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Hao Zhu
MARMET Workshop
University of Illinois
Assistant Professor
Power & Energy Systems Group
Dept. of Electrical & Computer Engineering
University of Illinois, Urbana-Champaign
March 30, 2015
Acknowledgements: George Moustakides (U. Patras) and Hao-Jan (Max) Liu (TCIPG, UIUC)
Fast Local Voltage Control under
Limited Reactive Power
- Optimality and Stability Analysis
Outline
§ Motivation and context
§ Modeling and problem formulation
§ Local control: optimality and stability analysis
§ Numerical tests
§ Conclusions and future research
The Green Credit: Rise of Solar Energy
§ Currently over 15.9 GW of cumulative solar electric capacity in the U.S.
Source: http://www.seia.org/research-resources/solar-industry-data
The Green Tax : Frequency Stability
§
During disturbances, the inertia of synchronous generators (as rotating
masses) is the key for balancing supply and demand
§
But solar PV generation is highly variable and has no inertia!
Transient freq vs. time
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0
Source: http://www.okiden.co.jp/english/r_and_d/
http://www.nrel.gov/electricity/transmission/variability.html
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The Green Tax & Credit: Voltage Stability
§
§
§
Ideally, the grid should be a constant voltage source (ANSI standard: 114-126V)
Reactive power (VAR) would affect voltage significantly
Solar PV inverters can control VAR output to help regulate voltage!!
Transformer
Tap Control
126V
Voltage
114V
PV Inverter
VAR Control
Voltage control using inverter VAR
§ Much faster compared to conventional voltage control devices such as
– Capacitor banks
– Voltage regulators
§ (De)centralized framework with an optimization objective
–
–
–
–
Cast as an optimal power flow (OPF) problem [Farivar et al’12]
Stochastic approximation with noisy and varying data [Kekatos et al’15]
Consensus averaging for balancing VAR resources [Robbins et al’13]
Distributed optimization using Alternating-Direction Method-of-Multipliers (ADMM)
[Dall’anese-Zhu-Giannakis’13] [Sulc et al’14]
§ Drawbacks of the (de)centralized var control
– Requiring high-quality high-throughput communication, not yet a reality though!
– Communication delay challenges the real-time implementation
– Consequently, stability issue could emerge under the online framework
Dynamic local voltage control
§ Inverter VAR output limited by its rating and real power output
§ VAR control by measuring local voltage [Turitsyn et al’10]
–
–
–
–
–
Can quickly and accurately respond to local voltage variation
Round-robin adaptive control [Yeh et al’12]
Droop control (IEEE 1547.8) [Farivar et al’13]
Delayed droop control to enhance stability [Jahangiri et al’13]
Integral control at unlimited inverter VAR [Zhang et al’12], [Li-Dahleh’14]
Our Goal: A general framework for developing fast local voltage control that
accounts for VAR limits, while offers optimality and stability analysis
Distribution flow model
§ Distribution network (𝑁𝑁, 𝐸𝐸)
– Tree topology system
– Line impedance (𝑟𝑟 + 𝑗𝑗𝑗𝑗)
– Bus complex power load (𝑝𝑝 + 𝑗𝑗𝑗𝑗)
§ DistFlow (DF) : single-phase power flow model [Baran-Wu ’89]
(DF-P)
(DF-Q)
(DF-V)
§ Three-phase flow model for unbalanced systems [Dall’anese-Zhu-Giannakis’13]
LinDistFlow: Linearized DistFlow model
§ Linearize the (DF) equations by assuming
– Minimal system loss (1-2% error)
– Almost flat voltage (1% error)
(LDF-P)
(LDF-Q)
(LDF-V)
§ Albeit an approximation, (LDF) has worked well for most distribution systems
– Viewed as the linearized sensitivity analysis at given operating point (Δ𝑃𝑃, Δ𝑄𝑄, Δ𝑉𝑉)
(LDF) using the graph incidence matrix
§ Concatenate into vector counterparts
𝑄𝑄𝑖𝑖𝑖𝑖 → 𝐐𝐐,
𝑞𝑞𝑗𝑗 → 𝐪𝐪,
𝑉𝑉𝑗𝑗 → 𝐕𝐕
§ Define the incidence matrix 𝐌𝐌𝑜𝑜 and its full-rank submatrix 𝐌𝐌 for graph (𝑁𝑁, 𝐸𝐸)
Line 1
Bus 0
Bus 1
Line 2
Bus 2
Line
1 0
𝑜𝑜
𝐌𝐌 = −1 1 Bus
0 −1
with
𝐌𝐌 =
−1 1
0 −1
(LDFm)
Voltage control under limited VAR
§ For given system and fixed loading, the optimal VAR control problem becomes
Inverter VAR limits
Prop 1: Both matrices
and
are positive definite (PD).
Proof: Diagonals of 𝐃𝐃𝑟𝑟 and 𝐃𝐃𝑥𝑥 all positive (Laplacian-1). (Induction pf in [Farivar et al’13])
In fact,
is the PD power network B matrix (dc couterpart of Y matrix).
§ Linear VAR-voltage implies voltage mismatch
Weighting for
local solvers
(VAR)
Cost of providing VAR
Gradient-projection method
§ (VAR) is a convex quadratic program (QP) with box constraints
§ Define the gradient
Gradient-projection (GP) method leads to iterative update [Bertsekas’99]
(GPm)
where 𝛼𝛼 𝑡𝑡 ∈ 0,1 , 𝐃𝐃 = diag 𝑑𝑑1 , … , 𝑑𝑑𝑁𝑁 > 𝟎𝟎, and
projects to
§ Completely decoupled into local update per bus j !
