Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdős-Rényi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n ) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction

Learning Graph Parameters from Linear Measurements: Fundamental Trade-offs and Application to Electric Grids

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We study fundamental trade-offs between the number of measurements (sample complexity), the complexity of the graph class, and the probability of error by first deriving a necessary condition (fundamental limit) on the number of measurements. Then, by considering a two-stage recovery scheme, we give a sufficient condition for recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdös-Rényi (n, p) class, the sample complexity derived from the fundamental trade-offs is tight up to multiplicative factors. In addition, we design and implement a polynomial-time (in n) algorithm based on the two-stage recovery scheme. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness of the proposed algorithm for accurate topology and parameter recovery

Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdős-Rényi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n ) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction

Learning Graph Parameters from Linear Measurements: Fundamental Trade-offs and Application to Electric Grids

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We study fundamental trade-offs between the number of measurements (sample complexity), the complexity of the graph class, and the probability of error by first deriving a necessary condition (fundamental limit) on the number of measurements. Then, by considering a two-stage recovery scheme, we give a sufficient condition for recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdös-Rényi (n, p) class, the sample complexity derived from the fundamental trade-offs is tight up to multiplicative factors. In addition, we design and implement a polynomial-time (in n) algorithm based on the two-stage recovery scheme. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness of the proposed algorithm for accurate topology and parameter recovery

Compressive Sensing-Based Harmonic Sources Identification in Smart Grids

Identifying the prevailing polluting sources would help the distribution system operators in acting directly on the cause of the problem, thus reducing the corresponding negative effects. Due to the limited availability of specific measurement devices, ad hoc methodologies must be considered. In this regard, compressive sensing (CS)-based solutions are perfect candidates. This mathematical technique allows recovering sparse signals when a limited number of measurements are available, thus overcoming the lack of power quality meters. In this article, a new formulation of the ell _{1} -minimization algorithm for CS problems, with quadratic constraint, has been designed and investigated in the framework of the identification of the main polluting sources in smart grids. A novel whitening transformation is proposed for this context. This specific transformation allows the energy of the measurement errors to be appropriately estimated, and thus, better identification results are obtained. The validity of the proposal is proven by means of several simulations and tests performed on two distribution networks for which suitable measurement systems are considered along with a realistic quantification of the uncertainty sources

Validation of algorithms to estimate distribution network characteristics using power-hardware-in-the-loop configuration

Distribution system operators (DSOs) require accurate knowledge of the status of the network in order to ensure the continuity and quality of power supply. In this context, the National Physical Laboratory (NPL) and the Power Network Demonstrations Centre (PNDC) have been working together in the development and validation of optimal sensor placement and network topology estimation algorithms. This paper presents the description of two of these algorithms as well as the topology configuration of the PNDC distribution network considered to gather measurements for the validation of the algorithms. A Power-Hardware-in-the-Loop (P-HiL) configuration has been used as the testbed, where a number of physical measurement devices are installed in the physical network and an extended number of devices are virtually installed in the simulate network. The applications of the proposed algorithms to the measurements along with results from the P-HiL tests are presented in the paper