Linearized equations of voltage, current and power at the AC terminals of FACTS Devices: general resultsFACTS Devices: general results

Small-signal models of synchronous generators, FACTS devices and the power system

4.3 Small-signal models of FACTS Devices

4.3.1 Linearized equations of voltage, current and power at the AC terminals of FACTS Devices: general resultsFACTS Devices: general results

As shown in Figure 4.12, a general FACTS device has multiple points of connection to the AC network. In the following the connection of just one terminal, k, of the FACTS device to its network bus is considered. The results are applicable to the connection of the other terminals of the device to their respective network buses. As noted in Section 4.2.3.5 the re- sults in this section are also applicable to calculating the quantities at the stator terminals of generator models.

Normally the principal base quantities for the ^kth terminal of the device and the network bus to which the terminal is connected are the RMS line-to-line voltage, the three-phase appar- ent power and the fundamental frequency as listed in Table 4.13. These base quantities are specified by the user (as indicated by the subscript ‘usb’). The base value of time, for both the device and network, , is chosen to be one-second.

1K_V^{ 1}

K_I^{ 1}

i

˜^RI d1

 

v˜^RI d1

 

i

˜^RI n1

 

v

˜^RI n1

 

1K_V^{ k}

K_I^{ k}

i˜^RI d k

 

v˜^RI d k

 

i

˜^RI n k

 

v˜^RI n k

 

1K_V^{ nt}

K_I^{ nt}

i

˜^RI d n _t

 

v

˜^RI d n _t

 

i˜^RI n n _t

 

v˜^RI n n _t

 

NETWORK

Bus to which terminal 1 is connected. Base quantities SB⁽¹⁾ Terminal 1.

Base quanti- ties MB⁽¹⁾

Bus to which terminal k is connected. Base quantities SB^(k)

Bus to which terminal n_t is connected. Base quantities SB^(nt) Terminal k.

Base quanti- ties MB^(k)

Terminal n_t. Base quanti- ties MB^(nt)

Device equations and parameters on device base quantities, MB(k) , k = 1,...,nt.

u

˜

Device inputs, SI units or pu on device

base quantities

y

˜

Device outputs, SI units or pu on device

base quantities

Device initial steady- state values, SI units or pu on device base

quantities Initial steady-state values of device termi- nal quantities from power flow solution nkknknk , ,, , VPQk1n=0000t

t_b

160 Generators, FACTS devices & system models Table 4.13 Principal base quantities for the k^th device terminal and the network bus to

which it is connected.

The set of base quantities for the k^th terminal of the device is denoted as and the cor- responding set of base quantities for the network bus to which the ^kth terminal is connected is denoted as .

The per-unit values of voltage, real and reactive power and current in the base system of the of the k^th terminal of the device (denoted by the superscript (d,k)) are related to the corre- sponding quantities in the network base system (denoted by the superscript (n,k)) as follows:

in which , (4.189)

in which , (4.190)

in which . (4.191)

It is assumed that the initial steady-state values , , and at the bus to which the ^kth terminal of the device is connected are obtained from the power flow solution.

Base Quantity

Units Description

Device Terminal

Network Bus

kV (RMS, line-to-line)

The base value of the fundamental-frequency positive- phase sequence component of the RMS line-to-line volt- age of the k^th terminal of the device. Normally

.

MVA

The base value of the three-phase apparent power of the k^th terminal of the device. Normally, is related to the device rating. The value of is normally the apparent power base of the power flow analysis.

Hz

The base value of fundamental frequency of the k^th termi- nal of the device. Normally . It is assumed that is the nominal operating frequency of the net- work.

V_usb^{d k } V_usb^{n k }

V_usb^{d k } = V_usb^{n k }

S_usb^{d k } S_usb^{n k } S_usb^{d k } S_usb^{n k }

f_usb^{d k } f_usb^{n k } f_usb^{d k } = f_usb^{n k } f_usb^{n k }

MB^{ k}

SB^{ k}

Vˆ^{n k } = K_V^{ k} Vˆ^{d k } K_V^{ k} = V_usb^{d k }V_usb^{n k }

P^{n k }+jQ^{n k } = K_S^{ k} P^{d k }+jQ^{d k } K_S^{ k} = S_usb^{d k }S_usb^{n k } Iˆ^{n k } = K_IIˆ^{d k } K_I^{ k} = K_S^{ k} K_V^{ k}

V₀^{n k } ^{ k} P₀^{n k } Q₀^{n k }

Sec. 4.3 Linearized equations of quantities at AC terminals of FACTS Devices 161 For the purposes of analysis the voltages and currents are represented by the real and imag- inary components of their phasor quantities in the synchronously rotating network frame of reference. Thus, in per unit on the base quantities of the k^th terminal of the device,

, (4.192) in which

, , and (4.193)

, and . (4.194)

The initial steady-state value of the voltage magnitude is per-unit on

; the initial value is then obtained by substitution of and in (4.194). By substituting superscript d with n in equations (4.192) to (4.194) the voltage magnitude and RI components are obtained in per-unit of the base voltage of the network bus to which the k^th terminal is connected.

