A global linearization approach to control and state ...

A global linearization approach to control and state estimation of a VSC-HVDC system ∗

Gerasimos Rigatos∗, Pierluigi Siano†, Patrice Wira§ and Carlo Cecati‡ Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece, email: [email protected] † Department of Industrial Engineering, University of Salerno, 84084, Fisciano, Italy, email: [email protected] § Laboratoire MIPS, Universit´e de Haute Alsace, 68093, Mulhouse, France, email: [email protected] ‡ Department of Informatics, University of L’ Aquila, 67100 , L’ Aquila, Italy, email: [email protected]

Abstract—The paper proposes a new control method for the VSC-HVDC system, that is for an AC to DC voltage source converter connected to the electricity grid through a high voltage DC transmission line terminating at an inverter. By proving that the VSC-HVDC system is a differentially Àat one, its transformation to the linear canonical form becomes possible. This is a global input-output linearization procedure that results into an equivalent dynamic model of the VSC-HVDC system for which the design of a state feedback controller becomes possible. Furthermore, to estimate and compensate for modeling uncertainty terms and perturbation inputs exerted on the VSCHVDC model it is proposed to include in the control loop a disturbance observer that is based on the Derivative-free nonlinear Kalman Filter. This ¿ltering method makes use of the linearized equivalent model of the VSC-HVDC system and of an inverse transformation which is based on differential Àatness theory and which ¿nally provides estimates of the state variables of the initial nonlinear model. The performance of the proposed VSC-HVDC control scheme is tested through simulation experiments.

I. I NTRODUCTION The integration of renewable energy sources in the electricity grid has imposed in several cases the development of methods for power transmission through AC to DC and DC to AC conversion, and with the intervention of high voltage DC (HVDC) lines [1-7]. The present paper proposes a global linearization approach, through differential Àatness theory, for the problem of nonlinear feedback control and state estimation of a Voltage Source Converter - HVDC line system. First, the dynamic model of this electric power transmission system is considered. This comprises the state-space equations of an AC to DC voltage source converter, that is connected to the rest of the grid through an HVDC line terminating at an inverter. This dynamic model describes the stages of transmission of electric power from a power generator to the rest of the grid, through the intervention of an HVDC line. Next, it is shown that this model is a differentially Àat one, which means that all its state variables and its control inputs can be expressed as differential functions of one speci¿c state variable of the model, which the so-called Àat output. By proving that differential Àatness properties hold for the electric power transmission system it is also shown that the initial nonlinear VSC-HVDC state-space model can be written into the linear canonical (Brunovsky) form. This is a global (exact) linearization procedure, that is valid throughout the entire state-space of the VSC-HVDC system and which is

978-1-4799-8704-7/15/$31.00 ©2015 IEEE

454

not based on approximate truncation of nonlinearities. For the latter description of the system’s dynamics the design of a feedback controller becomes possible, thus permitting to make the voltage output at the inverter’s side track any reference setpoint [8-10]. Another problem that has to be dealt with in the design of the feedback control scheme is that the control loop is subject to modelling uncertainty and external disturbances. To estimate and compensate for these perturbation terms it is proposed to include in the control loop a disturbance observer that is based on the Derivative-free nonlinear Kalman Filter. Actually, the Derivative-free nonlinear Kalman Filter consists of the Kalman Filter recursion applied to the linearized equivalent model of the VSC-HVDC and of an inverse transformation based again on differential Àatness theory which enables to obtain estimates for the state variables of the initial nonlinear model. The inclusion of an additional term in the control input, based on the identi¿cation of these exogenous or endogenous perturbations, makes possible their compensation and improves the robustness of the control loop [8-9]. The performance of the considered control scheme for the VSCHVDC system is evaluated through simulation experiments. II. M ODELLING OF

THE

VSC-HVDC

TRANSMISSION

SYSTEM

By connecting the three-phase voltage source converter, which is found at the generator’s side with the three-phase inverter which is found at the load’s side through an HVDC line (see Fig. 1) the high voltage DC transmission system is formed. Moreover, by applying Kirchhoff’s voltage and current laws the dynamic of the HVDC line is obtained. Thus at the converter’s side it holds C1 V˙ dc1 = − Z1L Vdc1 +

