Compute state-space model of linear electrical circuit

You must call `power_statespace`

with a minimum
of seven input arguments.

[A,B,C,D,states,x0,x0sw,rlsw,u,x,y,freq,Asw,Bsw,Csw,Dsw,Hlin] = power_statespace(rlc,switches,source,line_dist,yout,y_type,unit )

You can also specify optional arguments. To use these optional arguments, the number of input arguments must be 12, 13, 14 or 16.

[A,B,C,D,states,x0,x0sw,rlsw,u,x,y,freq,Asw,Bsw,Csw,Dsw,Hlin] = power_statespace(rlc,switches,source,line_dist,yout,y_type,unit , net_arg1,net_arg2,net_arg3,...,netsim_flag,fid_outfile, freq_sys,ref_node,vary_name,vary_val)

The `power_statespace`

command computes the
state-space model of a linear electrical circuit expressed as

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

where *x* is the vector of state-space variables
(inductor currents and capacitor voltages), *u *is
the vector of voltage and current inputs, and *y *is
the vector of voltage and current outputs.

When you build a circuit from Simscape™
Power Systems™ Specialized
Technology blocks, `power_statespace`

is automatically
called by the `power_analyze`

command. `power_statespace`

is
also available as a stand-alone command for expert users. This allows
you to generate state-space models without using the Simscape
Power Systems Specialized
Technology block modeling interface and to access options that are
not available through the Specialized Technology library. For example,
using `power_statespace`

, you can model transformers
and mutual inductances with more than three windings.

The linear circuit can contain any combination of voltage and current sources, RLC branches, multiwinding transformers, mutually coupled inductances, and switches. The state variables are inductor currents and capacitor voltages.

The state-space representation (matrices A,B,C,D, and vector `x0`

)
computed by `power_statespace`

can then be used in
a Simulink^{®} system, via a State-Space block, to perform simulation
of the electrical circuit (see Examples). Nonlinear elements (mechanical
or power electronic switches, transformer saturation, machines, distributed
parameter lines, etc.) can be connected to the linear circuit.

These Simulink models are interfaced with the linear circuit through voltage outputs and current inputs of the state-space model. You can find the models of the nonlinear elements provided with Simscape Power Systems software in the Specialized Technology library.

The number of input arguments must be 7, 12, 13, 14, or 16. Arguments 8 to 16 are optional. The first seven arguments that must be specified are

`rlc`

: Branch matrix specifying the network topology as well as the resistance R, inductance L, and capacitance C values. See format below.`switches`

: Switch matrix. Specify an empty variable if no switches are used. See format below.`source`

: Source matrix specifying the parameters of the electrical voltage and current sources. Specify an empty variable if no sources are used. See format below.`line_dist`

: Distributed parameter line matrix. Specify an empty variable if no distributed lines are used. See format below.`yout`

: Matrix of output, expressed as character vectors. See format below.`y_type`

: Integer vector indicating output types (`0`

for voltage output,`1`

for current output).`unit`

: Character vector specifying the units to be used for R, L, and C values in the`rlc`

matrix. If`unit = 'OHM'`

, R L C values are specified in ohms Ω at the fundamental frequency specified by`freq_sys`

(default value is 60 Hz). If`unit = 'OMU'`

, R L C values are specified in ohms (Ω), millihenries (mH), and microfarads (µF).

The last nine arguments are optional. The first three are used
to pass arguments from the `power_analyze`

command.
Hereafter, only the arguments to be specified when `power_statespace`

is
used as a stand-alone command are described:

`net_arg1`

,`net_arg2`

,`net_arg3`

: Used to pass arguments from`power_analyze`

. Specify an empty variable`[]`

for each of these arguments.`netsim_flag`

: Integer controlling the messages displayed during the execution of`power_statespace`

. Default value is`0`

.If

`netsim_flag = 0`

, the version number, number of states, inputs, outputs, and modes are displayed. Output values are displayed in polar form for each source frequency.If

`netsim_flag = 1`

, only version number, number of states, inputs, and outputs are displayed.If

`netsim_flag = 2`

, no message is displayed during execution.`fid_outfile`

: File identifier of the`power_statespace`

output file containing parameter values, node numbers, steady-state outputs, and special messages. Default value is`0`

.`freq_sys`

: Fundamental frequency (Hz) considered for specification of X_{L}and X_{C}reactances if`unit`

is set to`'OHM'`

. Default value is 60 Hz.`ref_node`

: Reference node number used for ground of PI transmission lines. If`−1`

is specified, the user is prompted to specify a node number.`vary_name`

: Matrix containing the symbolic variable names used in output, expressed as character vectors. These variables must be defined in your MATLAB^{®}workspace.`vary_val`

: Vector containing the values of the variable names specified in`vary_name`

.

