As industrial systems (such as smart grids) grow in size and complexity, transparency and explainability become increasingly important for supporting user understanding and trust. Causal models are key components to translate system behaviour into human reasoning. While causal graphs can be created from domain-expert knowledge about system dynamics, this method carries the risk of specifying only partially complete graphs. Yet, attempting to learn the entire causal graph from observational data is also known to be challenging and error-prone. In this paper, we propose a procedure for the data-driven augmentation of existing causal graphs defined by domain-experts. Specifically, we test Granger non-causalities implied by the existing graph on sensor measurement data. Under appropriate statistical and causal assumptions, the test results can then indicate missing edges in the existing graph. We evaluated our approach in a real-life smart charging garage scenario, testing the expert-defined causal graphs with real-world sensor data. The results show inconsistencies between the causal graph defined by experts and the observed data, demonstrating the capability of our approach to reveal missing causal relations. These findings highlight the potential of our approach in combining expert knowledge and data-driven analysis for validating and augmenting causal representations in complex systems, such as smart grids, contributing to the broader topic of transparent and interpretable industrial systems.
In cyber-physical systems (CPSs), computational elements tightly interact with physical processes
through sensors and actuators. This enables smart monitoring and control of these systems, as well as
increased automation. In a smart grid, as an example of a CPS, power flow can be handled eficiently
through real-time monitoring. However, it is important to keep system behaviour understandable for
users (e.g., grid operators responsible for a stable grid operation), especially when some unexpected
events (i.e., anomalies) occur. To ensure transparency of such systems for human stakeholders, causal
models of the system can be employed to make system processes more clear, and to trace anomalies
back to their root causes. Explanations based on causal knowledge are important to match users’
expectations and reasoning patterns [
Traditionally, causal graphs are constructed by domain experts, who propose a set of causal relations
between system components based on their extensive domain knowledge. This process is not only
time-consuming, but it is also error-prone. For instance, in the biomedical domain, Kook [
CEUR
ceur-ws.org
Battery 1 AP_EV1
In this work, we aim to support the process of causal knowledge acquisition using a data-driven approach. To this end, we investigate Granger non-causality tests to falsify and amend an expert-defined causal graph based on time series data. This method can be used to check if there are discrepancies between a proposed causal graph (e.g., causal knowledge acquired from domain experts) and the time series data, e.g., if there is evidence for additional causal relations in the data that were missed in defining the initial causal graph.
We evaluated this approach on measurement data of an electric vehicle (EV) charging garage from
our research project 1. The garage is equipped with a set of sensors, to measure the power consumption
and state of charge of diferent devices in the system. In Figure 1a, the garage with its devices and
sensors is shown. It contains 8 EV charging stations and a battery, which can be used for peak shaving
(i.e., reducing peak power consumption of the garage by providing power from the battery). There are
multiple sensors installed in the system (shown as blue circles). The garage experiences an overload if
the system draws too much power, indicated by AP_G1 exceeding a certain threshold. A causal graph
between sensors is needed to determine potential sources of such an overload. Over multiple workshops
and discussions, experts have proposed a candidate causal graph to be used for generating explanations
of anomalies. The expert-defined causal graph is represented in Figure 1b [
Our evaluation result shows that Granger non-causality tests can be used to identify inconsistencies between the provided causal graph and the observed sensor data. While the statistical tests indicate several possible scenarios of missing causal relations, this information can be presented to domain experts for further analysis and discussion, enabling the development of a more complete and consistent causal graph.
The rest of the paper is structured as follows. In Section 2, we discuss related work on CPS and causal discovery. Then, our proposed methods are explained in Section 3, followed by experimental results in Section 4. Finally, we discuss our contributions, limitations as well as planned future work in Section 5.
Explainability of CPS is a key factor to make existing systems more transparent for human users. This
aligns with the European Commission’s vision of Industry 5.0, to create human-centric industrial systems
beyond eficiency and productivity [
The Semantic Web research community proposed a diferent approach which focuses on modelling
only relations between sensors and variables that are interesting to system users, thus reducing the
eforts to model a fully engineered system [
Learning causal relationships from observational time series data (e.g., sensor data in CPS) can help
experts in the process of creating a semantic causal representation of their system [
Our work is positioned at the intersection of knowledge acquisition and statistical causality as solution domains, while we focus on CPS as a potential problem domain for our proposed methods. Specifically, we investigate the use of Granger non-causality tests as a method to falsify and amend a causal graph that was provided by domain experts. Thus, domain knowledge and expert assumptions are tested against existing sensor data to check if it is consistent with observed values. Furthermore, this method can also provide suggestions for further causal relations that have been overlooked by experts initially.