1 ZL Vdc2

+ 34 id u1 + 34 iq u2

(1)

where ZL is the impedance of the transmission line (Fig. 2). Here ZL is considered to consist of only a resistance part (if inductance part is also included in the transmission line’s dynamics then a higher order state-space model for the VSCHVDC system will be obtained). It is also possible to model the transmission line’s dynamics as a distributed parameters system, however this latter assumption will not be followed in this manuscript. At the inverter’s side it holds

C2 V˙ dc2 =

1 ZL Vdc1

−

1 ZL Vdc2

(2)

which can be also written in matrix form ⎞ ⎛ ⎞ ⎛ vd −R x˙ 1 L x1 + ωx2 + L vq ⎟ ⎜x˙ 2 ⎟ ⎜ −ωx1 − R ⎟ L x2 + L ⎜ ⎟=⎜ 1 1 + ⎜ ⎝x˙ 3 ⎠ ⎝ ZL Cdc x4 − ZL Cdc x3 ⎟ ⎠ 2 2 1 1 x˙ 4 − ZL Cdc x4 + ZL Cdc x3 1

⎛

1

(5)

⎞

x4 − 2L ⎜ 0 +⎜ ⎝ 0 3 4Cdc x1

0 x4 ⎟  − 2L ⎟ u1 0 ⎠ u2 3 4Cdc x2

1

1

while, as in the case of the stand-alone VSC the output of the voltage source converter is taken to be  h= Fig. 1. Transmission system comprising a voltage source converter (VSC), a high voltage DC line (HVDC) and an inverter

h1 h2

=

  3 ec L(x21 + x22 ) + = 4 iq x2

Cdc2 2

x23



Consequently, the VSC-HVDC system is written again in the state-space form x˙ = f (x) + G(x)u y = h(x)

(7)

where f (x), G(x) and h(x) are given by ⎛ ⎞ vd −R L x1 + ωx2 + L ⎜ −ωx1 − R x2 + vq ⎟ ⎜ ⎟ L L 1 1 f (x) = ⎜ ⎟ ⎝ ZL Cdc2 x4 − ZL Cdc2 x3 ⎠ 1 1 − ZL Cdc x4 + ZL Cdc x3 1

⎛

1

⎞

x4 − 2L

0 x4 ⎟ − 2L ⎟ 0 ⎠ 3 3 4Cdc1 x1 4Cdc1 x2 3 L(x21 + x22 ) + C2dc x23 4 h(x) = x3 ⎜ G(x) = ⎜ ⎝

Fig. 2.

Components of the high voltage DC line (HVDC)

1 Li˙ d = −Rid + Lωiq + vd − dc 2 u1 V dc Li˙ q = −Lωid − Riq + vq − 2 1 u2 C2 V˙ dc2 = Z1L Vdc1 − Z1L Vdc2 ˙ C1 Vdc1 = − Z1L Vdc1 + Z1L Vdc2 + 34 id u1 + 34 iq u2

V

(3)

By de¿ning the new state vector of the VSC-HVDC system as x = [x1 .x2 , x3 , x4 ]T = [id , iq , Vdc2 , Vdc1 ] one obtains the state-space description Lx˙ 1 = −Rx1 + Lωx2 + vd − x24 u1 Lx˙ 2 = −Lωx1 − Rx2 + vq − x24 u2 C2 x˙ 3 = Z1L x4 − Z1L x3 C1 x˙ 4 = − Z1L x4 + Z1L x3 + 34 x1 u1 + 34 x2 u2

THE

(9)

VSC-HVDC

SYSTEM

Next, it will be shown that the dynamic model of the voltage source converter is a differentially Àat one, i.e. it holds that all state variables and its control inputs can be written as functions of the Àat outputs and their derivatives [16-20]. Moreover, it will be shown that by expressing the elements of the state vector as functions of the Àat outputs and their derivatives one obtains a transformation of the VSC-HVDC model into the linear canonical (Brunovsky) form. The dynamic model of the joint VSC-HVDC dynamics has been de¿ned as x˙ 1 = − R L x1 + ωx2 +

vd L

−

x4 2L u1

(10)

x˙ 2 = −ωx1 −

vq L

−

x4 2L u2

(11)

x3

(12)

