Machine Learning and the Power Flow Equations

Hello! My name is Wesley Chan, and I’m a rising junior studying computer science and economics at Northwestern University. This summer I’m interning at Argonne in the CEEESA (Center for Energy, Environmental, and Economic Systems Analysis) division. I’m working with my PI, Dr. Daniel Molzahn, to research the topic of worst-case errors in linearizations of the power flow equations for electric power systems.

What does that even mean? Well to break it down, steady-state mathematical models of electric power grids are formulated as systems of nonlinear “power flow” equations. The power flow equations form the key constraints in many optimization problems used to ensure reliable and economically efficient operation of electric power grids. However, the nonlinearities in the power flow equations result in challenging optimization problems. In many practical applications, such as electricity markets, linear approximations of the power flow equations are used in order to obtain tractable optimization problems. These linear approximations induce errors in the solutions of the optimization problems relative to the nonlinear power flow equations. In addition, characterizing the accuracies of the power flow approximations has proved extremely challenging and test-case dependent.

A depiction of electric power generation, transmission, and distribution in our grid system.

As a result, the research effort Dr. Molzahn is trying to carry out aims to develop new characterizations of the problem features that yield large errors in the power flow linearizations through utilizing a variety of data analysis and machine learning methods. If accurate and reliable characterizations can be made, it would allow power system operators to identify when the results of their optimization problems may be erroneous, thus improving the reliability and economic efficiency of electric power grids.

So what I’ve been working on is building and implementing a number of different machine learning algorithms in order to help accomplish that. One of those algorithms I’ve developed is a multilayer perception neural network using Python and Tensorflow. Using the IEEE 14 bus test case, we were able generate actual optimization results for an AC and DC case using Matlab and Matpower. With the data from those results, I was able to create a dataset with enough samples and features to train on. I would use the neural network model to predict the difference between optimal cost generated from the AC model vs the DC model. The neural network takes in the data, splits it into training and testing sets, and then using forward and back propagation, will iterate through a specified number of epochs, and learn the data, minimizing error on each epoch using stochastic gradient descent.

Because I am still relatively new to machine learning and Tensorflow, I ran into some difficulties trying to build the model. For close to a full week, there was a bug in my code that was yielding an uncharacteristically large error no matter how many epochs I trained the model on. I tried countless different things in order to remedy this. Finally, I realized the bug lied in the fact that I was “normalizing” my input data (a technique I read somewhere online to help deal with varying scales in the features) when I should have been “scaling” it. A simple one word fix helped change my results drastically. With that change, my model went from making predictions with a mean squared error of 600, to a mean squared error of 0.8. Given that the range of optimal cost difference was between 300-600 dollars, a mean squared error of 0.8 was less than a 0.01% average error.

Following that, I’m now working on generalizing the neural network model to predict other relevant aspects such as the power generation from each bus, and the power generation cost of each bus. I’m excited to gain more hands on experience with machine learning, to work more on this topic, and to see what kind of results we can get!