The linearized forms of the voltage magnitude and angle of the ^kth terminal of the device are written, respectively, as

(pu), and (4.195) , (rad) (4.196) in which the superscript (d,k) is omitted from the voltage quantities.

The terminal current phasor is expressed in terms of the terminal voltage and the real and reactive power injected by the device from its k^th terminal into the network as follows. All quantities are in per-unit on the base quantities of either the k^th terminal of the device (d,k) or those of the network bus to which the k^th terminal is connected (n,k).

(pu) (4.197)

Vˆ^{d k } v_R^{d k }+jv_I^{d k } V^{d k }e^jk

 

V^{d k }cos^{ k} +jsin^{ k} 

= = =

V^{d k } = v_R^{d k }²+v_I^{d k }² ^{ k} = atan2v_I^{d k }v_R^{d k } v_R^{d k } = V^{d k }cos^{ k}  v_I^{d k } = V^{d k }sin^{ k}  v

˜^RI

d k

 

v_R^{d k } v_I^{d k }

= T

V₀^{d k } = V₀^{n k }K_V^{ k} V_usb^{d k } v

˜^RI0

d k

  V^{d k } = V₀^{d k }

^{ k} = ₀^{ k}

V^{d k } ^{ k}

V v_R

0V₀

 v_R v_I

0V₀

 v_I

+ 1V₀v

˜^RI0

T v

˜^RI

= =

 v_I

0V₀²

 v_R

– v_R

0V₀²

 v_I

+ 1V₀² v_I

– 0 v_R

0 v

˜^RI

= =

Iˆ i_R+ji_I P jQ+ Ve^j

---

 

 

= =

1 V---

  Pcos+Qsin+j P sin–Qcos

= 1 V² ---

  Pv_R+Qv_I+j Pv _I–Qv_R

=

162 Generators, FACTS devices & system models Equating the real and imaginary components of the preceding equation results in the follow- ing matrix relationship:

. (4.198)

The initial values of the current components , in the network per-unit system, are found by substituting the initial values , , and obtained from the power flow solution into equation (4.198) and then in the per-unit system of the device ter-

minal by setting .

The perturbations in the current components about their initial values are obtained by line- arizing equation (4.198) and by eliminating perturbations in the voltage magnitude using (4.195) to give:

(pu), in which , (4.199)

and . (4.200)

The current magnitude:

(4.201) is linearized about the initial steady-state operating point to yield:

(pu); (4.202)

is obtained from (4.201) by substitution of the initial values of the current components .

The apparent power is , where all quantities are in per-unit on the bases of either (i) the k^th terminal of the device (d,k); or (ii) the network bus to which the terminal is connected (n,k). From this relationship it follows that:

(pu) and (pu). (4.203)

The perturbations in the real and reactive power are thus given by:

i_R i_I

1 V---

   cos sin sin –cos

P Q

1 V² ---

   v_R v_I v_I –v_R

P

= = Q

˜i^RI0

n k

 

V^{n k } ^{ k} P₀^{n k } Q₀^{n k }

˜i^RI0

d k

  i

˜^RI0

n k

 K_I^{ k}

=

i

˜^RI J_isS

˜ J_ivv

˜^RI +

= S

˜ = P Q ^T

J_is 1 V₀² ---

  

  v_R

0 v_I

0

v_I

0 v_R – 0

= J_iv 1

V₀² ---

  

  P₀ 2i_R

0v_R – 0

  Q₀ 2i_R

0v_I – 0

 

Q₀ 2i_I

0v_R – 0

 

– P₀ 2i_I

0v_I – 0

 

=

I = i_R²+i_I²

I i_R

0I₀

 i_R i_I

0I₀

 i_I

+ 1I₀i

˜^RI0

T i

˜^RI

= =

I₀

˜i^RI0

P jQ+ = VˆIˆ

P = v_Ri_R+v_Ii_I Q = v_Ii_R–v_Ri_I

Sec. 4.3 Model of a Static VAR Compensator 163

(pu). (4.204)

Consider the general case of an AC current flow from terminal 1 to terminal 2 through a reactance X connected between the terminals, the terminal voltage phasors being and

; all quantities are in pu on the appropriate bases. Thus, from which the components of the terminal 1 voltage are:

, and . (4.205)

Because the equations (4.205) are linear, the linearized equations are formed by replacing the variables by their perturbed quantities.