(4) x˙ 3 =

455

(8)

0 0

III. D IFFERENTIAL FLATNESS OF

The state-space model of the VSC-HVDC transmission line, can be obtained using established models about the dynamics of the voltage source converter [11-15]:

(6)

1 ZL Cdc2

R L x2

+

x4 −

1 ZL Cdc2

be concluded that u2 = fg (yf , y˙ f ). x˙ 4 =

− ZL C1 dc 1

x4 +

1 ZL Cdc1

3x1 4Cdc1

x3 +

u1 +

3x2 4Cdc1

u2

(13) The Àat output of the system is taken to be yf = [yf1 , yf2 ]T , where yf1 =

3L 2 4 (x1

+ x22 ) +

Cdc1 2

x24

(14)

Consequently, all state variables and the control inputs of the VSC-HVDC model can be written as differential functions of the Àat output yf = [yf1 , yf2 ]T . Thus the VSC-HVDC system is a differentially Àat one. IV. F LATNESS - BASED CONTROL

VSC-HVDC

OF THE

SYSTEM

(15)

Next, a Àatness-based controller will be designed for the VSC-HVDC system. Using Eq. (16) and deriving y˙ f1 once more with respect to time one gets

˙ 1 + 3L ˙ 2 + Cdc1 x4 x˙ 4 ⇒ y˙ f1 = 3L 2 x1 x 2 x2 x vd x4 3L x + ωx y˙ f1 = 2 x1 [− R 2 + L − 2L u1 ]+ L 1 vq x4 3L R + 2 x2 [− L x2 − ωx1 + L − 2L u2 ]+ 3x1 3x2 +Cdc1 x4 [− ZL C1 dc x4 + ZL C1 dc x3 + 4C u1 + 4C u2 ] dc1 dc1 1 1 (16) Using that vd is constant (d-axis component of the grid voltage) and vq = 0, and after intermediate computations one obtains

y¨f1 = − 3R ˙ 1 + 2x2 x˙ 2 ) + 3 x˙21 vd − 2xZ4Lx˙ 4 + x˙Z3 xL4 + 2 (2x1 x x3 x˙ 4 vd x4 yf1 = (−3Rx1 + 3v2d )[− R ZL ⇒¨ L x1 + ωx2 + L − 2L u1 ] − vq x4 x4 x4 R 3Rx2 [−ωx1 + L x2 + L − 2L u2 ] + ZL ( ZL Cdc − ZLxC3dc ) + 2 2 x3 x4 x3 3x1 3x2 (− 2x+4 ZL + ZL )[− ZL Cdc1 + ZL Cdc1 + 4Cdc1 u1 + 4Cdc1 u2 ], which after intermediate computations gives

yf2 = x3 By deriving Eq. (14) with respect to time one obtains

L

2 2 y˙ f1 = − 3R 2 (x1 + x2 ) +

3x1 2 vd

−

x24 ZL

+

x3 x4 ZL

(17)

From Eq. (12) one has x4 = (ZL Cdc2 )[x˙ 3 + 1 = (ZL Cdc2 )[y˙ f2 + ZL C1 dc yf2 ]⇒x4 = ZL Cdc2 x3 ]⇒x4 2 fa (yf , y˙ f ), which means that state variable x4 is a function of the Àat output and of its derivatives. Moreover, from Eq. (14) one gets (x21

C 1 2 4 + x22 ) = 3L [yf1 − dc 2 x4 ]⇒ 2 2 (x1 + x2 ) = fb (yf , y˙ f )

(18)

2 3R 2 2 3vd {y˙ f1 + 2 (x1 +x2 )+ 2 yf2 x4 x4 x3 x4 2 3R ZL }⇒x1 = 3vd {y˙ f1 + 2 fb (yf , y˙ f ) + ZL − ZL }. x4 = fa (yf , y˙ f ) in the previous equation about x1

− Using one ¿nally gets that state variable x1 is a function of the Àat output and its derivatives x1 = fc (yf , y˙ f ).