`A,B,C,D`

: state-space matrices of the linear circuit with all switches open.A(nstates, nstates) , B(nstates, ninput), C(noutput, nstates) , D(noutput, ninput),

where

`nstates`

is the number of state variables,`ninput`

is the number of inputs, and`noutput`

is the number of outputs.`states`

: Matrix containing the names of the state variables. Each name has the following format:Inductor currents:

`Il_bxx_nzz1_zz2`

Capacitor voltages:

`Uc_bxx_nzz1_zz2`

where

xx = branch number zz1 = first node number of the branch zz2 = second node number of the branch

The last lines of the `states`

matrix, which
are followed by an asterisk, indicate inductor currents and capacitor
voltages that are not considered as state variables. This situation
arises when inductor currents or capacitor voltages are not independent
(inductors forming a cut set – for example, inductors connected
in series – or capacitors forming a loop). The currents and
voltages followed by asterisks can be expressed as a linear combination
of the other state variables:

`x0`

: Column vector of initial values of state variables considering the open or closed status of switches.`x0sw`

: Vector of initial values of switch currents.`rlsw`

: Matrix (`nswitch`

,`2`

) containing the R and L values of series switch impedances in ohms (Ω) and henries (H).`nswitch`

is the number of switches in the circuit.`u,x,y`

: Matrices`u`

(`ninput`

,`nfreq`

),`x`

(`nstates`

,`nfreq`

), and`y`

(`noutput`

,`nfreq`

) containing the steady-state complex values of inputs, states, and outputs.`nfreq`

is the length of the`freq`

vector. Each column corresponds to a different source frequency, as specified by the next argument,`freq`

.`freq`

: Column vector containing the source frequencies ordered by increasing frequency.`Asw,Bsw,Csw,Dsw`

: State-space matrices of the circuit including the closed switches. Each closed switch with an internal inductance adds one extra state to the circuit.`Hlin`

: Three-dimensional array (`nfreq`

,`noutput`

,`ninput`

) of the`nfreq`

complex transfer impedance matrices of the linear system corresponding to each frequency of the`freq`

vector.

Two formats are allowed:

Six columns: Implicit branch numbering. Branch numbers correspond to the RLC line numbers.

Seven columns: Explicit branch numbering. Branch number

`Nobr`

is assigned by the user.

Each line of the `RLC`

matrix must be specified
according to the following format.

`[node1, node2, type, R, L, C, Nobr]`

for RLC
branch or line branch

`[node1, node2, type, R, L, C, Nobr]`

for transformer
magnetizing branch

`[node1, node2, type, R, L, U, Nobr]`

for transformer
winding

`[node1, node2, type, R, L, U, Nobr]`

for mutual
inductances

`node1`

: First node number of the branch. The node number must be positive or zero. Decimal node numbers are allowed.`node2`

: Second node number of the branch. The node number must be positive or zero. Decimal node numbers are allowed.`type`

: Integer indicating the type of connection of RLC elements, or, if negative, the transmission line length:`type = 0`

: Series RLC element`type = 1`

: Parallel RLC element`type = 2`

: Transformer winding`type = 3`

: Coupled (mutual) windingIf

`type`

is negative, the transmission line is modeled by a PI section of length`|type|`

. See details below.

For a mutual inductor or a transformer having N windings, N+1
consecutive lines must be specified in `RLC `

matrix:

N lines with

`type = 2`

or`type = 3`

; (one line per winding). Each line specifies`R/L/U`

or`R/Xl/Xc`

where [`R/L, R/Xl`

= winding resistance and leakage reactance for a transformers or winding resistance and self reactance for mutually coupled windings.`U`

is the nominal voltage of transformer winding (specify 0 if`type = 3`

).One extra line with

`type = 1`

for the magnetizing branch of a transformer (parallel`Rm/Lm`

or`Rm/Xm`

) or one line with`type = 0`

for a mutual impedance (series`Rm/Lm`

or`Rm/Xm`

).

