EMT.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "generous-pennsylvania",
   "metadata": {},
   "source": [
    "# Numerical simulation of short-circuit at the terminals of a synchronous generator in EMT\n",
    "\n",
    "This notebook is a companion to the course [Control and Operation of Electric Power Systems](https://sps.cut.ac.cy/courses/een452/). It is based on material from the courses delivered by Prof. Thierry Van Cutsem at the University of Liege.\n",
    "\n",
    "## Background\n",
    "\n",
    "**Network represented by a simple Thevenin equivalent:**\n",
    "\n",
    "- resistance $R_e$ and inductance $L_e$ in each phase\n",
    "- no magnetic coupling between phases, for simplicity\n",
    "\n",
    "**Machine:**\n",
    "\n",
    "- only the field winding $f$ in the $d$ axis\n",
    "- only one damper winding $q1$ in the $q$ axis\n",
    "- rotor speed $\\dot{\\theta}_r$ assumed constant\n",
    "  - the focus is on short-lasting electromagnetic transients\n",
    "- constant excitation voltage $V_f$\n",
    "  - it is assumed that the automatic voltage regulator hasn't time to react"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "placed-cooking",
   "metadata": {},
   "source": [
    "<img src=\"img/sync_model.PNG\" width=\"600\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "significant-flour",
   "metadata": {},
   "source": [
    "### Network equations\n",
    "\n",
    "\\begin{alignat*}{3}\n",
    "&v_{a}-e_{a}=R_{e} i_{a}+L_{e} \\frac{d i_{a}}{d t} \\quad &&\\text{ with }\\quad &&e_{a}=\\sqrt{2} E \\cos \\left(\\omega_{N} t+\\theta\\right) \\\\ \n",
    "&v_{b}-e_{b}=R_{e} i_{b}+L_{e} \\frac{d i_{b}}{d t} \\quad &&\\text{ with }\\quad &&e_{b}=\\sqrt{2} E \\cos \\left(\\omega_{N} t+\\theta-\\frac{2 \\pi}{3}\\right) \\\\ \n",
    "&v_{c}-e_{c}=R_{e} i_{c}+L_{e} \\frac{d i_{c}}{d t} \\quad &&\\text{ with }\\quad &&e_{c}=\\sqrt{2} E \\cos \\left(\\omega_{N} t+\\theta-\\frac{4 \\pi}{3}\\right)\n",
    "\\end{alignat*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "promising-screen",
   "metadata": {},
   "source": [
    "### Park equations\n",
    "\n",
    "$$\\left[\\begin{array}{c}v_{d} \\\\ v_{q} \\\\ v_{o}\\end{array}\\right]=\\mathcal{P}\\left[\\begin{array}{c}v_{a} \\\\ v_{b} \\\\ v_{c}\\end{array}\\right] \\qquad\\left[\\begin{array}{c}i_{d} \\\\ i_{q} \\\\ i_{0}\\end{array}\\right]=\\mathcal{P}\\left[\\begin{array}{c}i_{a} \\\\ i_{b} \\\\ i_{c}\\end{array}\\right]$$\n",
    "\n",
    "where $$\\mathcal{P}=\\sqrt{\\frac{2}{3}}\\left[\\begin{array}{ccc}\\cos (\\theta_{r}) & \\cos \\left(\\theta_{r}-\\frac{2 \\pi}{3}\\right) & \\cos \\left(\\theta_{r}-\\frac{4 \\pi}{3}\\right) \\\\ \\sin (\\theta_{r}) & \\sin \\left(\\theta_{r}-\\frac{2 \\pi}{3}\\right) & \\sin \\left(\\theta_{r}-\\frac{4 \\pi}{3}\\right) \\\\ \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}}\\end{array}\\right]$$ is the Park transformation (power invariant version) and $\\theta_r = \\theta_r^o + \\omega_N t$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "tired-bobby",
   "metadata": {},
   "source": [
    "### Machine equations\n",
    "\n",
