David J Hill
The University of Hong Kong and
The University of Sydney
From MA Pai to Power Network Science ��
- reflections on the stability of power systems�
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Symposium in Honour of Prof MA Pai (PaiFest), University of Illinois Urbana-Champaign, 15 October 2015 �
Outline �
Prof MA Pai��A story of a simple model��DJH projects (stability)��Conclusion �
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Professor MA Pai �
§ Berkeley 1978-80 § Papers on Gij § Meeting about 1979 § BH papers 1981-3 § Books (Px2, SP) § Visits Illinois (+PK, PS) § Emails ever since § Editorship Springer
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Outline �
Prof MA Pai��A story of a simple model��DJH projects��Conclusion �
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Phase Angle Stability�
Ref: M.A.Pai, Energy Function Analysis for Power System Stability
Basic synchronism issue in power networks
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Network-Reduction Model�
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Lossless network Kron reduction • Lossy due to loads
• Network structure lost
• Theory for two-machine case rigorous
Multi-machine Network-Reduction Model�
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Chiang et al., Direct stability analysis of electric power systems using energy function, IEEE Proceedings, 1995
Stability Theory�
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�i = !i
!i = �Di
Mi+
Pmi
Mi� 1
Mi
X
j 6=i
ViVjBij sin(�i � �j) Gij = 0
• Use first integrals, Popov (Lure-Postnikov)
• But these are not well-defined, i.e. path dependent, unless make assumptions
• The Kron reduction creates the difficulty, i.e. loads become Gij
• Properties of the original physical network structure are eliminated
Common but bad assumption
Network-Preserving Model�
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Ref: Bergen and Hill, IEEE PAS,1981; Hill and Bergen, 1983
Network-Preserving Model�
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• Generators: second-order differential equations • Loads: first-order differential equations • Non-identical nodes with nonlinear interconnections • Angle stability: (synchronization)
xi = fi(xi) +NX
j=1
aijhi(xi, xj)
Di�i = P 0i �
X
j 6=i
bij sin(�i � �j)
�i = !i
Mi!i = �Di!i + P 0i �
X
j 6=i
bij sin(�i � �j)Generators
Loads/RES
Dynamical Network
!i = !j = !s
�i � �j = cij
No Gij problem!
Dynamic models - Ref: Hiskens and Hill, IEEE TPWRS, 1989�
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• Voltage dependence of loads important • Lyapunov theory for DA systems (Hill and Mareels, 1990)
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Lyapunov function (Ref: Hiskens and Hill, IEEE TPWRS, 1989) �
Discovered later: More basic version given as first integral by Vasin in St Petersburg journal, 1971 (ten years earlier than NBMs)
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Energy surfaces�
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Incompletely solved problems�
• Rigorous Lyapunov functions for NPMs where the real-power loads
are independent of voltages
• These Lyapunov (or energy) functions can be expressed in a topological form where the kinetic energy term consists of a sum of contributions from:
• all generators and the potential energy term similarly consists of a sum of terms across the lines
)(21),( 02
kkL
kiiG
WMVE σσωω −+= ∑∑
line angle differences
Network science ideas�
§ Behaviour determined by interaction of (graph) structure, coupling and node dynamics
§ Concepts of fragility, robustness, vulnerable nodes etc.
§ Results allow for scale, e.g. scale-free relates to granularity
§ Nature finds good motifs so networks can be robust to connection changes (work by Slotine, passivity results)
§ Fits with taxonomies of feeders, dynamics and standardising for plug-and-play (cf. Ilic ideas)
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Power Network Science Ref: Hill and Chen, 2006�
We should cross fertilise complex networks and power networks: • Control theory • Stability theory • Network science • Optimization • Game theory
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Research questions �
§ Unsolved: Rigorous Lyapunov functions for general voltage dependent real power loads
§ Graph based stability criteria, e.g. conditions on cutsets
§ Extensions to linking synch to structure § Robustness to all the uncertainty
§ Dependence on structure » Find the vulnerable points for collapse
§ How to guarantee stability from local checks » certificates (with some exchange) » can these be granulated?
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Stability from a (Physics) network science point of view �
§ Power system models �» Network-reduction model�» Network-preserving model�
§ Small-signal stability: network-reduction model��
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Ref: Motter et al., Spontaneous synchrony in power-grid networks, Nature Physics, 2013.
