Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting
.pdfXXXVI |
List of Examples |
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7.6 |
Power flow modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
174 |
7.7 |
Minimum singular value index . . . . . . . . . . . . . . . . . . . . . . . . . . |
176 |
8.1 |
OMIB di erential algebraic equations . . . . . . . . . . . . . . . . . . . . |
190 |
8.2 |
Runge-Kutta’s formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
194 |
8.3 |
Modified Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
194 |
8.4 |
Backward Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
196 |
8.5 |
Trapezoidal method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
196 |
8.6 |
Rosenbrock’s semi-implicit method . . . . . . . . . . . . . . . . . . . . . . |
197 |
8.7 |
Comparison of time domain integration methods . . . . . . . . . . |
202 |
8.8 |
Application of the SIME method . . . . . . . . . . . . . . . . . . . . . . . . |
206 |
8.9 |
Quasi-static integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
214 |
9.1 |
Two-axis synchronous machine model . . . . . . . . . . . . . . . . . . . . |
223 |
9.2 |
Initialization of the synchronous machine two-axis model . . . |
225 |
10.1 |
Enforcing generator reactive power limits . . . . . . . . . . . . . . . . . |
251 |
10.2 |
Constant power vs. constant impedance load models in |
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transient stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
258 |
11.1 |
Tie line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
268 |
11.2 |
E ect of frequency on line parameters . . . . . . . . . . . . . . . . . . . . |
270 |
11.3 |
Voltage-tap ratio characteristic of loads fed by an ULTC . . . |
277 |
11.4 |
Comparison of ULTC discrete and continuous models . . . . . . |
278 |
11.5 |
Three-winding transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
281 |
13.1 |
Bus frequency measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
312 |
14.1 |
PV curves considering load characteristics . . . . . . . . . . . . . . . . |
315 |
14.2 |
E ect of tap changer dynamics on transient analysis . . . . . . . |
319 |
15.1 |
Comparison of synchronous machine models of di erent |
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orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
336 |
15.2 |
Comparison of synchronous machine models of di erent |
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types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
336 |
15.3 |
One-axis model with stator flux dynamics . . . . . . . . . . . . . . . . |
338 |
15.4 |
E ect of Using the center of inertia . . . . . . . . . . . . . . . . . . . . . . |
343 |
15.5 |
Transient behavior of dynamics shafts . . . . . . . . . . . . . . . . . . . . |
344 |
15.6 |
Sub-synchronous resonance transient . . . . . . . . . . . . . . . . . . . . . |
347 |
15.7 |
Induction motor start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
352 |
16.1 |
E ect of turbine governor on generator frequency . . . . . . . . . . |
360 |
16.2 |
E ect of automatic voltage regulation on synchronous |
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machine bus voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
367 |
16.3 |
E ectiveness of power system stabilizers for removing |
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Hopf bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
373 |
18.1 |
Fuel cell controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
403 |
18.2 |
Solar photovoltaic cell controls . . . . . . . . . . . . . . . . . . . . . . . . . . |
404 |
18.3 |
Superconducting magnetic energy storage . . . . . . . . . . . . . . . . . |
406 |
18.4 |
Power flow HVDC-VSC model . . . . . . . . . . . . . . . . . . . . . . . . . . |
411 |
19.1 |
Comparison of SVC models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
416 |
19.2 |
Comparison of STATCOM models . . . . . . . . . . . . . . . . . . . . . . . |
423 |
20.1 |
Weibull’s distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
437 |
List of Examples |
XXXVII |
20.2 Composite wind model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 20.3 Mexican hat wavelet wind model . . . . . . . . . . . . . . . . . . . . . . . . 440 20.4 Comparison of wind turbine transient behaviors . . . . . . . . . . . 456 21.1 Data format example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 22.1 Temperature map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 22.2 3D visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 22.3 Italian system temperature map . . . . . . . . . . . . . . . . . . . . . . . . . 487
List of Scripts
3.1 |
First Python script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
43 |
3.2 |
Basis of a power system analysis program . . . . . . . . . . . . . . . . |
50 |
4.1 |
Jacobi’s and Gauss-Seidel’s methods . . . . . . . . . . . . . . . . . . . . . |
73 |
4.2 |
Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
75 |
4.3 |
Power flow Jacobian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
78 |
4.4 |
Robust Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
83 |
4.5 |
Fast-decoupled power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
89 |
4.6 |
Runge-Kutta’s formula for solving the power flow |
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problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
100 |
5.1 |
CPF predictor step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
119 |
5.2 |
CPF corrector step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
121 |
6.1 |
Interior Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
148 |
7.1 |
Small-signal stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . |
160 |
7.2 |
Participation factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
165 |
8.1 |
Computing the first time step . . . . . . . . . . . . . . . . . . . . . . . . . . . |
201 |
8.2 |
Complete time domain integration algorithm . . . . . . . . . . . . . . |
206 |
9.1 |
Conversion of parameter bases . . . . . . . . . . . . . . . . . . . . . . . . . . |
229 |
9.2 |
Meta-attributes of a base device class . . . . . . . . . . . . . . . . . . . . |
232 |
9.3 |
Methods of the synchronous machine two-axis model . . . . . . . |
241 |
11.1 |
Sparse matrix implementation of the admittance matrix . . . . |
284 |
11.2 |
Incidence matrix implementation . . . . . . . . . . . . . . . . . . . . . . . . |
285 |
11.3 |
Transmission system power flow Jacobian matrix . . . . . . . . . . |
286 |
11.4 |
Transmission system power flow Hessian matrix . . . . . . . . . . . |
286 |
11.5 |
Network connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
288 |
12.1 |
Implementation of supply o ers . . . . . . . . . . . . . . . . . . . . . . . . . |
295 |
13.1 |
Fault interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
305 |
21.1 |
Data parser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
472 |
22.1 |
Batch script for power flow analysis . . . . . . . . . . . . . . . . . . . . . . |
476 |
22.2 |
Parser for simulations results . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
479 |
C.1 |
Implementation of windup and anti-windup limiters . . . . . . . . |
518 |
Notation
To set up a complete list of symbols for a book of this kind is a complex task. The variety of models and physical quantities used in Part III leads to a huge list of symbols. On the other hand, Parts I and II present general mathematical concepts that require a careful notation consistency. Hence, two notation approaches are used throughout the book.