    "$$\\begin{aligned} \\psi_{d} &=L_{d d} i_{d}+L_{d f} i_{f} \\\\ \\psi_{q} &=L_{q q} i_{q}+L_{q q 1} i_{q 1} \\\\ \\psi_{0} &=L_{o o} i_{o} \\\\ \\psi_{f} &=L_{f f} i_{f}+L_{d f} i_{d} \\\\ \\psi_{q 1} &=L_{q 1 q 1} i_{q 1}+L_{q q 1} i_{q} \\\\ v_{d} &=-R_{a} i_{d}-\\dot{\\theta}_{r} \\psi_{q}-\\frac{d \\psi_{d}}{d t} \\\\ v_{q} &=-R_{a} i_{q}+\\dot{\\theta}_{r} \\psi_{d}-\\frac{d \\psi_{q}}{d t} \\\\ v_{0} &=-R_{a} i_{0}-\\frac{d \\psi_{0}}{d t} \\\\ V_{f} &=R_{f} i_{f}+\\frac{d \\psi_{f}}{d t} \\\\ 0 &=R_{q 1} i_{q 1}+\\frac{d \\psi_{q 1}}{d t} \\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "solid-bunny",
   "metadata": {},
   "source": [
    "### Remarks\n",
    "\n",
    "- The model is made up of Differential-Algebraic Equations (DAEs)\n",
    "- some of the variables and some of the equations could be eliminated but the additional computational effort of keeping all of them is negligible\n",
    "- $\\theta_r$ being known, the equations are linear with respect to the unknowns\n",
    "- some coefficients in these equations vary with time.\n",
    "\n",
    "### Equations in per unit\n",
    "\n",
    "**At the stator (a, b, c):**\n",
    "\n",
    "- base voltage $V_B$ = nominal RMS phase-to-neutral voltage (kV)\n",
    "- base power $S_B$ = single-phase apparent power (MVA)\n",
    "- base current $I_B = S_B/V_B$, base magnetic flux $\\psi_B = V_B/\\omega_N$, etc.\n",
    "\n",
    "**In each of the Park winding (d, q, o):**\n",
    "\n",
    "- base voltage = $V_B$\n",
    "- base power = $S_B$\n",
    "- base current = $I_B = S_B/V_B$, base flux = $\\psi_B = V_B/\\omega_N$, etc.\n",
    "\n",
    "The rotor variables and equations are also transformed in pu; the system is not detailed here.\n",
    "\n",
    "**After passing in per unit:**\n",
    "\n",
    "- $\\dot{\\theta}_r=1$ pu\n",
    "- each time derivative is multiplied by $1/\\omega_N$, since we keep the time t in seconds (not in pu)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "organic-island",
   "metadata": {},
   "source": [
    "\n",
    "**The equations become:**\n",
    "\n",
    "$$\\begin{aligned} \\frac{1}{\\omega_{N}} \\frac{d i_{a}}{d t} &=-\\frac{R_{e}}{L_{e}} i_{a}+\\frac{1}{L_{e}} v_{a}-\\frac{1}{L_{e}} e_{a} \\\\ \\frac{1}{\\omega_{N}} \\frac{d i_{b}}{d t} &=-\\frac{R_{e}}{L_{e}} i_{b}+\\frac{1}{L_{e}} v_{b}-\\frac{1}{L_{e}} e_{b} \\\\ \\frac{1}{\\omega_{N}} \\frac{d i_{c}}{d t} &=-\\frac{R_{e}}{L_{e}} i_{c}+\\frac{1}{L_{e}} v_{c}-\\frac{1}{L_{e}} e_{c} \\\\ 0 &=\\sqrt{\\frac{2}{3}}\\left[\\cos \\left(\\theta_{r}\\right) v_{a}+\\cos \\left(\\theta_{r}-\\frac{2 \\pi}{3}\\right) v_{b}+\\cos \\left(\\theta-\\frac{4 \\pi}{3}\\right) v_{c}\\right]-v_{d} \\\\ 0 &=\\sqrt{\\frac{2}{3}}\\left[\\sin \\left(\\theta_{r}\\right) v_{a}+\\sin \\left(\\theta_{r}-\\frac{2 \\pi}{3}\\right) v_{b}+\\sin \\left(\\theta_{r}-\\frac{4 \\pi}{3}\\right) v_{c}\\right]-v_{q} \\\\ 0 &=\\frac{1}{\\sqrt{3}}\\left(v_{a}+v_{b}+v_{c}\\right)-v_{o} \\\\ 0 &=\\sqrt{\\frac{2}{3}}\\left[\\cos \\left(\\theta_{r}\\right) i_{a}+\\cos \\left(\\theta_{r}-\\frac{2 \\pi}{3}\\right) i_{b}+\\cos \\left(\\theta_{r}-\\frac{4 \\pi}{3}\\right) i_{c}\\right]-i_{d} \\\\ 0 &=\\sqrt{\\frac{2}{3}}\\left[\\sin \\left(\\theta_{r}\\right) i_{a}+\\sin \\left(\\theta_{r}-\\frac{2 \\pi}{3}\\right) i_{b}+\\sin \\left(\\theta_{r}-\\frac{4 \\pi}{3}\\right) i_{c}\\right]-i_{q} \\\\ 0 &=\\frac{1}{\\sqrt{3}}\\left(i_{a}+i_{b}+i_{c}\\right)-i_{o} \\\\ 0 &=L_{d