Moreover, from Eq. (18) it holds (x21 + x22 ) = fb (yf , y˙ f ), while as shown above x1 = fc (yf , y˙ f ). Thus by solving with respect to x2 one gets x22 = fb (yf , y˙ f ) − fc2 (yf , y˙ f )⇒x2 =

|fb (yf , y˙ f ) − fc2 (yf .y˙ f )|⇒x2 = fd (yf , y˙ f ), which means that state variable x2 is also a function of the Àat output and of its derivatives. Next by solving Eq. (10) with respect to the control input u1 and Eq. (11) with respect to the control input u2 one obtains u1 =

2L ˙1 x4 {x

+

u2 =

2L ˙2 x4 {x

+ ωx1 +

R L x1

− ωx2 − R L x2

−

vd L}

(19)

vq L}

(20)

Using that x1 = fc (yf , y˙ f ) and x2 = fd (yf , y˙ f ) in Eq. (19) it can be concluded that u1 = fe (yf , y˙ f ). Similarly, using x1 = fc (yf , y˙ f ) and x2 = fd (yf , y˙ f ) in Eq. (20) it can

456

1

1

L

L

x2

2

2

L

(21)

− Zx2 3Cxdc4 + − Z 2 C3dc ]+ 1

L

L

1

1 x4 d x4 4 3 − 3v4L − 4Z6xL1Cxdc + 4Z3xL1Cxdc ]u1 + +[ 3Rx 2L 1 1 3Rx2 x4 6x2 x4 3x2 x3 +[ 2L − 4ZL Cdc + 4ZL Cdc ]u2 1 1

Eq. (21) can be also written in the concise form y¨f1 = L2f h1 (x) + Lg1 Lf h1 (x)u1 + Lg2 Lf h1 (x)u2

(22)

Additionally, from the second component of the Àat outputs vector given in Eq. (15) one gets

Next, using Eq. (17) one has x1 =

x24 ZL

3v 2

2

6Rx1 vd 3ωvd x2 2 2 + 4d + y¨f1 = [ 3R L2 (x1 + x2 ) − L 2 + 2 2x4 x4 2x1 x4 x3 x4 + Z 2 Cdc − Z 2 Cdc + Z 2 Cdc − Z 2 Cdc −

yf2 = x2 ⇒y˙ f3 = x˙ 3 ⇒ y˙ f2 = ZL C1 dc x4 − ZL C1 dc x3 2

(23)

2

By deriving once more the above relation with respect to time one gets y¨f2 =

1 ZL Cdc2

x˙ 4 −

1 ZL Cdc2

x˙ 3

(24)

Next, by substituting Eq. (12) and Eq. (13) into Eq. (24) one arrives at 3x1 3x2 1 1 1 ZL Cdc2 [− ZL Cdc1 x4 + ZL Cdc1 x3 + 4Cdc1 u1 + 4Cdc1 u2 ]− 1 1 1 ZL Cdc2 [ ZL Cdc2 x4 − ZL Cdc2 x3 ], which after intermediate op-

y¨f2 =

erations gives

x4 x3 y¨f2 = [− Z 2 Cdc + Z 2 Cdc − Z 2 Cx4dc 2 + C C L L L 1 dc2 1 dc2 2 1 2 u1 + 4ZL C3x u2 + 4ZL C3x dc Cdc dc Cdc 1

2

1

x3 2C 2 ]+ ZL dc2

2

(25)

which can be also written in the more concise form y¨f2 = L2f h2 (x) + Lg1 Lf h2 (x)u1 + Lg2 Lf h2 (x)u2

(26)