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X1 = (�01, �02 . . . , �
0n)
>
X2 = (�01, �02 . . . , �
0n)
>
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↵ > �2/4
21
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Transient (angle) stability analysis�
Dorfler, Chertkov and Bullo, Sync. in complex oscillator networks and smart grids, PNAS, 2013
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Questions�
• What are the conditions on the coupling and the
dissimilarity such that a synchronization behavior emerges?
• Under which condition on the network parameters and
topology, the current load profile, and power generations does there exist a synchronous operating point
• When is it optimal, stable, robust?
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Synchronization condition �
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Network-Preserving Model�
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• Generators: second-order differential equations • Loads: first-order differential equations • Non-identical nodes with nonlinear interconnections • Angle stability: (synchronization)
xi = fi(xi) +NX
j=1
aijhi(xi, xj)
Di�i = P 0i �
X
j 6=i
bij sin(�i � �j)
�i = !i
Mi!i = �Di!i + P 0i �
X
j 6=i
bij sin(�i � �j)Generators
Loads/RES
Dynamical Network
!i = !j = !s
�i � �j = cij
No Gij problem!
Role of graph�
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Korsak, “On the question of uniqueness of stable load-flow solutions, 1972
Nonlinearity gives multiple equilibria in angle and voltage��Power networks have another possibility: multiple equilibria arising from the graph�
Stability Theory II �
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What does stability theory look like in terms of dynamical networks and high renewables? People also looking at basic control questions in a more distributed form. Let’s call this Power Network Science?
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Structure Preserving Model�
Bergen and Hill’s model for angle stability analysis augmented
with inverter-based generators
where Pline refers to power flow across transmission lines, E refers
to incidence matrix of the power network. Subscripts M, R, L refer
to synchronous generators, inverter-based generators and loads,
respectively.
M ✓M +DM ✓M = PM �EMPline(ET✓)
DR✓R = PR �ERPline(ET✓)
DL✓L = PL �ELPline(ET✓)
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The Active Power Flow Graph�
The active power flow can be expressed as follows
Define G(θ) as the active power flow graph with the diagonal
matrix of edge weights W(θ)=∂Pline(ETθ)/ ∂(ETθ).
The Laplacian matrix of G(θ) is defined as
Pbus = EPline(ET✓)
L(G(✓)) = @Pbus
@✓= EW (✓)ET
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The Critical Lines and Critical Cutset �
A line is defined as a critical line if the angle difference across it satisfies
Otherwise it is called a non-critical line.
A cutset formed by critical lines is called a critical cutset.
The critical lines induce negative weighted lines in G(θ).
Divide G(θ) into the positive subgraph G+(θ) with positive
weighted lines W+ (θ) and negative subgraph G-(θ) with
negative weighted lines W- (θ) (the critical lines).
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Main Results�
1. Network topology indicates stability:
2. Critical-line based stability criterion: Define
where is the Moore-Penrose inverse, E- is the
incidence matrix of G-(θ), and is the normalized null space
of , the incidence matrix of a spanning forest of G+(θ) .
