (t), z k+1 (t)), ∀t ∈ I denotes the solution of this problem. ZT 1 k˜ x(t) − x ˜k (t)kdt is below some (3) Stop if T

0

s.t.

1 2 +H3k (t)w(t)k ˜ x(t) − x ˜k (t)k2 R−1 + k˜  1 2 +k˜ v (t)k2Q−1 + kw(t)k ˜ }dt + k˜ x(T )k2M −1 Q−1 1 2 2 x ˜˙ (t) = −F1k (t) − F2k (t)˜ x(t)

0

−E˜ v (t) + F3k (t)w(t), ˜ (17)

threshold. Otherwise, go back to (2), while setting k ← k + 1.

The interesting feature is solution to auxiliary problem (17) and problem (15) can be obtained explicitely as shown hereafter. Theorem 3.1. The optimal solution to problem (17) must satisfy the following canonical equations defined on I:

The (local) convergence of the algorithm will be obtained for small enough  > 0 at a vicinity of the optimal solution, by using that the gradient w.r.t. xk+1 (t), ∀t ∈ I, of the Hamiltonian associated to problem (17) is at least locally Lipschitz w.r.t. xk+1 (t), ∀t ∈ I,(see Cohen (1984) for similar arguments).

y˜(t) = −H1k (t) + H2k x ˜(t) − H3k w(t) ˜

T

−p˜˙ (t) = −H2k (t)R−1 (˜ ym (t) + H1k (t) −H2k (t)˜ x(t) + H3k (t)w(t)) ˜

Problem (17) is an Affine-Quadratic optimal control problem. An explicit solution to canonical equations (18)-(23) can be obtained by denoting

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IFAC ICPS'13 May 22-24, 2013. Cluj-Napoca, Romania

p˜(t) = P˜ (t)˜ x(t) + g˜(t),

(25)

where P˜ (t) is a symmetric non-negative definite matrix, under the following form T −P˜˙ (t) = −P˜ (t)(F2k (t) − F3k (t)Q−1 (t)H3k (t)R−1 H2k (t)) T −(F2k (t)

− 1 T Id + H2k (t)R−1 H2k (t)  T −P˜ (t)(EQ1 E T + F3k (t)Q−1 (t)F3k (t))P˜ (t)

P˜ (T ) = M

(26) (27)

T

−(F2k (t) − H2k (t)R−1 H3k (t)Q−1 (t)F3k (t))˜ g (t)

+EQ1 E T + F3k (t)Q−1 (t)F3k (t) + T

T

W (t)H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)R−1 H2k (t)

W (0) = M x ˆ˙ (t) = F k (t) + F k (t)ˆ x(t) 1

2 T T kT − W (t)(H2 (t) − H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)) ×R−1 (ym (t) + H1k (t)) (38)

(39) k

zˆ(t) = zˆ (t) −

(˜ ym (t) + H1k (t))) T T T −(H2k (t) − H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)) × R−1 (˜ ym (t) + H1k (t)) (28)

(29)

−1 Gkz (t) {G(ˆ xk (t), zˆk (t), l(t)) k

+Gkx (t)(ˆ x(t) − x ˆ (t))}.

1 T Id + H2k (t)R−1 H2k (t)  T

P˜ (0)˜ x(0) + g˜(0) = 0

(30)

since the Hamiltonian at time t = 0 is given by H(t = 0) = 1 x ˜(0)T P˜ (0)˜ x(0) + x ˜(0)T g˜(0) + α(0). 2 Coming back to positive time, we get T P˙ (t) = −P (t)(F2k (t) − F3k (t)Q−1 (t)H3k (t)R−1 H2k (t)) T

T

−(F2k (t) − H2k (t)R−1 H3k (t)Q−1 (t)F3k (t))P (t) 1 T Id + H2k (t)R−1 H2k (t) 

P (0) = M

T

(32)

