Master in Advanced Power Electrical Engineering Techno-economic aspects of power systems
Ronnie Belmans Dirk Van Hertem Stijn Cole
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Overview • • • • • • • • •
Lesson 1: Liberalization Lesson 2: Players, Functions and Tasks Lesson 3: Markets Lesson 4: Present generation park Lesson 5: Future generation park Lesson 6: Introduction to power systems Lesson 7: Power system analysis and control Lesson 8: Power system dynamics and security Lesson 9: Future grid technologies: FACTS and HVDC • Lesson 10: Distributed generation © Copyright 2005
Outline Power system analysis and control
• Power system analysis
Power flow Optimal power flow
• Power flow control
Primary control Secondary control Tertiary control Voltage control
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Control of active and reactive power Voltage regulation
• Voltage between sender and receiver S R = U R ⋅ I R = PR + j ⋅ QR • Voltage related to reactive power: *
X ΔU ≈ ⋅ QR UR • Angle related to active power: X δ ≈ 2 ⋅ PR U R R+ j⋅X Sender
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PS + j ⋅ QS
PR + j ⋅ QR
Receiver
Power flow • Normal conditions ==> steady state (equilibrium) • Basis calculations to obtain this state are called Power Flow
Also called Load Flow
• Purpose of power flow:
Determine steady state situation of the grid Get values for P, Q, U and voltage angle Calculate system losses First step for o o o o o o
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N-1 contingency study Congestion analysis Need for redispatch System development Stability studies ...
N-1 Example • Each line has capacity of 900 MW • Equal, lossless lines between nodes P = 1000 MW
G
P = 1000 MW
G
P = 166 MW Load = 500 MW P = 843 MW P = 666 MW
Load = 1500 MW © Copyright 2005
P = 1000 MW
G
P = 1000 MW
G
P = 500 MW Load = 500 MW P = 1500 MW P = 0 MW
Load = 1500 MW
Congestion and redispatch Example • Each line has capacity of 900 MW • Equal, lossless lines between nodes • The right generator is cheaper than the left, both have capacity 1500 MW
P = 1000 MW
P = 1000 MW
G
G
A
P = 166 MW Load = 500 MW P = 843 MW P = 666 MW
P = 800 MW
G A
P = 1200 MW
G B
B P = 200 MW Load = 500 MW P = 900 MW P = 500 MW congested Load = 1500 MW
Load = 1500 MW © Copyright 2005
If the load of gen B would increase, the profit would rise, but the line is congested
Power flow Three types of nodes • Voltage controlled nodes (P-U node) Nodes connected to a generator Voltage is controlled at a fixed value Active power delivered at a known value • Unregulated voltage node (P-Q node) A certain P and Q is demanded or delivered (non dispatched power plants, e.g. CHP) In practice: mostly nodes representing a pure `load' • Slack or swing bus (U-δ node) Variable P and Q Node that takes up mismatches © Copyright 2005
G
G
G G
Power flow Assumptions and representation • Properties are not influenced by small changes in voltage or frequency • Linear, localized parameters • Balanced system ==> Single line representation • Loads represented by their P and Q values • Current and power flowing to the node is positive • Transmission lines and transformers: πequivalent
Is
Ir Z
Y/2
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Y/2
1 ⎡ ⎤ 1 + YZ Z ⎥ ⎢ ⎡Vs ⎤ ⎡Vr ⎤ 2 ⎥⎢ ⎥ ⎢I ⎥ = ⎢ ⎛ 1 ⎞ ⎣ s ⎦ ⎢Y 1 + YZ 1 + 1 YZ ⎥ ⎣ I r ⎦ ⎢⎣ ⎜⎝ 4 ⎟⎠ 2 ⎥⎦
Power Flow Equations • I=Y.V is a set of (complex) linear equations • But P and Q are needed ==> S=V.I*
Set of non-linear equations
ΔPk
= PGk − PLk −
{
[
(
)
(
)]} = 0
(
)
(
)]} = 0
−Vk2Gkk +VkVm Gkm cos θk −θm + Bkm sin θk −θm
ΔQk
= QGk − QLk −
{
[
−Vk2Bkk +VkVm Gkm sin θk −θm − Bkm cos θk −θm
(i )
⎡ΔP ⎤ ⎢ΔQ ⎥ ⎣ ⎦
(i )
∂P ⎡ ∂P ⎤ V ⎡ Δθ ⎢ ∂θ ⎥ ∂V = − ⎢ ⋅⎢ΔV ⎥ ⎢ ∂Q ⎢ ∂Q ⎥ V ⎣ V ⎢⎣ ∂ θ ⎥ ∂ V
⎦ J
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(K (
i−1)
)
⎤ ⎥ ⎥ ⎦
(i )
Power flow Newton-Raphson • Newton-Raphson has a quadratic convergence • Normally +/- 7 iterations needed • Principle Newton-Raphson iterative method:
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Power Flow Alternative methods • Gauss-Seidel
Old method (solves I=Y.V), not used anymore Linear convergence