d} i_{d}+L_{d f} i_{f}-\\psi_{d} \\\\ 0 &=L_{q q} i_{q}+L_{q q 1} i_{q 1}-\\psi_{q} \\\\ 0 &=L_{f f} i_{f}+L_{d f} i_{d}-\\psi_{f} \\\\ 0 &=L_{q 1 q 1} i_{q 1}+L_{q q 1} i_{q}-\\psi_{q 1} \\\\ 0 &=L_{o o} i_{o}-\\psi_{o} \\\\ \\frac{1}{\\omega_{N}} \\frac{d \\psi_{d}}{d t} &=-R_{a} i_{d}-\\psi_{q}-v_{d} \\\\ \\frac{1}{\\omega_{N}} \\frac{d \\psi_{q}}{d t} &=-R_{a} i_{q}+\\psi_{d}-v_{q} \\\\ \\frac{1}{\\omega_{N}} \\frac{d \\psi_{f}}{d t} &=-R_{f} i_{f}+V_{f} \\\\ \\frac{1}{\\omega_{N}} \\frac{d \\psi_{q 1}}{d t} &=-R_{q 1} i_{q 1} \\\\ \\frac{1}{\\omega_{N}} \\frac{d \\psi_{0}}{d t} &=-R_{a} i_{o}-v_{o} \\end{aligned}$$\n",
    "\n",
    "### Variable - equations balance\n",
    "\n",
    "- 19 variables : $v_a,\\ v_b,\\ v_c,\\ i_a,\\ i_b,\\ i_c,\\ v_d,\\ v_q,\\ v_o,\\ i_d,\\ i_q,\\ i_o,\\ \\psi_d,\\ \\psi_q,\\ \\psi_o,\\ \\psi_f ,\\ \\psi_{q1},\\ i_f,\\ i_{q1}$\n",
    "- 19 equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "amazing-issue",
   "metadata": {},
   "source": [
    "### In compact form:\n",
    "\n",
    "$$\\begin{aligned}\\left(1 / \\omega_{N}\\right) \\dot{\\boldsymbol x} &= \\boldsymbol A_{x x} \\boldsymbol x+\\boldsymbol A_{x y} \\boldsymbol y+\\boldsymbol u_{x} \\\\ \\boldsymbol 0 &=\\boldsymbol A_{y x} \\boldsymbol x+\\boldsymbol A_{y y} \\boldsymbol y+\\boldsymbol u_{y} \\end{aligned}$$\n",
    "\n",
    "where\n",
    "\n",
    "$\\begin{aligned} \\boldsymbol x &=\\left[i_{a}\\ i_{b}\\ i_{c}\\ \\psi_{d}\\ \\psi_{q}\\ \\psi_{f}\\ \\psi_{q 1}\\ \\psi_{o}\\right]^{T} \\\\ \\boldsymbol y &=\\left[v_{a}\\ v_{b}\\ v_{c}\\ v_{d}\\ v_{q}\\ v_{0}\\ i_{d}\\ i_{q}\\ i_{o}\\ i_{f}\\ i_{q 1}\\right]^{T} \\\\ \\boldsymbol{u}_{x}&=\\left[-\\frac{e_{a}}{L_{e}}\\ -\\frac{e_{b}}{L_{e}}\\ -\\frac{e_{c}}{L_{e}}\\  0\\ 0\\  \\mathrm{~V}_{f}\\  0\\ 0\\right]^{T} \\\\ \\boldsymbol{u}_{y}&=\\left[0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0\\right]^{T} \\end{aligned}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cathedral-coating",
   "metadata": {},
   "source": [
    "### Numerical solution of the DAEs\n",
    "\n",
    "To simulate the equations above over a time-horizon $T$, we need to discretise them and solve with an integration method. A popular numerical integration formula is the _Trapezoidal Method_:\n",
    "\n",
    "$$x_{k+1}=x_k+\\frac{h}{2}(\\dot{x}_{k+1}+\\dot{x}_k)$$\n",
    "\n",
    "Replacing in the compact form, gives:\n",
    "\n",
    "$$\\boldsymbol{x}_{k+1}=\\boldsymbol{x}_{k}+\\frac{h}{2} \\omega_{N} \\boldsymbol{A}_{x x} \\boldsymbol{x}_{k+1}+\\frac{h}{2} \\omega_{N} \\boldsymbol{A}_{x y} \\boldsymbol{y}_{k+1}+\\frac{h}{2} \\omega_{N} \\boldsymbol{u}_{x k+1}+\\frac{h}{2} \\dot{\\boldsymbol{x}}_{k}$$\n",
    "\n",
    "Dividing by $\\frac{h\\omega_N}{2}$ and rearranging the various terms:\n",
    "\n",
    "$$\\left[\\boldsymbol{A}_{x x}-\\frac{2}{h \\omega_{N}} \\boldsymbol{I}\\right] \\boldsymbol{x}_{k+1}+\\boldsymbol{A}_{x y} \\boldsymbol{y}_{k+1}=-\\frac{2}{h \\omega_{N}} \\boldsymbol{x}_{k}-\\frac{1}{\\omega_{N}} \\dot{\\boldsymbol{x}}_{k}-\\boldsymbol{u}_{x k+1}$$\n",
    "\n",
    "where $I$ is the unit matrix of same dimension as $x$.\n",
    "\n",
    "From the compact form, we also get:\n",
    "\n",
    "$$\\boldsymbol{A}_{y x} \\boldsymbol{x}_{k+1}+\\boldsymbol{A}_{y y} \\boldsymbol{y}_{k+1}=-\\boldsymbol{u}_{yk+1}$$\n",