T g(t) ˙ = −P (t)(EQ1 E + F3k (t)Q−1 (t)F3k (t))g(t) T T T −(F2k (t) − H2k (t)R−1 H3k (t)Q−1 (t)F3k (t))g(t) T −P (t)(F1k (t) + F3k (t)Q−1 (t)H3k (t)R−1 × (ym (t) + H1k (t))) T T T −(H2k (t) − H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)) × R−1 (ym (t) + H1k (t)) (33) T

g(0) = 0

is a positive-definite matrix, ∀t ∈ I. The overall fixed-point algorithm is thus defined as follows: (1) At k = 0, start with some initial (ˆ xk (t), zˆk (t)), ∀t ∈ I. (2) At iteration k, compute the estimated trajectory of the linearized system using (36)-(40). Let (ˆ xk+1 (t), zˆk+1 (t)), ∀t ∈ I denotes the solution of this problem. ZT 1 kˆ x(t) − x ˆk (t)kdt is below some threshold. (3) Stop if T Otherwise, go back to (2), while setting k ← k + 1.

−H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)R−1 H2k (t) (31) −1

T

−H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)R−1 H2k (t)

0

T

−P (t)(EQ1 E T + F3k (t)Q−1 (t)F3k (t))P (t) T

(40)

Notice that existence of a solution to Riccati equation (36) is ensured provided that the linearized system is at least detectable and for a sufficiently small matrix Q2 , such that

Furthermore condition (20) may be expressed as follows

T

(37)

x ˆ(0) = 0

T

−P˜ (t)(F1k (t) + F3k (t)Q−1 (t)H3k (t)R−1 ×

g˜(T ) = 0

T

(36)

T

T

T

×W (t)

−g˜˙ (t) = −P˜ (t)(EQ1 E T + F3k (t)Q−1 (t)F3k (t))˜ g (t) T

T

+W (t)(F2k (t) − H2k (t)R−1 H3k (t)Q−1 (t)F3k (t)) 1 T −W (t)( Id + H2k (t)R−1 H2k (t))W (t)  T

T T H2k (t)R−1 H3k (t)Q−1 (t)F3k (t))P˜ (t)

T T −H2k (t)R−1 H3k (t)Q−1 (t)H3k (t)R−1 H2k (t) −1

˙ (t) = (F2k (t) − F3k (t)Q−1 (t)H3k T (t)R−1 H2k (t))W (t) W

The number of iterations k can be dramatically reduced when the intial guess is obtained from the solution of a Kalman filter of a linearized model around an equilibrium point for instance. 4. A RECEDING HORIZON APPROACH Receding horizon state estimation is a well-established approach (see Alamir (2007) for instance), which is based on the solution of a finite-horizon optimal estimation problem, whose horizon is defined in a receding or moving manner, i.e. in our case:

(34)

with

1 {ˆ x(t), zˆ(t)} = arg min 2 P (T )x(T ) + g(T ) = 0

(35)

Finally the equations of the equivalent Kalman observer are obtained by using W (t) = P −1 (t) and introducing x ˆ(t) = −W −1 (t)g(t) (see (35)):

110

Zt

{kym (τ ) − H(ˆ x(τ ), zˆ(τ ))k2R−1

t−T

+kv(τ )k2Q−1 + kw(τ )k2Q−1 }dτ 1

1 + kˆ x(0)k2M −1 2

2

(41)

IFAC ICPS'13 May 22-24, 2013. Cluj-Napoca, Romania

x ˆ˙ (t) = F (ˆ x(t), zˆ(t), u(t)) + Ev(t),

s.t.

0 = G(ˆ x(t), zˆ(t), l(t)) + Hw(t),

AREA 1

where T is the so-called measurement horizon. (41) is equivalent to problem (14).

