Data Driven Identification of Power Plant Operation States Using Clustering

Project Entities:

Chair of Prof. Dr. Christoph Weber (Management Sciences and Energy Economics)

Chair of Prof. Dr. Florian Ziel (Data Science in Energy and Environment)

  Project Goal:

Use publicly available data (particularly ENTSO-E Transparency Platform) to estimate parameters for energy system and energy market models.

Overview

Empirical identification of states

3-Step Approach:

• Prior Partitioning
  • We create preliminary clusters
  • They will be used to initialize the main clustering
• Main Clustering
  • Gaussian Model Based Clustering
• Label Assignment
  • We assign meaningful labels to the final clusters

Prior Partitioning

Divide the space in meaningful partitions:

Define the Capacity: \(\zeta = max(t0)\)

Define a threshold: \(\gamma = \frac{\zeta}{50}\)

\(\pm \gamma\) around the diagonal: Stable
\(t0 < 1\) & \(t1 < 1\): Zero
\(t0 < \gamma\) & \(t1 > 1\): Startup
\(t0 > 1\) & \(t1 < \gamma\): Shutdown
\(t1 > t0\): Ramp-Up
\(t1 < t0\): Ramp-Down

We project Stable observations onto the diagonal, Startup on \(t1\) and Shutdown on \(t0\) for the next step.

Prior Partitioning

Model-Based Clustering of the Regions using mclust::Mclust in R.

• Stable: 2-5 Clusters
• Ramp Up: 2-4 Clusters
• Ramp Down: 2-4 Clusters

Obtain finite mixture distribution:

\[\sum_{k=1}^{G}{\pi_k f_k (\mathbf{x}; \mathbf{\theta}_k)}\]

\(f_k\) Density of k’s component
\(\pi_k\) Mixture weights
\(\theta_k\) parameters of k’s density component

\[f(\mathbf{x}; \mathbf{\Psi}) = \sum_{k=1}^{G}{\pi_k \phi (\mathbf{x}; \mathbf{\mu}_k; \mathbf{\Sigma}_k)}\]

\(\phi(\cdot)\) Multivariate Gaussian density

Maximum Likelihood Estimation via Expectation Maximization (EM) algorithm

Likelihood for Gaussian Mixture Models (GMMs):

\[\begin{align} \ell(\Psi) = \sum_{i=1}^n \log \left\{ \sum_{k=1}^G \pi_k \phi(x_i; \mu_k, \Sigma_k) \right\} \end{align}\]

We Re-Formulate this likelihood to a complete-data likelihood to utilize the EM algorithm

\[\begin{align} \ell_{\mathcal{C}}(\Psi) = \sum_{i=1}^n \sum_{k=1}^G z_{ik} \left\{ \log \pi_k + \log \phi(x_i; \mu_k, \Sigma_k) \right\} \end{align}\]

\[\begin{align} z_{ik} = \begin{cases} 1 & \text{if } x_i \text{ belongs to component }k \\ 0 & \text{otherwise.} \end{cases} \end{align}\]

E-Step:

\[\begin{align} \hat{z}_{ik} = \frac{\hat{\pi}_k \phi(x_i; \hat{\mu}_k, \hat{\Sigma}_k)}{\sum_{g=1}^{G} \hat{\pi}_g \phi(x_i; \hat{\mu}_g, \hat{\Sigma}_g)}, \end{align}\]

M-Step:

\[\begin{align} \quad \hat{\mu}_k = \frac{\sum_{i=1}^{n} \hat{z}_{ik} x_i}{n_k}, \quad \text{where} \quad n_k = \sum_{i=1}^{n} \hat{z}_{ik}. \end{align}\]

Prior Partitioning

Initialization

We initialize the EM algorithm (E-Step) using the partitions obtained from model-based agglomerative hierarchical clustering (MBAHC)

Estimation

The Bayesian information criterion (BIC) is used for model selection

Prior Partitioning Results

Right graph shows prior clusters.

Main Clustering

MBAHC

Prior Clusters are used in MBAHC

The results of the MBAHC are used to initialize the EM Algorithm in the main Gaussian Model Based Clustering

Main Clustering Results

Right graph shows Maximum A Posteriori (MAP) Classification

Colour indicates cumulated log(density) of all components.

We assign labels to the clusters using their mean \(\mu\) and correlation \(\rho\)

Multiple clusters may describe one Generation State (e.g., along the diagonal)

Label Assignment

Right graphs show assigned states

The points are coloured according to

• MAP
• Probability (each pure colour reflects a probability of 1)

Some points below /above the diagonal are assigned to Ramp Up / Ramp Down

• Can be easily fixed for MAP
• Fixing probabilistic predictions not that easy

Label Assignment

Fixing assignments

Relabeling Ramp Up and Ramp Down MAP predictions is trivial:

\[\begin{align} \text{State} = \begin{cases} \text{Ramp Up} & x_{t1} > x_{t0}, \\ \text{Ramp Down} & x_{t1} < x_{t0}. \end{cases} \end{align}\]

Fixing the probability array is more involved:

Find observations \(x_{t1} < x_{t0}\) that can not be “Ramp Up”:

Set probability of all Ramp Up clusters to \(0\).

Normalize the probabilities.