PROBABILISTIC LOAD FLOW MODELINGElectric Grid, Renewable Energy & Plug-in Electric Vehicles

Trevor Williams - PICS, University of Victoria, Department of Mechanical EngineeringSupervising Professor: Dr. Curran Crawford, Department of Mechanical Engineering

Research Abstract Motive: Grid complexity is increasing with technological innovation, the

growth of Distributed Generation (DG) such as combined heat and power, solar, wind turbine farms, run-of-the-river generation and unpredictable loads such as Plug-in Hybrid Electric Vehicles (PEVs).

With limited control, little curtailment or reactive power control, and low capacity factors, DG could adversely affect power quality, generation efficiency or impact existing infrastructure capacity limits and reliability.

Renewable DG sources are often probabilistic, with regional, seasonal and daily variations, complicating operational grid control, and requiring increased power system reserve.

Task: develop and implement a probabilistic multivariate modelling methodology to study the integration of renewable energy sources and PEVs into the electrical grid.

Probabilistic Load Flow (PLF) vs MonteCarlo (MC)

Existing probabilistic techniques often rely upon computer resource inefficient MC techniques.

PLF modeling is a more efficient alternative that is gaining popularity as power grid complexity increases due to intermittent renewable energy generation and unpredictable loads such as PEVs.

Capabilities and ResultsThe PLF methods have been coded in MATLABTM, for both Tranmission and Distribution Grids which have distinctly different functions and physical characteristics.

The code is called FEPES (Finite Element Probabilistic Energy System Simulation) and provides:

• DC and AC Transmission Grid probabilistic modeling (with Normal, Weibull and Binomial Distributions)

• 3-Phase AC Distribution Grid - applying a Forward/Backward Sweep algorithm

• Large scale data handling and graphical output

• PDF generation (Gram Charlier, two Maximum Entropy methods and Cornish Fisher1)

• Outage analysis, voltage stability and comparative metrics

• Much faster than MC methods

• Accurate mean, standard deviation and tail region PDFs

Contribution and FutureFEPES is currently being improved to:

• Incorporate a second order linearisation to improve covariance calculations and better standard deviation predictions for non-linear power load flows.

• Include multi-linearisation to improve non-gaussian calculations.

Methodology Summary Scope: In PLF, conventional generation dispatch and grid

configurations are treated as discrete random variables while loads and variable generation (like wind) are treated as continuous random variables.

Outages, network changes, load variation, and DG are probabilistic, so deterministic LF analysis is inadequate.

System analysis requires PLF or MC techniques to obtain Probability Density Functions (PDFs) of state vectors (voltage magnitude and angle) and line flows (power, current, losses) of a statistically varying electrical network in a single shot solution.

Solve: (1) Linearisation of non-linear AC power injection and load flow equations around expected mean solution point.

(2) Convert probabilistic input to cumulants, convolve and generate PDFs.

Cumulants

Define the probabilistic inputs.

Simpler mathematically than moments.

Are semi-invariants: 1st cumulant is the mean, 2nd is variance, 3rd is skew, 4th is kurtosis, etc.

Applications

The FEPES analysis tool can be applied to:

• Analyse renewable energy generation, random loads (PEVs) and storage (PEV vehicle-to-grid (V2G)), avoid distribution grid overload, make future load predictions, to improve unbalance and reduce power loss, as well as predict voltage instability.

• Analyse renewable energy generation (wind farm) effects on transmission systems, power flows and voltage stability.

• Combine complex and large-scale transmission and distribution systems to optimise renewable energy and storage in an efficient power generation grid.

• Provide predictions for decision-making on grid renewable energy, storage and PEV policy development.

Fig.1 Probabilistic Load Flow PDF versus increasing number of MC simulations

Fig.2 Power injections equations

Fig.6 Real and reactive power flow equations

Fig.7 BC Hydro predicted Vancouver PEV ownership

Fig.5 Transmission Grid schematic showing probabilistic generation and load definition

Table.1 AC analysis and PDF reconstruction speed comparison

Table.2 AC analysis accuracy

Fig.3 Convolve cumulants Fig.4 Cumulant order

footnotes: (1) collaborative code from Eric Hoevenaars (UVic., Dept. Mech. Eng.) (2) image and schematic courtesy of IEEE PES

SSDL Sustainable SystemsDesign Laboratory

slack

Bus1

72 MW 27 Mvar

Bus 4

Bus 5

125 MW 50 Mvar

Bus 2

163 MW 7 Mvar

Bus 7 Bus 8 Bus 9 Bus 3

85 MW -11 Mvar

100 MW 35 Mvar

Bus 6

90 MW 30 Mvar

slack

Bus1

72 MW 27 Mvar

Bus 4

Bus 5

125 MW 50 Mvar

Bus 2

163 MW 7 Mvar

Bus 7 Bus 8 Bus 9 Bus 3

85 MW -11 Mvar

100 MW 35 Mvar

Bus 6

90 MW 30 Mvar

Ratios GC ME-NM ME-FL MATPOWERMATPOWER/Other (seconds)

AC Normal 7.8 9.4 2 407AC Weibull 10.8 10.9 1.2 413

AC Binomial 1.9 8.3 2 409

ARMS Voltage Magnitude (mean/worst case (%))

AC GC ME-NM ME-FLNormal 0.059 / 0.090 0.050 / 0.071 0.067 / 0.097Weibull 0.527 / 0.596 0.500 / 0.588 0.454 / 0.568

Binomial 0.033 / 0.051 0.039 / 0.056 0.033 / 0.053

ARMS Voltage Angle (mean/worst case (%))

AC GC ME-NM ME-FLNormal 0.020 / 0.030 0.050 / 0.080 0.020 / 0.030Weibull 0.200 / 0.280 0.200 / 0.260 0.150 / 0.210

Binomial 0.020 / 0.030 0.020 / 0.030 0.020 / 0.030

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The Pacific Institute of Climate Solutions (PICS) makes funding available to support this research and their contributions are gratefully acknowledged.