    "\n",
    "Grouping the equations, the linear system to solve at each time step is:\n",
    "\n",
    "$$\\left[\\begin{array}{cc}\\boldsymbol{A}_{x x}-\\frac{2}{h \\omega_{N}} \\boldsymbol{I} & \\boldsymbol{A}_{x y} \\\\ \\boldsymbol{A}_{y x} & \\boldsymbol{A}_{y y}\\end{array}\\right]\\left[\\begin{array}{c}\\boldsymbol{x}_{k+1} \\\\ \\boldsymbol{y}_{k+1}\\end{array}\\right]=\\left[\\begin{array}{c}-\\frac{2}{h \\omega_{N}} \\boldsymbol{x}_{k}-\\frac{1}{\\omega_{N}} \\dot{\\boldsymbol{x}}_{k}-\\boldsymbol{u}_{x k+1} \\\\ -\\boldsymbol{u}_{y k+1}\\end{array}\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "beneficial-haiti",
   "metadata": {},
   "source": [
    "## Case study\n",
    "\n",
    "### Network and machine data\n",
    "\n",
    "- $f_N=50$ Hz\n",
    "- $L_e=0.2$ pu, $R_e=0.01$ pu\n",
    "- $R_a=0.005$ pu\n",
    "- $L_{dd} = 2.4$ pu, $L_{df}=2.2$ pu, $L_{ff}=2.42$ pu\n",
    "- $L_{qq} = 2.4$ pu, $L_{qq1}=2.2$ pu, $L_{q1q1}=2.2512$ pu\n",
    "- $R_f = 0.0011$ pu, $R_{q1}=0.0239$ pu\n",
    "- $L_{00}=0.1$ pu\n",
    "\n",
    "\n",
    "### Initial operating point\n",
    "\n",
    "- $P=0.5$ pu\n",
    "- $Q=0.1$ pu\n",
    "- $\\underline{V}_a=1.0\\angle{0}$ pu\n",
    "\n",
    "### Simulation case study\n",
    "\n",
    "A three-phase short-circuit is simulated by setting E to zero at $t = 0.05$ s.\n",
    "\n",
    "### Important remarks\n",
    "\n",
    "The fault is not cleared in order to show the various time constants present in the current evolution.\n",
    "However, in practice:\n",
    "\n",
    "- the fault must be cleared fast enough, e.g. after $5 - 10$ cycles ($0.1 - 0.2$ s)\n",
    "- beyond that time, the model is no longer valid:\n",
    "  - rotor speed would not remain constant\n",
    "  - $v_f$ would be adjusted by the Automatic Voltage Regulator\n",
    "  - etc."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "least-infrared",
   "metadata": {},
   "source": [
    "## Main simulation procedure\n",
    "\n",
    "### Linearization function\n",
    "\n",
    "The function linmodel defines the procedure to calculate the matrix $A$ of the model and the input vector $u$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "selected-deficit",
   "metadata": {},
   "outputs": [],
   "source": [
    "function [A,u] = linmodel(t) \n",
    "\n",
    "global wN Re Le Ldd Ldf Lff Lqq Lqq1 Lq1q1 Loo Ra Rf Rq1 theta0 phasE Vf ;\n",
    "\n",
    "A=zeros(19,19) ;\n",
    "theta=theta0+wN*t ;\n",
    "\n",
    "A(1,1)  = -Re/Le ;   % d ia/dt\n",
    "A(1,4)  = 1/Le ;\n",
    "A(2,2)  = -Re/Le ;   % d ib/dt\n",
    "A(2,5)  = 1/Le ;\n",
    "A(3,3)  = -Re/Le ;   % d ic/dt\n",
    "A(3,6)  = 1/Le ;\n",
    "A(4,4)  = (sqrt(2)/3)*cos(theta) ;         % vd \n",
    "A(4,5)  = (sqrt(2)/3)*cos(theta-2*pi/3) ;\n",
    "A(4,6)  = (sqrt(2)/3)*cos(theta+2*pi/3) ;\n",
    "A(4,7)  = -1 ;\n",
    "A(5,4)  = (sqrt(2)/3)*sin(theta) ;         % vq \n",
    "A(5,5)  = (sqrt(2)/3)*sin(theta-2*pi/3) ;\n",
    "A(5,6)  = (sqrt(2)/3)*sin(theta+2*pi/3) ;\n",
    "A(5,8)  = -1 ; \n",
    "A(6,4)  = 1/3 ;                            % vo\n",
    "A(6,5)  = 1/3 ; \n",
    "A(6,6)  = 1/3 ; \n",
    "A(6,9)  = -1 ;\n",
    "A(7,1)  = (sqrt(2)/3)*cos(theta) ;         % id \n",