Next, Eq. (22) and Eq. (21) are written in matrix form

⎛ ⎞ z1 ⎟ 0 ⎜ ⎜z 2 ⎟ ⎝ 0 z3 ⎠ z4

 m  z1 1 0 = 0 0 z3m

0  (34) u1 1 u2 (27) V. C OMPENSATION OF DISTURBANCES USING THE By de¿ning the new control inputs v1 = L2f h1 (x) + D ERIVATIVE - FREE NONLINEAR K ALMAN F ILTER 2 Lg1 Lf h1 (x)u1 + Lg2 Lf h1 (x)u2 and v2 = Lf h2 (x) + Lg1 Lf h2 (x)u1 + Lg2 Lf h2 (x)u2 one arrives at the linearized Next, it is assumed that the VSC-HVDC model is subjected and decoupled description of the dynamics of the VSC-HVDC to additive input disturbances, that is model y¨f1 = L2f h1 (x) + Lg1 Lf h1 (x)u1 + Lg2 Lf h1 (x)u2 + d˜1 y¨f1 = v1 (28) y¨f2 = L2f h2 (x) + Lg1 Lf h2 (x)u1 + Lg2 Lf h2 (x)u2 + d˜2 y¨f2 = v2 (35) ˜ or equivalently y ¨ = v + d and y ¨ = v + d˜2 . f1 1 1 f2 2 Moreover, by de¿ning the following vectors and matrices Moreover, it is considered that the disturbance inputs can be described by their n-th order derivative. This is because each  2  Lf h1 (x) yf1 ˜ function can be described either by its mathematical equations f= y˜f = yf2 L2f h2 (x) or equivalently by its derivatives and the associated initial conditions. However, in the studied case there will be ¿nally no   v1 Lg1 Lf h1 (x) Lg2 Lf h1 (x) use of initial conditions because the disturbance function will ˜ v˜ = M= Lg1 Lf h2 (x) Lg2 Lf h2 (x) v2 be estimated by the Kalman Filter which is not dependent on (29) the processing of initial conditions. Without loss of generality one obtains the following matrix-form description of the it is assumed that the additive disturbance terms d˜ , i = 1, 2 i system’s dynamics are described by the signal’s second order derivative, that is    2 Lf h1 (x) Lg1 Lf h1 (x) y¨f1 + = Lg1 Lf h2 (x) y¨f1 L2f h2 (x)

Lg2 Lf h1 (x) Lg2 Lf h2 (x)

˜u ˜ −1 [˜ y¨˜f = f˜ + M ˜⇒˜ u=M v − f]

where u ˜ is the control input that is actually exerted on the VSC-HVDC system, while the transformed control input v˜ is given by v˜ = y¨˜fd − Kd (y˜˙ f − y˜˙ fd ) − Kp (˜ yf − y˜fd )

(31)

For the linearized model of the VSC-HVDC dynamics one can also obtain a description in the linear canonical (Brunovsky) form. The following state variables are de¿ned: z1 = yf1 , z2 = y˙ f1 , z3 = yf2 and z4 = y˙ f2 . This results in the following state-space description z˙1 = z2 z˙2 = L2f h1 (x) + Lg1 Lf h1 (x)u1 + Lg2 Lf h1 (x)u2 z˙3 = z4 z˙4 = L2f h2 (x) + Lg1 Lf h2 (x)u1 + Lg2 Lf h2 (x)u2

(32)

The previous state-space description results also in the linear canonical (Brunovsky) form: ⎛ ⎞ ⎛ z˙1 0 ⎜z˙2 ⎟ ⎜0 ⎜ ⎟=⎜ ⎝z˙3 ⎠ ⎝0 0 z˙4

1 0 0 0

0 0 0 0

⎞⎛ ⎞ ⎛ z1 0 0 ⎜z2 ⎟ ⎜1 0⎟ ⎟⎜ ⎟ + ⎜ 1⎠ ⎝z3 ⎠ ⎝0 0 0 z4

⎞

0  0⎟ ⎟ v1 0⎠ v2 1

¨ d˜1 = fd1

(30)

(33)

and since the measurable variables for this system are taken to be z1 = yf1 and z3 = yf2 the associated measurement equation becomes

457

¨ d˜2 = fd2

(36)

