1. Machine Learning-Based Power Flow Solver
GridLAB-D TAC Meeting
September 6, 2019
Hi my name is lily Buechler, I’m a PhD student at Stanford, studying mechanical engineering
I’m going to talk about our preliminary work on developing a machine learning based power flow solver that will be integrated into Gridlab-D
A lot of this work was done by another phd student in our group, Siobhan powell, who has put a lot of work into this over the last few months.

2. The Power Flow Problem
At the core of any grid simulation engine, like gridlabd, is a power flow solver
Whose purpose is to solve the so-called powerflow equations
This system of equations, relates the real and reactive power injections at each bus in the network to the magnitude and phase of the voltage at each bus
Generally, in grid simulations, the real and reactive power, are the inputs to the problem
As well at the topology and characteristics of the network (admittance matrix)
And the voltage magnitude and phase, are the variables that you are trying to solve for
These equations implicitly assume that the voltages at any given time instant, are only dependent on the power injections at that time instant, and not dependent on any previous history
That assumption, which is called quasi-static power flow, is a key assumption used in many simulation engines that simplifies this power flow solution
By decoupling the solver solutions in time
Now these equations are highly nonlinear, which means that it is impossible to derive an explicit expression for the outputs, which are the voltages, in terms of the inputs

3. Standard Approach: Newton Raphson Iteration
Newton Raphson method:
Iterative root finding method
Seed initial guess with last solver solution
New approach: replace Newton-Raphson solver with data-driven approximation:
Initial guess
Evaluate function
Linearize
model
Update guess
So in order to solve the equations, you have to take an iterative approach
There are various iterative methods for solving the powerflow equations
Standard one is called newton Raphson iteration
Which is an iterative root finding method
Basic idea, is that you want to find the solution to an equation
So you first guess a solution
Then you evaluate the function at that solution
Then you linearize the model at that point, as an approximation
Which in this visualization is just the tangent line to the curve
And then you use that linearization, to make a better guess
And you repeat that until your guess is close enough to the actual solution
Newton Raphson has been used extensively for powerflow
And it very efficient
Each iteration might only take a couple of milliseconds and the algorithm can typically converge in only a few iterations
This issue is that generally an extremely large number of solutions are needed, each time you use the grid simulator
And those add up
But say you are using Gridlabd to do a grid integration study
And you have tens or hundreds of scenarios you want to evaluate
And you want to run each simulation over say several months or years
With a timestep on the order of seconds to minutes to hours
That is going to require tens to hundreds of millions of NR solver solutions
And that can be very computationally expensive
So the idea that we are proposing here, is instead of iteratively solving these equations
We can use methods from statistical learning and data science to learn the relationship between these variables
And then solve for the solution explicitly (requiring no iteration), which is much more efficient

4. Literature on Data-Driven Power Flow Approximation
Yu, Jiafan, Yang Weng, and Ram Rajagopal. "Robust mapping rule estimation for power flow analysis in distribution grids." 2017 North American Power Symposium (NAPS). IEEE, 2017.
Development of SVM-based power flow approximation
Forward and backward power flow mappings
Comparison with regression-based methods
Our group at Stanford has done some past work in this area
PhD student Jiafan Yu, who recently graduated, did some work a couple years ago on developing a support vector machine based approximation to the powerflow equations
Compared that to regression based methods
Generally found that SVMs are more accurate than linear regression
But are more computationally expensive
If you use the right kernel functions, you can actually recover the actual power flow equations with high degree of accuracy
This study was a great proof of concept

5. Challenges for Implementation in GridLAB-D
Model formulation
3-phase unbalanced power flow
Current injection method
ZIP load representation
Scalability
Which modeling methods work well in high dimension?
How much training data is needed?
Accuracy
How often should the model be re-trained?
Can model accuracy be predicted?
Computation speed
Can the number of Newton Raphson iterations be predicted?
How does an ML-approach compare to existing methods for speeding up power flow simulations?
ZIP load model
But there are a number of challenges that have to be addressed to actually implement this type of methodology in a simulation engine like gridlabd
Some of these we have analyzed already, but there are still many more we will be investigating in the future
First, gridlabd simulates 3-phase unbalanced powerflow, as opposed to single phase, which is what has been looking at in the past, for this data-driven approximation
Which makes the model formulation more complicated
Also, In the original study, the loads were assumed to be constant power loads
Which means they are not dependent on voltage
But gridlabd, many load models use a ZIP load representation, where the power consumption is a polynomial function of the voltage
So our algorithm has to be adapted for that as well
Scalability is a key requirement
Certain methods like svm, may have high accuracy, but don’t scale well to high dimension
We want these methods to be useful for large networks
Different methods also require different amounts of training data to learn a good model
Another key requirement is accuracy
The general idea is that you would use the outputs from the standard NR solver, to train the model
What training data is optimal to learn a model that we know is going to be in a certain operating regime most of the time
Standard NR method has a guaranteed degree of accuracy
The question is can we do the same for a data-driven model
Can we estimate the model accuracy for certain model inputs, without having to use NR to verify
Can we design methods to decide when the model should be retrained? For example when the topology of the network changes
And finally, computational speed
Can we estimate the number of newton Raphson iterations that it is going to take to solve for a solution, before doing so, and use that information to decide wither or not to use a data-driven model?
In terms of computation time, how does this method compare to other methods in the literature on speeding up power flow solvers
Variable timestepping, parallelization, vector quanitization, which is where you reuse solutions that have similar inputs