    "A(7,2)  = (sqrt(2)/3)*cos(theta-2*pi/3) ;\n",
    "A(7,3)  = (sqrt(2)/3)*cos(theta+2*pi/3) ;\n",
    "A(7,10) = -1 ;\n",
    "A(8,1)  = (sqrt(2)/3)*sin(theta) ;         % iq \n",
    "A(8,2)  = (sqrt(2)/3)*sin(theta-2*pi/3) ;\n",
    "A(8,3)  = (sqrt(2)/3)*sin(theta+2*pi/3) ;\n",
    "A(8,11) = -1 ; \n",
    "A(9,1)  = 1/3 ;     % io\n",
    "A(9,2)  = 1/3 ; \n",
    "A(9,3)  = 1/3 ;\n",
    "A(9,12) = -1 ;\n",
    "A(10,10)= Ldd ;     % psid\n",
    "A(10,13)= Ldf ;\n",
    "A(10,15)= -1 ;\n",
    "A(11,11)= Lqq ;     % psiq\n",
    "A(11,14)= Lqq1 ;\n",
    "A(11,16)= -1 ; \n",
    "A(12,13)= Lff ;     % psif\n",
    "A(12,10)= Ldf ;\n",
    "A(12,17)= -1 ;\n",
    "A(13,14)= Lq1q1 ;   % psiq1\n",
    "A(13,11)= Lqq1 ;\n",
    "A(13,18)= -1 ;\n",
    "A(14,12)= Loo ;     % psio\n",
    "A(14,19)= -1 ;\n",
    "A(15,16)= -1 ;      % d psid/dt\n",
    "A(15,7) = -1 ;\n",
    "A(15,10)= -Ra ;\n",
    "A(16,15)=  1 ;      % d psiq/dt\n",
    "A(16,8) = -1 ;\n",
    "A(16,11)= -Ra ;\n",
    "A(17,13)= -Rf ;     % d psif/dt\n",
    "A(18,14)= -Rq1 ;    % d psiq1/dt\n",
    "A(19,12)= -Ra ;     % d psio/dt\n",
    "A(19,9) = -1 ;\n",
    "\n",
    "% Short-circuit application on the terminals of the sync. machine\n",
    "if t<= 0.05  || t>= 99\n",
    "    ea=sqrt(2)*abs(phasE)*cos(wN*t+angle(phasE)) ;\n",
    "    eb=sqrt(2)*abs(phasE)*cos(wN*t+angle(phasE)-2*pi/3) ;\n",
    "    ec=sqrt(2)*abs(phasE)*cos(wN*t+angle(phasE)+2*pi/3) ;\n",
    "else\n",
    "    ea=0. ;\n",
    "    eb=0. ;\n",
    "    ec=0. ;\n",
    "end\n",
    "\n",
    "u= [ -ea/Le -eb/Le -ec/Le 0 0 0 0 0 0 0 0 0 0 0 0 0 Vf 0 0]' ; % Output vector u\n",
    "\n",
    "endfunction"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "loaded-wednesday",
   "metadata": {},
   "source": [
    "### Simulation procedure\n",
    "\n",
    "Below is the main simulation procedure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "false-mapping",
   "metadata": {},
   "outputs": [],
   "source": [
    "global wN Re Le Ldd Ldf Lff Lqq Lqq1 Lq1q1 Loo Ra Rf Rq1 theta0 phasE Vf ;\n",
    "\n",
    "j=sqrt(-1) ;\n",
    "\n",
    "% --------------- DATA ---------------\n",
    "% network and operating point\n",
    "% P and Q are in per unit on a single-phase base power (three-phase model)\n",
    "\n",
    "Xe=0.20 ;\n",
    "Re=0.01 ;\n",
    "P=0.5 ;\n",
    "Q=0.1 ;\n",
    "V=1 ;\n",
    "\n",
    "% machine parameters\n",
    "\n",
    "Ra=0.005 ;\n",
    "Xd=2.4 ;\n",
    "Xq=2.4 ;\n",
    "Xl=0.2 ;\n",
    "Xpd=0.40 ;\n",
    "Xpq=0.25 ;\n",
    "Loo=0.1 ;\n",
    "fN=50. ;\n",
    "Tpdo_s=7. ;\n",
    "Tpqo_s=0.3 ;\n",
    "\n",
    "% time step size and simulation interval\n",
    "\n",
    "h= 100e-06 ;\n",
    "Tsim= 3 ;\n",
    "\n",
    "% -------------- END OF DATA --------------\n",
    "\n",
    "% derived parameters\n",
    "\n",
    "wN=2*pi*fN ;\n",
    "TB=1/(2*pi*fN) ;\n",
    "Le=Xe ;\n",
    "Ldd=Xd ;\n",
    "Ldf=Xd-Xl ;\n",
    "Lff=Ldf^2/(Xd-Xpd) ;\n",
    "Tpdo=Tpdo_s/TB ;\n",
    "Rf=Lff/Tpdo ;\n",
    "Lqq=Xq ;\n",
    "Lqq1=Xq-Xl ;\n",
    "Lq1q1=Lqq1^2/(Xq-Xpq) ;\n",
    "Tpqo=Tpqo_s/TB ;\n",
    "Rq1=Lq1q1/Tpqo ;\n",
    "\n",
    "% allocate space to store trajectory\n",
    "\n",
    "nbsteps=ceil(Tsim/h) ;\n",
    "time=zeros(1,nbsteps+1) ;\n",