(GPs)
Optimality analysis
�, the fixed-point of the iterations (GPm) , or
Prop 2: For given system and static 𝐕𝐕
equivalently (GPs) , will be the optimum 𝐪𝐪⋆ to the VAR control problem (VAR).
Proof: First-order optimality of constrained optimization implies
Hence,
and it is a fixed-point to (GPm).
§ Local control achieves the optimum of the weighted VAR problem
– The PD matrix B facilitates the GP iterations that can be implemented locally
– But it may inevitability affect the performance of local control as compared to original
uniform voltage regulation problem
Stability analysis
§ As for any iterative schemes, choices of stepsize is critical for convergence in a
�)
deterministic setting, or equivalently stability in dynamic setting (time-varying 𝐕𝐕
§ Consider first the case of unit 𝛼𝛼 𝑡𝑡 = 1, which boils down to the scaled update
(SCA)
§ Since projection operation is non-expansive
Jacobian matrix
�, local update (SCA) converges to 𝐪𝐪⋆ if
Prop 3: Under the static scenario of fixed 𝐕𝐕
all eigenvalues within unit circle, i.e.,
Scaled VAR control
(SCA)
§ Motivated by Newton’s method, diagonally scale with
§ To ensure stability, the positive 𝜖𝜖 has to satisfy
§ Generalizes earlier result on using integral control under unlimited VAR
§ Convergence speed is related to the conditioning of matrix (X+C), which just
depends on the distribution network case information
§ Hence, the stepsize can be tuned off-line for a given network
Special case: Droop VAR control
§ Interestingly, (SCA) also generalizes the droop control by setting
§ A modified droop update using instantaneous voltage
(DRP)
– Droop slope of −𝑑𝑑𝑗𝑗
– No deadband
§ To ensure stability, we need to have
– Generalizes the results of [Farivar et al’13]
– Droop slope has to be small enough, slowing down the convergence
– The optimization problem (VAR) needs a large enough penalty C on providing VAR
Delayed VAR control
§ General delayed update scheme with forgetting factor 1 − 𝛼𝛼
§ Small 𝛼𝛼 can help stability even for very large D
– Reduces the effective Jacobian to be 𝛼𝛼𝐃𝐃(𝐗𝐗 + 𝐂𝐂)
§ A delayed + droop scheme proposed in [Jahangiri et al’13]
(DD)
§ Our GP-based control framework offers the optimality and stability
analysis of the delayed control method
(DEL)
Features of the local-control framework
§ Requiring minimal coordination with a distribution system operator (DSO)
– Just need to determine the update stepsize off-line
– No communications with DSO or neighboring buses in real time
§ Generalizable to the realistic (three-phase) voltage measurements
– Analysis based on linear (LDF) approximation
– But the algorithms can work with the voltage from ac power flow flow
§ Allowing for asynchronous control updates
– Not all buses need to update VAR at the same rate
– Buses can disable/enable their local update at any time (plug-and-play)
Numerical static tests
§ A radial 16-bus single-phase case with r/x ratio ≈ 0.635 (highly resistive)
§ Each bus loading ~ (70+j30) kVA; VAR limits at ± 100 kVAR (abundant VAR)
§ Static scenario with cost C=0.2; droop control C=0.5 for [0.95, 1.05] voltage bounds
§ NOTE: voltage obtained by solving ac power flow (not the linear approximation)!
Scaled VAR control
§ For the scaled update, best stepsize is around 0.3
Delayed VAR control
§ For the delayed scaled update, stepsize 𝛼𝛼 controls the convergence rate
Dynamic tests
§ Daily profile of house real power load and solar PV generation at every minute
§ Heavy loading during the evening (18:00-22:00)
§ High solar variability in the afternoon (12:00-17:00)
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Load
Solar Generation
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Time (min)
Source: https://archive.ics.uci.edu/ml/datasets/Individual+household+electric+power+consumption
Voltage series with no VAR
§ Over-voltage during the day, and low-voltage in the evening
§ More significant at the end of feeder
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Voltage (p.u.)
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Dynamic voltage control
§ Proposed delayed schemes effectively reduce the voltage mismatch
§ Local control updates every 5 seconds
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No VAR
Delayed Scaled
Delayed Droop
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Three-phase system tests
§ Proposed local updates easily extended to three-phase systems
§ IEEE 123-bus test cases with 30% solar penetration
Three-phase voltage profile
§ Real system voltage obtained from OpenDSS (distribution system solver)
Voltage (p.u.)
– Phase a: black
– Phase b: red
– Phase c: blue
Distance from substation
Dynamic voltage mismatch
§ Proposed method can handle high variability in load and generation
Conclusions
Thank you!
Hao Zhu
[email protected]
§ PV inverters can provide VAR to enhance distribution network voltage stability
§ Extremely fast disturbances motivates the local control framework
–
–
–
–
Gradient-projection iterations decouple into local update
Optimal to a modified version of the voltage optimization problem
Stability conditions through choice of stepsize
Generalizes earlier work on droop, delayed, or integral control
§ Realistic numerical tests show trade-off between speed and stability
§ Ongoing work
– Accelerate the local GP update as a special case of the proximal algorithms
– To a decentralized update using local and neighbor bus voltage measurements
– This can resolve the original voltage optimization problem with no B weighting