We propose a procedure to find inconsistencies between an existing causal graph, derived (e.g., through expert knowledge) and sensor measurements. As shown in Figure 2, the initial causal graph implies assumptions about Granger causalities in the system which can be tested against data from system sensors. If statistical tests of Granger non-causality indicate discrepancies between the graph and the data, the graph is considered falsified. The results of these tests can suggest missing edges in the graph, which the domain expert can use to amend the existing graph and create a graph that is more consistent with the data.
Expert knowledge about causal relationships in a CPS can be encoded in a causal graph, in which edges indicate the direct influence of the history of a process on the future of another process (such as in Figure 1b). Assuming the graph is induced by a structural causal model (as in Equation 1), the graph encodes causal assumptions in the form of graphical separations, called -separations. By the causal Markov property, these separations imply conditional independencies (or Granger non-causalities) in the observed data distribution [14, 15]. By using data from the CPS, testing those implications allows us to falsify and amend the given causal graph. We briefly introduce the necessary tools from graphical modelling and Granger causality, before describing the proposed falsification procedure in detail. sensor data ⋯ 1 ⋯ 2 ⋯ 3
2 +11 +21 +31 1 3 causal models for time series [16, Chapter 10.1]. Let ( )∈ denote a multivariate time series, in which each coordinate of ∈ ℝ corresponds to a measurement of a sensor in the CPS and ⊆ ℤ the discrete time domain. We assume the time series is generated by a structural causal model (SCM), denotes such that, for all ∈ , initial causal graph amended causal graph system domain expert domain expert
Granger non-causality
tests +13 ⋯ +23 ⋯ +33 ⋯ 1 3 +12 +22 +32 2 induces a so called full-time graph (Figure 3a) in which there is an edge from to where denotes the causal mechanism (assumed to be constant over time), denotes the history of the process before time point and are independent and identically distributed noise terms. The SCM ′ if appears in ′. Based on the full-time graph, we define the summary graph (henceforth referred to as the causal graph) with node set {1, … , } and in which there is an edge from to , if and = ( , ), (1) ′ (for arbitrary time points < ′) in the full-time graph the structural equation of
(Figure 3b). only if there is at least one edge from to
We next give the necessary prerequisites on graphical models based on the more detailed expositions in [15, 17]. The causal graph = ( , ) , with node set = {1, … , } and edge set ⊆ × , encodes the dependence structure of the coordinates of using directed edges (←, →). A walk, 0 ∼1 1 ∼2 … ∼ +1 , is an alternating sequence of nodes and edges in , where ∼ is a placeholder for any edge. A node on a walk is a collider, if −1 nodes and and let ⊆ {1, … , } → ← +1 and a non-collider otherwise. Consider a walk between
. We call a -connecting walk from to given , if (i) ∉ , (ii) every non-collider on is not in , (iii) every collider on is among the ancestors of , and (iv) has a head at . Finally, for , , ⊆
, we say is -separated from given if there exists no -connecting walk from any ∈
to any ∈ given , and write ↛ ∣ .
Granger causality is a predictive notion of causality in time series [18, 16, Chapter 10.3.3]. Informally, a process is Granger causal for another process given another process , if improves the prediction of when already accounting for [19]. Formally, we define the negation of Granger causality, i.e., Granger non-causality, as the following conditional independence, for all , ⟂⟂ ∣ . are testable conditional independencies in the observed data distribution.
∪{} Under mild technical conditions, there is a close connection between Granger causality and -separations in causal graphs: If is -separated from given ∪ {} , then is Granger non-causal for given [14, 15]. The Granger non-causalities implied by the -separations in a candidate causal graph
We list all -separations of the form ↛ ∣ ∪ {} implied by the candidate causal graph for || ≤ . Here, is a user-defined constant that can be chosen based on computational tractability, as a larger results in a larger number of tests. As outlined above, each -separation implies a testable Granger non-causality, 0(, , ) ∶ for all ,
⟂⟂ ∣
∪{} . Assuming linear relationships between Δ = − −1 and , ∪{} and stationarity of Δ , we perform Granger non-causality tests as a paired -test comparing the out-of-sample mean squared prediction error contributions of two autoregressive linear (AR() ) models in which one model includes while the other does not. The chance of falsely rejecting at least one null hypothesis is exacerbated when multiple tests are performed. Therefore, we apply a Bonferroni-Holm correction to control the familywise error rate across all tested Granger non-causalities. Small adjusted -values indicate discrepancies between the data and the candidate causal graph and indicate missing causal connections. The graph can then be amended by an expert based on the results of the Granger non-causality tests.