    "xhist=zeros(17,nbsteps+1) ;\n",
    "\n",
    "% initialization of state variables\n",
    "\n",
    "phasI=(P-j*Q)/V ;             % network voltages and currents\n",
    "va=sqrt(2)*V ;\n",
    "vb=sqrt(2)*V*cos(-2*pi/3) ;\n",
    "vc=sqrt(2)*V*cos( 2*pi/3) ;\n",
    "ia=sqrt(2)*abs(phasI)*cos(angle(phasI)) ;\n",
    "ib=sqrt(2)*abs(phasI)*cos(angle(phasI)-2*pi/3) ;\n",
    "ic=sqrt(2)*abs(phasI)*cos(angle(phasI)+2*pi/3) ;\n",
    "phasE=V-(Re+j*Xe)*phasI ;\n",
    "\n",
    "phi=angle(V+Ra*phasI+j*Xq*phasI);     % machine\n",
    "theta0=phi+pi/2 ;\n",
    "\n",
    "id=(sqrt(2)/3)*(cos(theta0)*ia+cos(theta0-2*pi/3)*ib+cos(theta0+2*pi/3)*ic) ;\n",
    "iq=(sqrt(2)/3)*(sin(theta0)*ia+sin(theta0-2*pi/3)*ib+sin(theta0+2*pi/3)*ic) ;\n",
    "io=0 ;\n",
    "vd=(sqrt(2)/3)*(cos(theta0)*va+cos(theta0-2*pi/3)*vb+cos(theta0+2*pi/3)*vc) ;\n",
    "vq=(sqrt(2)/3)*(sin(theta0)*va+sin(theta0-2*pi/3)*vb+sin(theta0+2*pi/3)*vc) ;\n",
    "vo=0 ;\n",
    "\n",
    "psid= vq+Ra*iq ;\n",
    "psiq=-vd-Ra*id ;\n",
    "psio=0 ;\n",
    "ifd=(psid-Ldd*id)/Ldf ;\n",
    "psif=Lff*ifd+Ldf*id ;\n",
    "Vf=Rf*ifd ;\n",
    "iq1=0 ;\n",
    "psiq1=Lqq1*iq ;\n",
    "\n",
    "% identify differential and algebraic states\n",
    "% note: the i-th differential equation is  d x(i)/dt = ...\n",
    "\n",
    "dif= [1 2 3 15 16 17 18 19] ;\n",
    "alg= [4 5 6 7 8 9 10 11 12 13 14] ;\n",
    "\n",
    "% initialisation of integration\n",
    "\n",
    "time(1)=0 ;\n",
    "%  1  2  3  4  5  6  7  8  9  10 11 12 13  14  15    16   17    18   19\n",
    "x=[ia ib ic va vb vc vd vq vo id iq io ifd iq1 psid psiq psif psiq1 psio]' ; \n",
    "xhist(1:19,1)=x ;\n",
    "\n",
    "[A,u]=linmodel(0) ;\n",
    "\n",
    "% integration method\n",
    "\n",
    "for k=1:nbsteps\n",
    "    c= A*x + u ;  \n",
    "    \n",
    "    t=k*h ;\n",
    "    time(k+1)=t ;\n",
    "    \n",
    "    [A,u]=linmodel(t) ;\n",
    "    \n",
    "    b= -u ;\n",
    "    b(dif)= b(dif)- (2/h/wN)*x(dif) - c(dif) ;\n",
    "    \n",
    "    M= A;\n",
    "    M(dif,dif)= M(dif,dif) - (2/h/wN)*eye(size(dif,2)) ;\n",
    "    x= M\\b ;\n",
    "    xhist(1:19,k+1)=x ;\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "unlike-recognition",
   "metadata": {},
   "source": [
    "## Analysis of results and discussion\n",
    "\n",
    "### Voltage on phase a\n",
    "\n",
    "- decrease of AC voltage magnitude under the effect of the fault\n",
    "- presence of a residual voltage due to some emf inside the generator\n",
    "- remark:\n",
    "  - such an emf does not exist in generators connected to the network through power electronic interfaces (dispersed generation in MV distribution grid)\n",
    "  - the latter do not participate to the short-circuit capacity!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "twelve-jason",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(4,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "improved-magnitude",
   "metadata": {},
   "source": [
    "### Current on phase a\n",
    "\n",
    "- increase of amplitude of alternating current under the effect of the fault\n",
    "- the envelop of the current wave varies with time (more details in the sequel)\n",
    "- presence of a small aperiodic or unidirectional or DC component\n",