• Decoupled Newton-Raphson
Strong coupling between Q and V, and between P and δ Weak coupling between P and V, and between Q and δ ==> 2 smaller systems to solve ==> faster (2-3 times faster)
⎡ΔP ⎤ ⎢ΔQ ⎥ ⎣ ⎦
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(i )
⎡ ∂P ⎢ ∂θ = −⎢ ⎢ 0 ⎣
⎤ 0 ⎥ ∂Q ⎥ V⎥ ∂V ⎦
(i )
⎡ Δθ ⋅ ⎢ ΔV ⎢⎣ V
⎤ ⎥ ⎥⎦
(i )
Power Flow Alternative methods (II) • Fast decoupled Newton-Raphson
Neglects coupling as in decoupled Newton-Raphson Approximation: Jacobian considered constant
• Newton-Raphson with convergence parameter
Step in right direction (first order) multiplied by factor
• DC load flow
Consider only B (not Y) Single calculation (no iterations needed) Very fast ==> 7-10 times faster than normal Newton-Raphson In high voltage grids: 1 pu Sometimes used as first value for Newton-Raphson iteration (starting value) Economic studies and contingency analysis also use DC load flow
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Power flow: Available computer tools
• Available programs:
PSS/E (Siemens) DigSILENT ETAP Powerworld (demo version available for download) Matpower (free download, matlab based) PSAT: power system analysis toolbox (free download, matlab based) ...
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Optimal power flow (OPF) • Optimal power flow = power flow with a goal • Optimizing for highest objective
Minimum losses Economic dispatch (cheapest generation) ...
• Problem formulation minimize F(x, u, p) Objective function subject to g(x, u, p) = 0 Constraints
• Build the Lagrangian function
L = F(x, u, p) + λT g(x, u, p)
• Other optimization algorithms can also be used © Copyright 2005
Optimal power flow Flow chart Estimate control parameters
Solve Normal Load Flow
Compute the gradient of control variables
Check if gradient is sufficiently small
Terminate process, solution reached © Copyright 2005
Adjust control parameters
Optimal power flow Example Iter 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
F-count
Directional f(x)
1 3 5.28e+003 6 9 11 14 17 19 22 24 26 28 30 32 34 36
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First-order constraint Step-size
4570.1 9656.06
1.63 0.3196
1
7345.79 5212.76 5384.17 5305.59 5439.61 5328.32 5267.51 5301.72 5300.88 5295.95 5296.69 5296.69 5296.69 5296.69
0.2431 0.1449 0.02825 0.08544 0.07677 0.08351 0.1398 0.05758 0.004961 0.003562 4.436e-005 8.402e-007 4.487e-009 3.16e-011
0.5 0.5 1 0.5 0.5 1 0.5 1 1 1 1 1 1 1
max derivative
optimality 1.35e+004 506 1.41e+003 367 -132 958 144 -82.7 63.8 17.3 -0.325 1.15 0.0222
1.98e+003 4.32e+004 2.83e+003 696 859 1.04e+003 730 282 406 116 30.8 0.000728 2.75e-006
4.99 0.431 0.0113
Outline Power system analysis and control
• Power system analysis
Power flow Optimal power flow
• Power flow control
Primary control Secondary control Tertiary control Voltage control
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Control problem
• Complex MIMO system
Thousands of nodes Voltage and angle on each node Power flows through the lines (P and Q) Generated power (P and Q), and voltage OLTC positions ... Not everything is known! o o o o o
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Not every flow is known Local or global control Cross-border information Output of power plants Metering equipment is not always available or correct
Control problem Requirements
• Voltage must remain between its limits
1 p.u. +/- 5 or 10 %
• Power flow through a line is limited
Thermal limit depending on section
• Frequency has to remain between strict limits • Economic optimum
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Control problem Assumptions
• P-f control and Q-U control can be separated τ Q−V 1Hz, a part of the load is shed Basic principle: P-control feedback to counter power fluctuations Primary control uses spinning reserves Each control area within the synchronous area (UCTE) has to maintain a certain reserve, so that the absolute frequency shift in case of a 3 GW power deviation remains below 200 mHz 3 GW are two of the largest units within UCTE • If Δf is too high ==> islanding
• • • •
t
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Secondary control Definition/principle • • • • • •