For a transformer magnetizing branch or a mutual impedance, the first node number is an internal node located behind the leakage reactance of the first winding. The second node number must be the same as the second node number of the first winding.

To model a saturable transformer, you must use a nonlinear inductance
instead of the linear inductance simulating the reactive losses. Set
the `Lm/Xm `

value to `0`

(no linear
inductance) and use the Saturable Transformer block,
set with proper flux-current characteristics.

This block can be found in the Fundamental Blocks/Elements library. It must be connected to the linear part of the system (State-Space block or S-function) between a voltage output (voltage across the magnetizing branch) and a current input (current source injected into the transformer internal node). See the Examples.

If type is negative, its absolute value specifies the length (km) of a transmission line simulated by a PI section. For a transmission line, the R/L/C or R/Xl/Xc values must be specified in Ω/km, mH/km, and µF/km, or in Ω/km.

Parameter | Description |
---|---|

| Branch resistance (Ω) |

| Branch inductive reactance (Ω at |

| Branch inductance (mH) |

| Branch capacitive reactance (Ω at |

| Capacitance (µF) |

| Nominal voltage of transformer winding. The same units
(volts or kV) must be used for each winding. For a mutual inductance
( |

Zero value for |

The following restrictions apply for transformer winding R-L
values. Null values are not allowed for secondary impedances if some
transformer secondaries form loops (as in a three-phase delta connection).
Specify a very low value for R or L or both (e.g., `1e-6`

pu
based on rated voltage and power) to simulate a quasi-ideal transformer.
The resistive and inductive parts of the magnetizing branch can be
set to infinite (no losses; specify `Xm = Rm = inf`

).

Three formats are allowed:

Five columns: All sources are generating the same frequency specified by

`freq_sys`

.Six columns: The frequency of each source is specified in column 6.

Seven columns: The seventh column is used to specify the type of nonlinear element modeled by the current source.

Each line of the source matrix must be specified according to the following format:

[ node1, node2, type, amp, phase, freq, model ]

`node1`

,`node2`

: Node numbers corresponding to the source terminals. These are the polarity conventions:Voltage source:

`node1`

is the positive terminal.Current source: Positive current flowing from

`node1`

to`node2`

inside the source.

`type:`

Integer indicating the type of source:`0`

for voltage source,`1`

for current source.`amp:`

Amplitude of the AC or DC voltage or current (V or A).`phase`

: Phase of the AC voltage or current (degree).`freq`

: Frequency (Hz) of the generated voltage or current. Default value is 60 Hz. For a DC voltage or current source, specify`phase`

=`0`

and`freq`

=`0`

.`amp`

can be set to a negative value. The generated signals are`amp * sin(2π*freq*t + phase)`

for AC,`amp`

for DC.`model:`

Integer specifying the type of nonlinear element modeled by the current source (saturable inductance, thyristor, switch,...). Used by`power_analyze`

only.

The commands that compute the state-space representation of a system expect the sources in a certain order. You must respect this order in order to obtain correct results. You must be particularly careful if the system contains any switches. This is the proper ordering of sources:

The currents from all switches that have a null inductance (Lon = 0), if any.

The currents from all nonlinear models that have a finite inductance (switches with Lon > 0, the magnetizing inductance in saturable transformers, etc.), if any.

All other voltage and current sources in any order, if any.

Refer to the Example section below for an example illustrating proper ordering of sources for a system containing nonlinear elements.

Switches are nonlinear elements simulating mechanical or electronic devices such as circuit breakers, diodes, or thyristors. Like other nonlinear elements, they are simulated by current sources driven by the voltage appearing across their terminals. Therefore, they cannot have a null impedance. They are simulated as ideal switches in series with a series R-L circuit. Various models of switches (circuit breaker, ideal switch, and power electronic devices) are available in the Simscape Power Systems Specialized Technology library. They must be interconnected to the linear part of the system through appropriate voltage outputs and current inputs.

The switch parameters must be specified in a line of the switches matrix in seven different columns, according to the following format.