The first notation concerns common quantities and general mathematical functions and variables. This notation is common to the whole book and is followed as rigorously as possible.
The second notation concerns local parameters that are needed to describe each device model included in Part III. These parameters are defined the first time they appear or gathered in tables. The organization of the book in several chapters helps avoid confusion and allows keeping each device model or group of models well separated from the others.
General Notation Rules
General notation rules are as follows.
1. Scalar functions, variables and parameters expressed in pu are in lower case Latin fonts. For example: x, v, p.
2. Angles and angular speeds are given in lower case Greek fonts. For example: θ, δ, α.
3.Upper case Latin fonts indicate scalar variables and parameters expressed in absolute values. For example: Sn [MVA], T1 [s].
4.Function and variable vectors are in lower case, bold face fonts. For example: f , g, x.
5.Jacobian matrices are in lower case, bold face fonts with a sub-index that indicates the variables with respect to which derivatives are computed. For example: gy = Ty g.
6.Hessian matrices are indicated as Jacobian ones but with two sub-indexes. For example: gyy = Tyy g.
XLII |
Notation |
7.Other matrices are in upper case, bold face fonts. For example: A, In, W .
8.Sets are indicated by calligraphic fonts. For example: I, N .
9.Iterations are indicated with a superscript in brackets. For example: x(i), y(i+1), v(0). The latter indicates the initial guess.
10.The subscript 0 indicates the equilibrium point or the initial value. For example: x0 = x(t0).
11.A bar on top of a symbol indicates a phasor. For example: v¯ = vejθ .
12. In case of complex quantities, a superscript asterisk indicates the conjugate. For example: v¯ = ve−jθ . In case of real quantities, a superscript asterisk indicates the optimal value. For example: pw.
13.A superscript T indicates transpose. For example: AT .
14.Verbatim fonts are used to indicate Python scripts. For example: list, class. The symbol >>> indicates the Python interactive command line prompt.
Frequent Symbols
The following is only a selection of most common quantities used in the book.
Functions and Equations
f di erential equations. g algebraic equations.
hinequality constraints.
LLagrangian function.
qˆ(i) vector used for implicit numerical methods.
rreduced gradient.
s di erential operator in block diagrams (s = dtd ).
continuation equation.
ϕobjective function.
ϕgeneral vector of di erential equations.
ψ homotopy map.
Variables and Parameters
am |
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converter modulating amplitude. |
E |
energy. |
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kG |
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distributed slack bus variable. |
transmission line length. |
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m transformer tap ratio. |
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ph, |
p |
active powers. |
qh, |
q |
reactive powers. |
sslack variables.
ttime.
udiscrete variables.
Notation |
XLIII |
vh, v bus voltage magnitudes. w left eigenvector.
xstate variables.
x˙ first time derivatives of state variables.
yalgebraic variables.
z compound vector of variables (e.g., z = [xT , yT ]T ).
αfiring angle.
δj , δ generator rotor angles.
η controllable variables or parameters.
θh, θ bus voltage angles.
Θtemperature.
λeigenvalue.
μloading level (continuation parameter).
μˆ barrier parameter for the interior point method.
μ independent variables or uncontrollable parameters.
νright eigenvector.
ξgeneral vector of state variables.
π dual variables associated with inequality constraints. ρ dual variables associated with equality constraints.
τtangent vector.
τe |
electrical torque. |
τm |
mechanical torque. |
φtransformer phase shift.
ωj , ω generator rotor speeds.