    "  - typical of transients in an RL circuit due to switchings\n",
    "  - much more visible in the other two phases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "consistent-cemetery",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(1,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "faced-profile",
   "metadata": {},
   "source": [
    "### Current on phases b and c\n",
    "\n",
    "- the magnitude of the aperiodic components decrease with a time constant $\\approxeq$ 0.15 s (in this example)\n",
    "- the aperiodic components are not the same in all three phases, because the rotor is not in the same position with respect to each stator winding\n",
    "- once they have vanished, the three phase currents become again sinusoidal and balanced"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "related-aluminum",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(2,1:4000))\n",
    "hold on\n",
    "plot(time(1:4000),xhist(3,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "convinced-latin",
   "metadata": {},
   "source": [
    "### Current on phase a -- long-term\n",
    "\n",
    "- the magnitude of the alternating component of ia shows two time constants:\n",
    "  - a short one, lasting a few cycles, resulting in a slightly higher initial amplitude of the current: caused by damper winding q1\n",
    "  - a much longer one ($\\approxeq$ 1.5 s in this example): caused by field winding f\n",
    "- the current that the breakers have to interrupt is much higher than the one which would prevail in steady-state!\n",
    "- the machine behaves initially as if it had a smaller internal reactance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "vertical-breast",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time,xhist(1,:))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "incident-selection",
   "metadata": {},
   "source": [
    "### Magnetic fields\n",
    "\n",
    "The alternating components of the stator currents $i_a$, $i_b$ and $i_c$\n",
    "- are shifted by $\\pm 2\\pi/3$ rad. They create a magnetic field $H_{AC}$ which rotates at the angular speed $\\omega_N$\n",
    "- this field is fixed with respect to the rotor windings\n",
    "- under the effect of the fault, the amplitude if $i_a$, $i_b$ and $i_c$ increases significantly. So does the magnetic field $H_{AC}$\n",
    "- this induces aperiodic current components in the rotor windings.\n",
    "\n",
    "The aperiodic components of the stator currents $i_a$, $i_b$ and $i_c$\n",
    "- create a magnetic field $H_{AC}$ which is fixed with respect to the stator\n",
    "- hence it rotates at angular speed $\\omega_N$ with respect to the rotor windings\n",
    "- this induces alternating components of angular frequency $\\omega_N$ in the rotor windings"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bulgarian-centre",
   "metadata": {},
   "source": [
    "### Field winding flux\n",
    "\n",
    "- the flux $\\psi_f$ in the field winding changes very little in spite of the large increase of stator currents!\n",
    "- large \"magnetic inertia\" due to the long time constant $L_{ff}/R_f$ (= 7 s in this example)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "historical-reproduction",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(17,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "younger-albert",