6. Cluster-based Linear Regression Model
Learn separate model for each bus
Regularized least squares regression:
Cluster training data into different operating modes and train a different model for each
Clustering methods:
Gaussian mixture model
K-means clustering
K-means clustering of residential load profiles
We started out by looking at the simplest and fastest methods, to form a performance baseline
We started using a very simple linear regression
And looked at ways to improve accuracy
Here the features are the real and reactive powers at nominal voltage
Samples are different timesteps
And the variables we are trying to predict are the voltage magnitude and phase
Learned a different model for each bus in the network
Estimate the parameters using regularized least squares regression
Couple observations we made
One is that for typical operation of the network, the power flow equations are actually fairly linear and linear regression does a pretty good job
The second is that load profiles generally have different operating modes, driven by typical load profile behavior that varies throughout the day
Perhaps learn a better model, by training different models for different operating regimes
We developed this cluster-based linear regression model
Where we cluster the training data into different operating regimes
And then train a different model for each
Tried 3 different methods – GMM, K-means clustering, and a heuristic, where we clustered by day of the week
This is an example of clustering the data from about 15 residential load profiles into 4 different clusters, with respect to time, using K-means clustering
Generally those operating regimes end up corresponding with different times of day

7. Error Checking and Solver Integration
Voltage vector:
Error metric: normalized vector error:
Interaction with NR solver:
Error check
Distance check
Step change check
We have also looked at the best way to decide when to use the data-driven model and when to use the NR solver based on the observed prediction error
Error metric that we used was the normalized vector error expressing the voltage in rectangular coordinates taking the maximum over all buses
First step is to split the data into a training and test set, and train the model on the training set
For each test sample, use the trained model to make a prediction
Using a set of tests, decide whether or not to use the estimate
Tried 3 different tests
Error check – every time you use the NR solver, also test the model approximation.
And to use the model approximation again, the last computed error needs to be below a certain threshold
This method assumes that the inputs to the model move somewhat slowly through different operating regimes
Distance check – Observed that high prediction error tends to occur for points that are at the edge of a cluster
Require that the model inputs are within a certain percentile distance away from the cluster center to use the model
Step change check – checks if there is a significant jump between the last simulation solution and the current model estimate
Only uses the model if that change in time is below some threshold
Also assumes that the solution won’t change much from one solution to another

8. Test Case: IEEE 123 Bus Network
IEEE 123 bus model
Replace spot loads with time-varying real and reactive power profiles for 344 residential homes
4.2 kV nominal voltage
4 voltage regulators
GridLAB-D simulation
7 day training
21 day testing
1-minute simulation timestep
Training and testing data breakdown
We tested these methods using the IEEE 123 bus network model in gridlabd
Replaced the spot loads with time-varying real and reactive power profiles for 344 homes
Network operates at 4.2 kV
4 voltage regulators
We tested these methods offine, but running a 31 day simulation and breaking it down into a training set and a test set
We discarded the first 3 days, because it generally takes a little while for the system to equilibrate, because of the simulation initial conditions
7 days of training
21 days of testing
Used a 1 minute timestep

9. Results: Cluster-Based Modeling
Training error
Testing error
Mean prediction error: 0.2%
Base Case: no clustering
K-Means test error
GMM test error
Cluster-based models (k=7):
Here are some of the results for prediction accuracy, using the model approximation 100% of the time
The top 2 plots here are for the base case with no clustering
Training error and testing error distributions look relatively similar
Mean test error is about 0.2%
That’s not as low as the NR solutions, but it is reasonable
The cluster-based models did reduce the mean error even further
Most significantly for K-means clustering
Although there still is a long tail to the distribution

10. Results: Error checking
Final model performance:
Avoided solves
86.7%
0.0168
Error above 0.01
0.05%
Parameters of error checks were tuned using sensitivity analysis
Best performance results from a combination of error checks
Significant reduction in number of Newton Raphson solves
Final error distribution using error checks:
For the error checking algorithm, we used a sensitivity analysis to tune the parameters and thresholds of the error checks
And found that the best performance results from a combination of the error check, distance check and step change check
Using the optimized model, we saw an 86% reduction in the number of NR solves
Median test error was 0.04 %
Maximum test error was about 1.6%
But with a very small number of samples with errors above 1%

11. Results: Load Composition and Loading Level
ZIP load model
We also did a sensitivity analysis to analyze how the prediction error is dependent on the loading level, as well as the load composition
One observation of the pervious study, is that model accuracy is significantly dependent on loading level, for both SVMs and regression-based methods
To evaluate loading level, we proportionally scaled the load profiles over a wide range
To evaluate load composition, we tested the model for 3 cases: constant current, constant power, and constant impedance loads
Which are the 3 extremes in terms of load behavior
We found that voltage magnitude error is highly correlated with loading level
The impact of load composition is less significant
But the error does tend to be more significant for constant power loads

12. Example: Voltage magnitude prediction
Low loading level
High loading level
Constant impedance loads
This is just an example of the predicted and actual voltage profiles for a specific bus, with a low loading level and high loading level, the only difference being the scaling factor applied to the real and reactive power profiles
For the low loading level you can’t even distinguish the difference between the two
The difference starts to become appartent for the high loading level, but the approximation still emulates the general behavior

13. Results: Effects of Training Set Size

14. Next steps:
Comparison of SVM and other ML-based methods with linear regression
Scalability – test accuracy on larger networks (IEEE 8500 node network)
Optimal training set design
Algorithm implementation with online interaction with GridLAB-D Newton Raphson solver

15. Questions?

16. Extra Slides