Matrices
A |
generic constant element matrix. |
AC |
complete system Jacobian matrix. |
A(i) |
matrix defined for implicit numerical methods. |
c |
state matrix. |
AS |
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B admittance matrix used in dc power flow. |
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B |
and B admittance matrices used in fast decoupled power flow. |
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C incidence matrix. |
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f |
x |
Jacobian matrix |
T f . |
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x |
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f y |
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T |
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Jacobian matrix y f . |
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G nodal conductance matrix. |
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g |
x |
Jacobian matrix |
T g. |
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x |
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gy |
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T |
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Jacobian matrix y g. |
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N |
matrix of right eigenvectors. |
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Tconnectivity matrix.
W |
matrix of left eigenvectors. |
¯ |
admittance matrix. |
Y |
Λ diagonal matrix of eigenvalues.
Σdiagonal matrix of singular values.
XLIV |
Notation |
Constants
Asurface area.
B, bh0, bk0, bhk susceptances. cp specific heat.
Drotor damping.
fn |
nominal frequency in Hz. |
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G, gh0, gk0, ghk |
conductances. |
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H, Hm, Ht |
inertia constants. |
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K, K0, Ki, Kp |
gains. |
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t total transmission line length. |
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mg |
mass. |
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R, rhk , rT |
resistances. |
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Sb |
base power. |
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T , Ta, T1 |
time constants. |
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Vb |
base voltage. |
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X, xhk, xT |
reactances. |
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z¯, z¯T |
impedances. |
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y¯hk, y¯T admittances. |
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Ωb |
nominal synchronous speed in rad/s. |
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ωs |
synchronous speed in pu. |
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Numbers |
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nb |
number of ac buses. |
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nc |
number of series connections or branches. |
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ng |
number of algebraic functions g. |
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nG |
number of generators. |
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nu |
number of discrete variables u. |
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nx |
number of state variables x. |
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ny |
number of algebraic variables y. |
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nz |
number of variables z. |
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nη |
number of controllable variables η. |
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nμ |
number of uncontrollable variables μ. |
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nξ |
number of general state variables ξ. |
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Sets
Bset of ac buses.
C set of series connections or branches.
D set of demands.
G set of generators.
Nset of dc nodes.
Rset of generator power reserves.
Notation |
XLV |
S set of supplies.
Tset of times.
Ωset of all devices.
Ωh set of devices connectes to bus h.
Device Model Notation
In Part III, the notation is aimed to maintain light expressions. The use of indexes is avoided wherever is possible. Device equations imply the device index i in all variables and parameters. Thus, vh and θh indicate the bus voltage magnitude and phase angle, respectively, while ph and qh indicate the active and power injections, respectively. For series devices connected to two buses, the indexes h and k are used.
Bases for Per Unit Values
Throughout the whole book, the bases used for ac values are:
Sbac |
three-phase power in MVA. |
V ac |
phase-to-phase voltage in kV. |
b |
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Thus, the current base Ibac and the impedance base Zbac are:
ac |
= |
√ |
Sbac |
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Ib |
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· Vbac |
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3 |
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ac |
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Vbac |
(Vbac)2 |
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Zb |
= |
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= |
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√ |
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ac |
Sac |
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3 |
· Ib |
b |
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For dc values, the following bases are used:
Sbdc |
power in MW. |
V dc |
voltage in kV. |
b |
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Thus, the current base Ibdc and the resistance base Rbdc are obtained as follows:
Idc = |
Sbdc |
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b |
V dc |
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b |
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Rdc = |
Vbdc |
= |
(Vbdc)2 |
b |
Idc |
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Sdc |
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b |
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b |
For systems where there are both ac and dc devices, it is assumed that
Sbdc = Sbac.
Part I
Introduction
Chapter 1
Power System Modelling
This chapter introduces basic modelling concepts that are used throughout the book. Section 1.1 defines a power system and provides most relevant references related to power system analysis. Section 1.2 states the philosophical background of the book and general motivations. Section 1.3 presents programmatic assumptions and the proposed methodological approach for power system modelling. Finally, Section 1.4 defines the general equations that are used for modelling power systems.
1.1Background
In essence, an electrical power systems is a set of interacting devices that transform primary energy sources, e.g., heat, into electricity and then transform electricity into another form of energy, e.g., the mechanical one. The electricity is transmitted at various voltage levels, through an adequate series of transformers, transmission lines and cables, from generators to loads. It is noteworthy to observe that electricity is never used as is. Therefore, the expression “electric power system” is somewhat incomplete, since a power system is essentially an energy conversion system. For this reason, the expression “power system” is used in this book, without specifying the form of the energy involved.
Energy is what engineering is all about. Thus, it is not surprising that interconnected power systems are the largest and most complex systems ever built by man. For example, Figure 1.1 depicts a simplified scheme of the UCTE high-voltage transmission system [318]. Fortunately, one does not need a real-world interconnected network to approach the study of power systems. Simple benchmark systems are often enough to understand the basic functioning as well as advanced topics of power systems. This remarkable property is exploited in the book to simplify as much as possible the examples.
However, even the simplest example requires some background. In the remainder, it is assumed that basic power system concepts are known by the reader. The level required for fully taking advantage of this book is that
F. Milano: Power System Modelling and Scripting, Power Systems, pp. 3–17. springerlink.com c Springer-Verlag Berlin Heidelberg 2010