Then by de¿ning the new state variables z1 = yf1 , z2 = y˙ f1 , ˙ ˙ z3 = yf2 , z4 = y˙ f2 , z5 = d˜1 , z6 = d˜1 , z7 = d˜2 , z8 = z6 = d˜2 one obtains the following extended state-space description of the system ⎞⎛ ⎞ ⎛ z1 0 0 ⎜z2 ⎟ ⎜1 0⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 0⎟ ⎟ ⎜z3 ⎟ ⎜0 ⎜ ⎟ ⎜ 0⎟ ⎟ ⎜z4 ⎟ + ⎜0 ⎜ ⎟ ⎜ 0⎟ ⎟ ⎜z5 ⎟ ⎜0 ⎜ ⎟ ⎜ 0⎟ ⎟ ⎜z6 ⎟ ⎜0 1 ⎠ ⎝z7 ⎠ ⎝0 0 0 z8

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ v1 0⎟ ⎟ v2 0⎟ ⎟ 0⎠ 0 (37) while by denoting the extended state vector ze = [z1 , z2 , z3 , z4 , z5 , z6 , z7 , z8 ]T , the measurement equation for the extended system becomes ⎛ ⎞ ⎛ 0 z˙1 ⎜z˙2 ⎟ ⎜0 ⎜ ⎟ ⎜ ⎜z˙3 ⎟ ⎜0 ⎜ ⎟ ⎜ ⎜z˙4 ⎟ ⎜0 ⎜ ⎟=⎜ ⎜z˙5 ⎟ ⎜0 ⎜ ⎟ ⎜ ⎜z˙6 ⎟ ⎜0 ⎜ ⎟ ⎜ ⎝z˙7 ⎠ ⎝0 0 z˙8

1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

 m  z1 1 = 0 z3m

0 0 1 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 1

0 0 0 1 0 0 0 0

0 0 0 0

0 0 0 0

0 z 0 e

(38)

According to the above the extended state-space model of the VSC-HVDC system is written in the linear matrix from z˙e = Aze + Bv zem = Cze

(39)

For the extended state-space model of the VSC-HVDC system it is possible to perform simultaneously states and

2

q

1.4

1

0

dc1

p.u.

2

p.u.

2

V

1.5

0

(41)

10

1

0

5 time (sec)

10

(a) 3 2 1

0

2

4

6

8

10

6

8

10

6

8

10

time (sec) 5

where Q(k) and R(k) stand for process and measurement noise covariance matrices. After estimating the disturbance ˆ ˆ inputs , that is zˆ5 = d˜1 and zˆ7 = d˜2 the control input that is applied to VSC-HVDC system is modi¿ed as follows: v1∗ = v1 − zˆ5 and v2∗ = v2 − zˆ7 . Thus, with respect to Eq. (35), the closed-loop dynamics becomes y¨f1 = v1 − zˆ5 + d˜1 and y¨f2 = v2 − zˆ7 + d˜2 . VI. S IMULATION

5 time (sec)

1.5

10

(40) Tm1 í Tm1íest p.u.

P (k + 1) = + Q(k) zˆ− (k + 1) = Ad zˆ(k) + Bd v(k)

5 time (sec)

0

u1 p.u.

Ad P (k)ATd

1

10

0

0

2

4 time (sec)

5 u2 p.u.

−

5 time (sec)

2.5

−

time update:

2

2.5

1

K(k) = P (k)CdT [Cd P (k)CdT + R(k)]−1 zˆe (k) = zˆe− (k) + K(k)[zem (k) − Cd zˆe− (k)] P (k) = P − (k) − K(k)Cd P − (k)

3

i p.u.

d

i p.u.

1.6

1.2

V

measurement update:

4

1.8

dc2

disturbances estimation using the Derivative-free nonlinear Kalman Filter. The latter consists of the Kalman Filter recursion on the linearized equivalent model of the VSCHVDC system and of an inverse transformation, based on the results of section III which allows to obtain estimates of the state variables of the initial nonlinear model described in Eq. (10) to Eq. (13). The ¿lter’s algorithm uses the discrete time equivalents of the previously de¿ned matrices A, B and C, which are denoted as Ad , Bd and Cd respectively. The proposed ¿ltering method consists of a measurement update and a time update stage [21-23]:

0

0

2

4 time (sec)