G1

G2

G3

The receding-horizon approach is finally defined as follows: (1) At t = 0, collect initial measurements ym (τ ), ∀τ ∈ [−T, 0]. Compute x ˆ(0) and zˆ(0). (2) At sampling time t = kdt, collect new measurement ym (kdt) and use measurement set {ym (τ ), τ ∈ [kdt − T, kdt]}; then solve (41) to get x ˆ(kdt) and zˆ(kdt), using the previously-defined fixed-point algorithm. (3) Go back to (2), while setting k ← k + 1. 5. A DECENTRALIZED ALGORITHM

G4

G5

I1=I2

G6

In this section, a decentralized version of the previouslydefined receding horizon observer is discussed. A power system consisting in N interconnected subsystems (corresponding for instance to N area of a wide-area power system) can be modeled as follows: x˙ i (t) = Fi (xi (t), zi (t), ui (t)) 0 = Gi (xi (t), zi (t), li (t), Ii (t)), i = 1, ..., N N X

Ci Ii (t) = 0

(42) (43)

AREA 2

(44)

i=1

where (xi , zi ), ui , li , Ii respectively denote the states, input controls, loads and interconnection variables of subsystem i. Ci are some interconnection row vectors whose coefficients are equal either to −1, 0 or +1. For instance for the two-area system described by Fig. 1, we will obtained the following model: x˙ 1 (t) = F1 (x1 (t), z1 (t), u1 (t)) 0 = G1 (x1 (t), z1 (t), l1 (t), I1 (t)) x˙ 2 (t) = F2 (x2 (t), z2 (t), u2 (t))

(45) (46) (47)

0 = G2 (x2 (t), z2 (t), l2 (t), I2 (t)), i = 1, ..., N (48) I1 (t) − I2 (t) = 0

(49)

If the interconnection vectors Ii are not measured, it makes sense to consider a decentralized approach capable of estimating the interconnection variables. Another important goal is to reduce the computational complexity of the overall system monitoring by allowing parallel computations of reduced dimension estimation problems. An extension of the fixed-point algorithm can be proposed by using an augmented Lagrangian formulation and mainly theorem 13 in Cohen (1984), which has been applied with a similar manner in Georges (2006). Due to the lack of place, the algorithm derivation will not be provided in this paper. The cost functional of (14) is simply transform into an augmented Lagrangian by adding ZT

Fig. 1. A two-area system with one interconnection. ZT +

(µ(t) + c

N X

Ci Iik (t))T

i=1

0

N X

Ci Ii (t)dt

(50)

i=1

where µ(t) is the Lagrange multiplier vector associated to the interconnection constraint. c > 0 is the so-called augmentation coefficient introduced for regularization purpose. Finally a sketch of the decentralized algorithm is defined by (1) Initialization (k = 0): Choose x ¯0i (t), Ii0 (t) and µ0 (t), ∀t ∈ [0, T ], i = 1, ..., N . (2) At iteration k: For each subsystem i, i = 1, ..., N : Integrate on the interval [0, T ], each optimal LQ state observer i, i = 1, ..., N to get the estimated trajectory x ˆk+1 (t) and zˆik+1 (t), by using a modified i version of (36)-(40) taking variables Iik (t) and µk (t) into account. Update I(t): Iik+1 (t) = Iik (t) − CiT (µk (t) + c

N X

Ci Iik (t)),

i=1

∀t ∈ [0, T ], i = 1, ..., N (51) (3) Coordination: Update µ: µk+1 (t) = µk (t) + ρ(

N X

Ci Iik+1 (t)), ∀t ∈ [0, T ].(52)

i=1

1 kIi (t) − Iik (t)k2 dt 2

k+1

k

(4) if kµ − µ k < c /ρ, i = 1, ..., N then stop, else k + 1 → k and go to step 2).

0

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IFAC ICPS'13 May 22-24, 2013. Cluj-Napoca, Romania

6. APPLICATION TO A 5-BUS POWER SYSTEM We consider now the 5-bus power system proposed in Bergen (2000), p. 357 (see Fig. 2).

G1

G2

G3

Bus 2

Bus 1

Bus 4

Fig. 3. Generator angle and relative speed estimation (time in seconds).