Then
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Remarks�
3. Stability and the type of UEP is determined by the
locations and weights of critical lines
1. Small-disturbance stability related to the properties of
power network topology
2. Instability results from the appearance of critical lines
4. The appearance of critical cutsets directly leads to
instability
Brief History of PNS�§ Early: Russian schools (Gorev in 1930’s, St Petersburg; Venikov in Moscow, ..);
Magnusson, 1947 Energy functions, dynamics, voltage stability § Lyapunov, Popov methods 1960-70’s: Gorev, Vasin, Pai, Willems, …
§ California 1970’s (Korsak, Smith..) Power flow theory § USA DOE Systems Eng For Power (Fink) 1980's: (Wu, Varaiya in Berkeley;
Baillieu, Zaborszky, etc) .. Differential geometry, stability, control theory
§ 1990's, 2000's: Voltage stability (Andersson, Hill, Varaiya etc) Bifurcation methods
§ 2010's: ‘Smart grids’ a. Modelling issues, stabilization (Ortega, etc) New Lyapunov b. Cascading collapse, synchronism (Dorfler, Motter etc) Network science c. Power flow theory (Low, Tse etc) Convexity d. Distributed control (Johannsson, Hill etc) Control theory
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Outline �
Prof MA Pai��A story of a simple model��DJH projects��Conclusion �
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Projects�§ HKU URC PDF/RAP Scheme�
» Cognitive Technical Systems and Networks�» Distributed Methods for Stability Analysis and Control of Large
Power Networks �
§ HK RGC GRF�» Network-based Stability Analysis and Control of Future Power
Grids ($692,894)�» Problems in Stability Theory for Complex Systems and Networks
($696,029)�
§ HK RGC TBRS�» Smart Solar Energy Harvesting, Storage, and Utilization (Co-I:
total $60.33M across CUHK, PolyU, HKU) �» Sustainable Power Delivery Structures for High renewables (PC:
total $47.12M across HKU, HKUST, PolyU)�36
Sustainable power delivery structures for high renewables
PC: Prof David HILL HKU Leader: Future networks team Co-PI: Prof Ron HUI HKU Leader: Smart loads team Co-PI: Prof Victor LI HKU Leader: Information networks team Co-PI: Prof Li QIU HKUST Leader: Decision and control team Co-I Dr. S.C. TAN HKU Member: Smart loads team
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Proposal�
Mission ��To derive a sustainable structure for the operation, control and protection of future electricity networks delivering harvested renewable energy to consumers who themselves play a demand-side role in the overall system.�
Goals ��§ Develop an integrated approach based on all four key
technical areas, including a novel ‘demand response’ balancing paradigm…; �
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§ Develop PE based electric springs into universal type smart load controllers..basic balancing (including a storage facility) and stability (frequency and voltage); �
§ Integrate DC and IC strategies to study distributed control algorithms … of the sustainable smart grid including self-healing, reduced peak demand and improved security with optimised wide-area communication and sensor networks; �
§ Integrate the outcomes … on improving the energy efficiency and reliability of power system operation, including the needs for Hong Kong delivery to high buildings, islands and an interconnection to a large changing grid; �
§ Further enabling increased consumer participation in power system operation ….; �
§ Establish a regional research and educational hub in energy harvesting and sustainable grid technology ..�
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Modelling emphasis�
§ High renewables
§ Full network (granular to loads)
§ New dynamics
§ Distributed control (wider and granular)
§ Connect to cyber network
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Figure 1 An example of power system (reproduced from [11])
Usual power networks model�
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Basic components of EPS More precisely - Ref: Glover, Sarma and Overbye�
Granulated networks
Cyber-physical analysis�
§ Requires�» Flexible aggregation (with delays for building on-line)�» Communications (type, broadcast, P2P etc)�» Encryption processes�» Sampling rates�» Network latency�» Failures, packet loss etc�» Raw, filtered or aggregated data�
§ All will interact with physical dynamics and affect stability and control performance�
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Australian transmission network �
Source: ABARES - Australian Energy Resource Assessment 45
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Scenarios and Sensitivities�
§ Example (CSIRO FG Forum, 2014)
Ø Scenario: Renewables thrive Ø Sensitivities:
• Farms vs rooftop • Demand-response • EV uptake • …
• How to analyse, plan for all these?
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FG interactions�
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Research projects�
1. Demand-side aggregate modeling
2. Scenario stability analysis
3. Sensitivity analysis (inertia etc)
4. Advanced techniques
a) Vulnerable points b) Stability margins (risk-based) c) Stability scanning
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Role of network structure�
S
C
W
S
C
W
• Generation types�• Transfers�• Inertia distribution �• Load types �
Etc�
FG stability studies�
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Other subjects�
§ Distributed control over network-based models (frequency, voltage)
§ Tsinghua Research Institute for Energy Internet
§ HKU Big Data Initiative – closing the loop idea (Ref: Le Xie work)
§ Future Grid project in China-HK? » Beyond planning to scenario analysis
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Grid2050 Architecture (Bakken et al.)�
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Consensus in power systems�
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What structures work for power systems?�How will they be different for the various tasks?�
Outline �
Prof MA Pai��A story of a simple model��DJH projects��Conclusion �
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From Pai�
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Conclusions�
Thankyou MA Pai
Power network science