   "metadata": {},
   "source": [
    "### Field current\n",
    "\n",
    "- Lenz law: additional current components appear in the field winding in order to keep $\\psi_f$ (almost) constant\n",
    "- the oscillatory component is due to the magnetic field $H_{DC}$\n",
    "  - check: time constant of decay = time constant of aperiodic component of stator currents ≃ 0.15 s\n",
    "- the aperiodic component is due to the magnetic field $H_{AC}$\n",
    "  - time constant ≃ 1.5 s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "atomic-transsexual",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(13,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "shared-biodiversity",
   "metadata": {},
   "source": [
    "### Flux in damper winding $q1$\n",
    "\n",
    "- the flux in the damper winding $q1$ is comparatively more \"volatile\"\n",
    "- indeed, the field and the damper windings are constructively very different: field coil vs. damper bars in rotor slots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "respected-response",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(18,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "geological-powder",
   "metadata": {},
   "source": [
    "### Current in damper winding $q1$\n",
    "\n",
    "- the damper current $i_{q1}$ has a zero initial (and final) value\n",
    "- the oscillatory component is due to the magnetic field $H_{DC}$\n",
    "- the aperiodic component decreases much faster than the aperiodic component of $i_f$\n",
    "- it corresponds to the initial, fast decaying, increment of the stator current amplitude"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "monthly-concept",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(14,1:4000))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "civilian-reduction",
   "metadata": {},
   "source": [
    "### Currents $i_d$ and $i_q$\n",
    "\n",
    "- the oscillatory component of $i_d$ (resp. $i_q$) corresponds to the oscillatory component of $i_f$ (resp. $i_{q1}$) which lies on the same axis\n",
    "- it can be shown that $i_q$ goes to almost zero due to the predominantly inductive nature of the short-circuit\n",
    "- the aperiodic component of $i_d$ evolves with the long time constant observed for the aperiodic component of $i_f$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "personalized-theology",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(:),xhist(10,:))\n",
    "hold on\n",
    "plot(time(:),xhist(11,:))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "balanced-bangladesh",
   "metadata": {},
   "source": [
    "### Currents $\\psi_d$ and $\\psi_q$\n",
    "\n",
    "- fluxes $\\psi_d$ and $\\psi_q$ in Park windings vary comparatively much faster\n",
    "- since $i_q$ and $i_{q1}$ tend to zero, so does $\\psi_q$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "vocal-sussex",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(time(1:4000),xhist(15,1:4000))\n",
    "hold on\n",
    "plot(time(1:4000),xhist(16,1:4000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fewer-plain",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "5.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}