[ node1, node2, status, R, L/Xl, no_I , no_U ]

Parameter | Description |
---|---|

| Node numbers corresponding to the switch terminals |

| Code indicating the initial status of the switch at t
= 0: |

| Resistance of the switch when closed (Ω) |

| Inductance of the switch when closed (mH) or inductive reactance (Ω at freq_sys) |

For these last two fields, you must use the same units as those specified for the RLC matrix. Either field can be set to 0, but not both. |

The next two fields specify the current input number and the voltage output number to be used for interconnecting the switch model to the State-Space block. The output number corresponding to the voltage across a particular switch must be the same as the input number corresponding to the current from the same switch (see Example section below):

`no_I`

: Current input number coming from the output of the switch model`no_U`

: Voltage output number driving the input of the switch model

The distributed parameter line model contains two parts:

A linear part containing current sources and resistances that are connected at the line sending and receiving buses together with the linear circuit.

A nonlinear part available in the Distributed Parameter Line block of the Elements library. This block performs the phase-to-mode transformations of voltage and currents and simulates the transmission delays for each mode. The

*distributed_param_line*block must be connected to appropriate voltage outputs and current inputs of the linear part of the system. The line parameters have to be specified in the`line_dist`

matrix and also in the Distributed Parameter Line block.

Each row of the `line_dist`

matrix is used to specify a distributed parameter
transmission line. The number of columns of `line_dist`

depends on the
number of phases of the transmission line.