Bus 3

state observer suitable for both reducing the computational cost and taking advantage of the distributed nature of wide-area power systems was also considered. The application of the decentralized algorithm and some comparisons with other existing approaches such as EKF should be performed in future works.

Bus 5

Fig. 2. A 5-bus power system with 3 generators.

REFERENCES

Without restriction and for simplification purpose, we assume that the series line impedances are zL = rL + jxL = 0.0099 + j0.099 and we neglect the capacitive (shunt) impedances. Bus 1 is the slack bus. Table 1 shows the generator and exciter data. Table 2 shows the other Table 1. Generator and exciter data. Generator 2 3

H 5.0 5.0

D 1.0 0.0

0 Td0 5.0 6.0

xd 1.5 1.4

xq 1.2 1.35

x0d 0.4 0.3

Ka 20 20

Ta 0.05 0.05

bus data. Table 2. Bus data.

Bus 1 2 3 4 5

Specified Voltage 1.0 1.0 1.0

Load

0.2+j0.1 1.7137+j0.5983 1.7355+j0.5496

Shunt

Generation 0.8830 0.2076

j1.0 j1.0

The centralized receding-horizon observer defined by (41) has been tested with the configuration of sensors given in Georges (2012), when the power system reacts to a sudden load drop of 0.8+j0.8 p.u at each buses, without considering frequency control mechanisms. Fig. 3 shows how well the observer behaves in response to this disturbance, in presence of noisy measurements (zero-mean noise with variance equal to 1e−3 ). 7. CONCLUSIONS AND PERSPECTIVES In this paper, a methodology for design an optimal finitehorizon state observer has been proposed. The main advantage of this approach is that a true optimal state estimation is provided unlike using Extended Kalman Filtering (EKF). A distributed version of the here-proposed

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Cohen, G., and Zhu D.L. (1984). Decomposition coordination methods in large scale optimization problems. The nondifferentiable case and the use of augmented Lagrangians. J.B. Cruz (Ed.), Advances in Large Scale Systems, Vol. I, 203–266, JAI Press, Greenwich, Connecticut. Georges, D. (2006). Distributed model predicted control via decomposition-coordination techniques and the use of an augmented lagrangian. IFAC Workshop on NMPC for Fast Systems, 111–116, Grenoble, France. Alamir, M. (2007). Nonlinear Moving Horizon Observers: Theory & Real-Time Implementation. In Nonlinear Observers and Applications, Gildas Besancon (Ed). Lecture Notes on Communication and Information Science. Springer-Verlag-Series. Bornard, G., Celle-Couenne, F., and Gilles, G. (1995). Observability and observers. Nonlinear Systems - T.1, Modeling and Estimatio, Chapman & Hall, London. Bergen, A.R., and Vittal, V. (2000). Power system analysis. Second edition. Prentice Hall, 2000. Georges, D. (2012). Optimal Design of a PMU-Based Monitoring Architecture for Power Systems. Proceedings of the IFAC Workshop on Power Plants and Power Systems Control, Toulouse, France, September 2-5, 2013, submitted to Control Engineering Practice - Special Issue on Power Systems Control. Gibbs, B. (2011). Advanced Kalman filtering, Leastsquares and modeling. J. Wiley and Sons, Inc. 2011. Huang, Z., Schneider, K., and Nieplocha, J. (2007). Feasibility studies of applying Kalman filter techniques to power system dynamic state estimation. Proceedings of the International Power Engineering Conference 2007, 2007. Ilic, M., and Zaborszky, J. (2000). Dynamics and control of large electric power systems. John Wiley and Sons, Inc., 2000. Phadke, A., Thorp, J., and Adamiak, M. (1983). A new measurement technique for tracking voltage phasors, local system frequency, and rate of change of frequency. IEEE Trans. on Power Apparatus and Systems, vol. 102, no. 5, pp. 1025-1038, may 1983.