System frequency is brought back to the scheduled value Balance between generation and consumption within each area Primary control is not impaired Centralized `automatic generation control' adjusts set points Power sources are called secondary reserves PI controlled:
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1 ΔPdi = − K ⋅ ∈ − ∈ ⋅dt ∫ Tsec
Primary and secondary control Example
50 Hz
P: X MW C: X MW
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pre-fault
0 MW
50 Hz
P: Y MW C: Y MW
Primary and secondary control Example (II)
49,8 Hz
P: X MW C: X MW
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Initial
500 MW
49,8 Hz
P: Y MW C: Y+1000 MW
Primary and secondary control Example (III)
49,9 Hz
P: X + 250 MW C: X MW
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primary control
500 MW
49,9 Hz
P: Y +250 MW C: Y+1000 MW
Primary and secondary control Example (IV)
49,9+ Hz
P: X + 250 MW C: X MW
Secondary control
500 - A/2 MW
49,9+ Hz
P: Y +250 + A MW C: Y+1000 MW +A MW
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G
Primary and secondary control Example (V)
50,1 Hz
P: X + 250 MW C: X MW
End secondary control
0 MW
50,1 Hz
P: Y + 1250 MW C: Y+1000 MW +1000 MW
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G
Primary and secondary control Example (VI) This phase happens simultaneously with the secondary control, and the “50.1 Hz” in reality doesn't occur 50 Hz
P: X MW C: X MW
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Second primary control
0 MW
50 Hz
P: Y + 1000 MW C: Y+1000 MW
Tertiary control Definition
• Automatic or manual set point change of generators and/or loads in order to:
Guarantee secondary reserves Obtain best power generation scheme in terms of economic considerations Cheap units (low marginal cost such as combined cycle or nuclear) o Highest security/stability o Loss minimalization o ... o
• How?
Redispatching of power generation Redistributing output of generators participating in secondary control Change power exchange with other areas Load control (shedding)
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Sequence overview
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Time control
• Limit discrepancies between synchronous •
time and universal time co-ordinated (UTC) within the synchronous zone Time difference limits (defined by UCTE)
Tolerated discrepancy: +/- 20 s Maximum allowed discrepancy under normal conditions: +/- 30 s Exceptional range: +/- 60 s
∫ Δf (t )⋅ dt < 20s
• Sometimes `played' with (week – weekend) © Copyright 2005
Voltage control • Voltage at busbar:
Voltage is mainly controlled by reactive power Can be regulated through excitation, tap changers, capacitors, SVC, ... Reactive power has a local nature
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Voltage control • Can the same control mechanism be used?
YES
• But
Good (sensitive) Q-production has to be available Synchronous compensator: expensive o Capacitors: not accurate enough o SVC/STATCom: possible, but not cheap o
U is `OK' between 0,95 and 1,05 p.u. Reactive power is less price (fuel) dependent (some losses)
• Voltage is locally controlled © Copyright 2005
Voltage control Control scheme
• Automatic voltage regulator (e.g. IEEE AVR 1)
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Conclusions
• Power flow analysis
Performed through iterative method (NewtonRaphson) Basis for many power system studies Optimal power flow
• Power flow control happens in several independent stages
Inter-area ties make the grid more reliable Voltage control is independent of power (frequency) control
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References • Power System Stability and control, Prabha • • • • •
Kundur,1994, McGraw-Hill Operational handbook UCTE, http://www.ucte.org/ohb/cur_status.asp Power system dynamics: stability and control, K. Padiyar, Ansham, 2004 Power system analysis, Grainger and Stevenson Power system control and stability, 2nd ed., Andersson and Fouad Dynamics and Control of Electric Power Systems, Goran Andersson
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