For an `nphase`

line, the first (```
4
+ 3 * nphase + nphase^2
```

) columns are used. For example,
for a three-phase line, 22 columns are used.

[nphase, no_I, no_U, length, L/Xl, Zc, Rm, speed, Ti]

Parameter | Description |
---|---|

| Number of phases of the transmission line |

| Input number in the source matrix corresponding to the
first current source Is_1 of the line model. Each line model uses
2*nphase current sources specified in the source matrix as follows: |

| Output number of the state-space corresponding to the
first voltage output Vs_1 feeding the line model. Each line model
uses 2*nphase voltage outputs in the source matrix as follows: |

| Length of the line (km) |

| Vector of the nphase modal characteristic impedances (Ω) |

| Vector of the nphase modal series resistances (Ω/km) |

| Vector of the nphase modal propagation speeds (km/s) |

| Transformation matrix from mode to phase currents such
that Iphase = Ti * Imod. The nphase * nphase matrix must be given
in vector format, |

The desired outputs are specified by the matrix `yout`

.
Each line of the `yout`

matrix must be an algebraic
expression containing a linear combination of states and state derivatives,
specified according to the following format.

Parameter | Description |
---|---|

| Capacitor voltage of branch |

| Inductor current of branch |

| Derivative of |

| Source voltage or current specified by line |

| Voltage between nodes x1 and x2 = Ux1 −Ux2 |

| Current in branch |

| Current flowing into node x of a PI transmission line
specified by line |

Each output expression is built from voltage and current variable
names defined above, their derivatives, constants, other variable
names, parentheses and operators `(+ − * / ^)`

,
in order to form a valid MATLAB expression. For example

yout = char(['R1*I_b1+Uc_b3-L2*dIl_b2','U_n10_20','I2+3*I_b5']);

If variable names are used (`R1`

and `L2`

in
the above example), their names and values must be specified by the
two input arguments `vary_name`

and `vary_val`

.

Parameter | Sign Convention |
---|---|

| Branch current, inductor current of branch |

| Current at one end (node x) of a PI transmission line.
If x = |

| Voltage across capacitor or source voltage |

| Voltage between nodes x1 and x2 = Ux1 − Ux2. |

The commands that compute the state-space representation of a system expect the outputs to be in a certain order. You must respect this order in order to obtain correct results. You must be particularly careful if the system contains any switches. The following list gives the proper ordering of outputs:

The voltages across all switches that have a null inductance (Lon = 0), if any

The currents of all switches that have a null inductance (Lon = 0), if any, in the same order as the voltages above

The voltages across all nonlinear models that have a finite inductance (switches with Lon > 0, the magnetizing inductance in saturable transformers, etc.)

All other voltage and current measurements that you request, in any order

Refer to the Example section below for an example illustrating proper ordering of outputs for a system containing nonlinear elements.

The following circuit consists of two sources (one voltage source
and one current source), two series RLC branches (`R1-L1`

and `C6`

),
two parallel RLC branches (`R5-C5`

and `L7-C7`

),
one saturable transformer, and two switches (`Sw1`

and `Sw2`

). `Sw1`

is
initially closed whereas `Sw2`

is initially open.
Three measurement outputs are specified (I1, V2, and V3). This circuit
has seven nodes numbered 0, 1, 2, 2.1, 10, 11, and 12. Node 0 is used
for the ground. Node 2.1 is the internal node of the transformer where
the magnetization branch is connected.

You can use the `power_statespace`

command
to find the state-space model of the linear part of the circuit. The
nonlinear elements `Sw1`

, `Sw2`

,
and `Lsat`

must be modeled separately by means of
current sources driven by the voltages appearing across their terminals.
Therefore you must provide three additional current sources and three
additional voltage outputs for interfacing the nonlinear elements
to the linear circuit.

You can find the state-space model of the circuit by entering
the following commands in a MATLAB script file. The example is
available in the `power_circ2ss.m`

file.
Notice that an output text file named `power_circ2ss.net`

containing
information on the system is requested in the call to `power_statespace`

.

unit='OMU'; % Units = ohms, mH, and uF rlc=[ %N1 N2 type R L C(uF)/U(V) 1 2 0 0.1 1 0 %R1 L1 2 0 2 0.05 1.5 100 %transfo Wind.#1 10 0 2 0.20 0 200 %transfo Wind.#2 2.1 0 1 1000 0 0 %transfo mag. branch 11 0 1 200 0 1 %R5 C5 11 12 0 0 0 1e-3 %C6 12 0 1 0 500 2 %L7 C7 ]; source=[ %N1 N2 type U/I phase freq 10 11 1 0 0 0 %Sw1 11 12 1 0 0 0 %Sw2 2.1 0 1 0 0 0 %Saturation 1 0 0 100 0 60 %Voltage source 0 10 1 2 -30 180 %Current source ]; switches=[ %N1 N2 status R(ohm) L(mH) I# U# # 10 11 1 0.01 0 1 1 %Sw1 11 12 0 0.1 0 2 2 %Sw2 ]; %outputs % % Both switches have Lon=0, so their voltages must be the first outputs, % immediately followed by their currents (in the same order as the voltages). % The voltage across all nonlinear models that don't have L=0 follow % (in this case the saturable transformer's magnetizing inductor). % The measurements that you request follow, in any order. % y_u1='U_n10_11'; %U_Sw1= Voltage across Sw1 y_u2='U_n11_12'; %U_Sw2= Voltage across Sw2 y_i3='I1'; %I1= Switch current Sw1 y_i4='I2'; %I2= Switch current Sw2 y_u5='U_n2.1_0'; %U_sat= Voltage across saturable reactor y_i6='I_b1'; %I1 measurement y_u7='U_n11_0'; %V2 measurement y_u8='U_n12_0'; %V3 measurement yout=char(y_u1,y_u2,y_i3,y_i4,y_u5,y_i6,y_u7,y_u8); % outputs y_type=[0,0,1,1,0,1,0,0]; %output types; 0=voltage 1=current % Open file that contains power_statespace output information fid=fopen('power_circ2ss.net','w'); [A,B,C,D,states,x0,x0sw,rlsw,u,x,y,freq,Asw,Bsw,Csw,Dsw,Hlin]= power_statespace(rlc,switches,source,[],yout,y_type,unit,[],[], [],0,fid);