TESTS

The performance of the control loop for the VSC-HVDC system was tested through simulation experiments. The associated results are presented in Fig. 3 to Fig. 4. Actually, in Fig. 3(a) to Fig. 4(a) the convergence of the real state variables of the system xi , i = 1, · · · , 4 to their reference setpoints is presented. Additionally, in Fig. 3(b) to Fig. 4(b) the estimated disturbance inputs of the VSC-HVDC model are plotted. Moreover, diagrams of the variation of the system’s control inputs are provided. In the aforementioned diagrams, the per unit (p.u.) measurement system has been used. It can be observed that through the proposed control scheme, the state variables of the VSC-HVDC system converge fast to the reference setpoints. Furthermore, the performance of the control loop is improved against external disturbances by including in it the Derivative-free nonlinear Kalman Filter. By functioning as a disturbance observer the ¿lter enables to estimate in real-time additive disturbance inputs that affect the VSC-HVDC model and subsequently to compensate for them. VII. C ONCLUSIONS A new method of nonlinear control of high voltage direct current (HVDC) links in the electric power system has been developed using differential Àatness theory. It has been proven that the model of the joint dynamics between the AC to DC

458

(b) Fig. 3. Convergence of the state vector elements xi , i = 1, · · · , 4 to the associated reference setpoints No 3 for the initial nonlinear model of the VSC-HVDC system, (b) estimation of disturbance input d˜ on the VSC-HVDC model and variation of control inputs ui , i = 1, 2.

voltage source converter and the DC transmission line that constitute the VSC-HVDC link is a differentially Àat one. This means that all state variables and the control inputs of this model (IGBT currents at the converter) can be ¿nally written as differential functions of an algebraic variable which is the Àat output. The differential Àatness property of the VSCHVDC model enables its transformation to the linear canonical (Brunovsky) form. This is a global linearization approach that remains valid for the complete state space of the VSC-HVDC system and which does not introduce any numerical errors or approximations. For the latter description of the system’s dynamics the design of a state feedback controller becomes easy. Another problem that had to be dealt with in the design of the feedback control loop was that of estimation and compensation of the disturbances that affected the VSC-HVDC model. To this end a disturbance observer has been included in the control scheme. The observer was based on the Derivative-

2.5

2.5 i p.u.

3

i p.u.

3

2

q

d

2 1.5 1

1.5 0

5 time (sec)

1

10

p.u.

3 2.5

p.u.

3 2.5

5 time (sec)

10

0

5 time (sec)

10

dc1

2

V

V

dc2

2

0

1.5 1

1.5 0

5 time (sec)

10

1

Tm1 í Tm1íest p.u.

(a) 3 2 1

0

2

4

6

8

10

6

8

10

6

8

10

time (sec)

u1 p.u.

5

0

0

2

4 time (sec)

u2 p.u.

5

0

0

2

4 time (sec)

(b) Fig. 4. Convergence of the state vector elements xi , i = 1, · · · , 4 to the associated reference setpoints No 4 for the initial nonlinear model of the VSC-HVDC system, (b) estimation of disturbance input d˜ on the VSC-HVDC model and variation of control inputs ui , i = 1, 2.

free nonlinear Kalman Filter, that is on the Kalman Filter recursion that was applied on the linear equivalent of the VSCHVDC model. Moreover, the ¿lter it comprised an inverse transformation that was based on differential Àatness theory and which enabled to obtain estimates for the state variables of the initial nonlinear VSC-HVDC model. The ef¿ciency of the proposed state estimation-based control scheme for VSCHVDC links was tested through simulation experiments. The proposed method is a reliable solution to this type of control problems appearing in the smart grid and in the connection of renewable energy sources with the electricity network. R EFERENCES [1] L. Arioua, B. Marinescu and E. Monmasson, Control of high voltage direct current links with overall large-scale grid objectives, IET Generation, Transmission and Distribution, vol. 8, no. 5, pp. 945-956, 2014. [2] L. Arioua and B. Marinescu, Large-scale implementation and validation of multivariable control with grid objectives of an HVDC link, 18th Power Systems Corporations Conference, Wroclau, Poland, Aug. 2014.

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