We applied the proposed falsification procedure on a real-life dataset of a smart charging garage. In total, a set of 39 sensors is installed at the facility, collecting data in 1 minute intervals. The sensors measure diferent variables, including active power, voltage, state of charge, energy and temperature. In our experiments, we included 12 sensors (measuring active power and state of charge), which have been identified by domain experts to be relevant for potentially identifying root causes of a power overload (see the garage setup in Figure 1a).
We used the candidate causal graph from Figure 1b and conditioning sets of size at most = 1 to limit the number of tests, considering computational tractability and power of the procedure. In total, the graph implies 1025 -separations at the chosen . We applied the corresponding tests of Granger non-causality using AR(10) models across four disjoint time periods of at least 8000 minutes and combined the -values for each hypothesis by taking the smallest -value after a Bonferroni-Holm correction over all four time periods. We then adjusted the resulting 1025 -values again using a Bonferroni-Holm correction. The results for all -separations rejected at the 5% level are shown in
The procedure suggests that the -separations (i) AP_B1 ↛ SoC_B1 given any of the EV chargers and (ii) AP_G1 ↛ SoC_B1 given the sum of all EV chargers are not in line with the data at the 5% level. Therefore, there is strong evidence that either AP_B1 and AP_G1 cause SoC_B1 directly, or that there are unobserved processes that influence AP_B1 and SoC_B1 or AP_G1 and SoC_B1. The amended graph in
Adjusted -value bidirected edge (orange) was added between and . the ambiguity cannot be resolved based on the given data, a domain expert may be able to judge the plausibility of these suggested edges.
From a domain-perspective (i) suggests that active power from the battery (AP_B1) could have an efect on the state of charge of the battery ( SoC_B1) according to observational data. Since power consumption of the battery is used to either discharge, or charge the battery, this efect is plausible.
Furthermore, (ii) would suggest that power consumption of the garage (AP_G1) is related to the state of charge of the battery (SoC_B1), when controlling for EV charging overall (AP_AEV1). The state of charge of the battery usually changes either by discharging (usually through providing power to EV chargers), or by charging (power is consumed from the external grid). Thus, this yet undefined relation between (AP_G1) and (SoC_B1) should be further investigated by domain experts.
Causal relations are a key component for human users to be able to interpret CPS behaviour. In this paper, we have addressed the challenge of constructing a causal graph for such systems. While causal graphs that are informed by domain experts are grounded in extensive domain knowledge, they can be incomplete as experts might miss certain causal relations. Data-driven approaches can uncover such relations. However, learning entire causal graphs from observational data alone can be dificult and unreliable.
To bridge this gap, we propose testing implied conditional independencies that are created by a causal graph from system experts on observational sensor data. Specifically, we apply the proposed tests to a real-life use case of a smart charging garage, where active power and state of charge sensor measurements are available that correspond to the proposed causal graph. Granger non-causality tests are applied to check if the causal graph aligns with existing sensor data.
In our experiments, we have successfully identified some inconsistencies between the provided causal graph and sensor data. While the statistical tests suggest multiple scenarios of which causal relations are missing, this knowledge can be presented to domain experts for further discussion, to define a more complete and consistent causal graph.
Currently, we have applied our approach only to observational data, testing for statistical independence. While we can, in this way, investigate predictive relationships between variables, causal relations can be investigated with more confidence and in more detail by actively manipulating the system (e.g. disconnecting an EV charger, or starting fast-charging sessions). Obtaining data from diferent interventional settings could allow the use of causal discovery methods that rely on diferent data sources [e.g., 20] and thereby greatly improve the causal understanding of the system. Additionally, the Granger non-causality tests used in this work rely on parametric modelling assumptions. Under model misspecification, the test results may be unreliable and reflect inflated type I error rates, rendering expert-based evaluation of the results invaluable.
In future work, we will investigate how the proposed falsification procedure performs on data from the system under interventions. Furthermore, we will investigate the potential of expert validation as a more integrated causal discovery process, potentially integrating multiple causal discovery methods from experts, data and LLMs. Finally, nonparametric tests of Granger non-causality can be considered to alleviate the problem of misspecification.
This work was supported by the FFG research project SENSE (project Nr. FO999894802), and the Austrian Science Fund (FWF) Bilateral AI project (Grant Nr. 10.55776/COE12).
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