While `power_statespace`

is executing, the
following messages are displayed.

Computing state space representation of linear electrical circuit (V2.0)... (4 states ; 5 inputs ; 7 outputs) Oscillatory modes and damping factors: F=159.115Hz zeta=4.80381e-08 Steady state outputs @ F=0 Hz : y_u1= 0Volts y_u2= 0Volts y_i3= 0Amperes y_i4= 0Amperes y_u5= 0Volts y_i6= 0Amperes y_u7= 0Volts y_u8= 0Volts Steady state outputs @ F=60 Hz : y_u1 = 0.009999 Volts < 3.168 deg. y_u2 = 199.4 Volts < -1.148 deg. y_i3 = 0.9999 Amperes < 3.168 deg. y_i4 = 0 Amperes < 0 deg. y_u5 = 99.81 Volts < -1.144 deg. y_i6 = 2.099 Amperes < 2.963 deg. y_u7 = 199.4 Volts < -1.148 deg. y_u8 = 0.01652 Volts < 178.9 deg. Steady state outputs @ F=180 Hz : y_u1 = 0.00117 Volts < 65.23 deg. y_u2 = 22.78 Volts < 52.47 deg. y_i3 = 0.117 Amperes < 65.23 deg. y_i4 = 0 Amperes < 0 deg. y_u5 = 11.4 Volts < 53.48 deg. y_i6 = 4.027 Amperes < 146.5 deg. y_u7 = 22.83 Volts < 52.47 deg. y_u8 = 0.0522 Volts < 52.47 deg.

The names of the state variables are returned in the `states`

matrix.

states states = Il_b2_n2_2.1 Uc_b5_n11_0 Uc_b6_n11_12 Il_b7_n12_0 Il_b1_n1_2* Uc_b7_n12_0*

Although this circuit contains a total of six inductors and
capacitors, there are only four state variables. The names of the
state variables are given by the first four lines of the `states`

matrix.
The last two lines are followed by an asterisk indicating that these
two variables are a linear combination of the state variables. The
dependencies can be viewed in the output file `power_circ2ss.net`

.

The following capacitor voltages are dependent: Uc_b7_n12_0 = + Uc_b5_n11_0 - Uc_b6_n11_12 The following inductor currents are dependent: Il_b1_n1_2 = + Il_b2_n2_0

The A,B,C,D matrices contain the state-space model of the circuit
without nonlinear elements (all switches open). The `x0`

vector
contains the initial state values considering the switch Sw1 closed.
The `Asw`

, `Bsw`

, `Csw`

,
and `Dsw`

matrices contain the state-space model
of the circuit considering the closed switch `Sw1`

.
The `x0sw`

vector contains the initial current in
the closed switch.

A A = -4.0006e+05 0 0 0 0 -4995 0 -499.25 0 -4992.5 0 4.9925e+05 0 2 -2 0 Asw Asw = -80.999 -199.99 0 0 4.9947e+05 -5244.7 0 -499.25 4.9922e+05 -5242.1 0 4.9925e+05 0 2 -2 0

The system source frequencies are returned in the `freq`

vector.

freq freq = 0 60 180

The corresponding steady-state complex outputs are returned
in the (6-by-3) `y`

matrix where each column corresponds
to a different source frequency.

For example, you can obtain the magnitude of the six voltage and current outputs at 60 Hz as follows.

abs(y(:,2)) ans = 0.0099987 199.42 0.99987 0 99.808 2.0993 199.41 0.016519

The initial values of the four state variables are returned
in the `x0`

vector. You must use this vector in the
State-Space block to start the simulation in steady state.

x0 x0 = 2.3302 14.111 14.07 3.1391e-05

The initial values of switch currents are returned in `x0sw`

.
To start the simulation in steady state, you must use these values
as initial currents for the nonlinear model simulating the switches.

x0sw x0sw = 0.16155 0

The Simulink diagram of the circuit is available in the `power_circ2ss_slk`

model.
If no resistive switches had been used, the linear part of the circuit
could have been simulated with the State-Space block of the Simulink/Continuous
library. However, as resistive switches are used, the `sfun_psbcontc`

S-function
is used instead of the State-Space block. This S-function reevaluates
the state-space matrices during simulation when the circuit topology
is changing (after a switch is opened or closed). Appropriate inputs
and outputs are used to connect the switch and saturable reactance
models to the linear system. Notice that the status of each switch
is fed back from the breaker to the S-function, after the inputs mentioned
earlier. You can find the Breaker and Saturable Transformer blocks
in the Fundamental Blocks/Elements library containing all the nonlinear
continuous models used by Simscape
Power Systems software. As the breaker
model is vectorized, a single block is used to simulate the two switches `Sw1`

and `Sw2`

.

If you use the Simscape
Power Systems Specialized Technology library
to build your circuit, the same Simulink system is generated
automatically by the `power_analyze`

command. The Simscape
Power Systems Specialized
Technology version of this system is also available in the `power_circ2ss_sps`

model.

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