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This is the second part of a two-part paper on data-based distributionally robust stochastic optimal power flow. The general problem formulation and methodology have been presented in Part I (Y. Guo, K. Baker, E. Dall’Anese, Z. Hu, and T.H. Summers, “Data-based distributionally robust stochastic optimal power flow—Part I: Methodologies,” IEEE Trans. Power Syst., 2018.). Here, we present extensive numerical experiments in both distribution and transmission networks to illustrate the effectiveness and flexibility of the proposed methodology for balancing efficiency, constraint violation risk, and out-of-sample performance. On the distribution side, the method mitigates overvoltages due to high photovoltaic penetration using local energy storage devices. On the transmission side, the method reduces $N-1$ security line flow constraint risks due to high wind penetration using reserve policies for controllable generators. In both cases, the data-based distributionally robust model-predictive control algorithm explicitly utilizes forecast error training datasets, which can be updated online. The numerical results illustrate inherent tradeoffs between the operational costs, risks of constraints violations, and out-of-sample performance, offering systematic techniques for system operators to balance these objectives. [ABSTRACT FROM AUTHOR]

We propose a data-based method to solve a multi-stage stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The framework explicitly combines multi-stage feedback policies with any forecasting method and historical forecast error data. The objective is to determine power scheduling policies for controllable devices in a power network to balance operational cost and conditional value-at-risk of device and network constraint violations. These decisions include both nominal power schedules and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming that the uncertainties across the networks follow prescribed probability distributions, we consider ambiguity sets of distributions centered around a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real unknown data-generating distribution, we formulate a multi-stage distributionally robust OPF problem to compute control policies that are robust to both forecast errors and sampling errors inherent in the dataset. Two specific data-based distributionally robust stochastic OPF problems are proposed for distribution networks and transmission systems. [ABSTRACT FROM AUTHOR]

This paper develops a tractable formulation for optimal placement and sizing of inverter-based renewable systems in multi-phase distribution networks. The goal of the formulation is to minimize the cost of inverter installation, average power import, and average distributed generation curtailment. Three-phase and single-phase inverter models are presented that preserve the underlying mappings between renewable uncertainty to power injection. The uncertainty of distributed generators (DGs) and loads are characterized by a finite set of scenarios. Linear multi-phase power flow approximations are used in conjunction with scenario reduction techniques to arrive at a tractable two-stage stochastic formulation for optimal DG placement and sizing. First-stage decisions are locations for DG deployment and capacity sizes, and second-stage decisions include DG real power curtailment, reactive power support, as well as feeder voltage profile. The resulting formulation is a mixed-integer second-order cone program and can be solved efficiently either by existing optimization solvers or by relaxing the binary variables to the [0,1] interval. Simulation studies on standard multi-phase IEEE test feeders promise that optimal stochastic planning of DGs reduces costs during validation, compared to a scheme where uncertainty is only represented by its average value. [ABSTRACT FROM AUTHOR]

This paper considers unbalanced multiphase distribution systems with generic topology and different load models, and extends the $Z$ -bus iterative load-flow algorithm based on a fixed-point interpretation of the AC load-flow equations. Explicit conditions for existence and uniqueness of load-flow solutions are presented. These conditions also guarantee convergence of the load-flow algorithm to the unique solution. The proposed methodology is applicable to generic systems featuring i) wye connections; ii) ungrounded delta connections; iii) a combination of wye-connected and delta-connected sources/loads; and iv) a combination of line-to-line and line-to-grounded-neutral devices at the secondary of distribution transformers. Further, a sufficient condition for the nonsingularity of the load-flow Jacobian is proposed. Finally, linear load-flow models are derived, and their approximation accuracy is analyzed. Theoretical results are corroborated through experiments on IEEE test feeders. [ABSTRACT FROM AUTHOR]

Decentralized methods for computing optimal real and reactive power setpoints for residential photovoltaic (PV) inverters are developed in this paper. It is known that conventional PV inverter controllers, which are designed to extract maximum power at unity power factor, cannot address secondary performance objectives such as voltage regulation and network loss minimization. Optimal power flow techniques can be utilized to select which inverters will provide ancillary services and to compute their optimal real and reactive power setpoints according to well-defined performance criteria and economic objectives. Leveraging advances in sparsity-promoting regularization techniques and semidefinite relaxation, this paper shows how such problems can be solved with reduced computational burden and optimality guarantees. To enable large-scale implementation, a novel algorithmic framework is introduced—based on the so-called alternating direction method of multipliers—by which optimal power flow-type problems in this setting can be systematically decomposed into subproblems that can be solved in a decentralized fashion by the utility and customer-owned PV systems with limited exchanges of information. Since the computational burden is shared among multiple devices and the requirement of all-to-all communication can be circumvented, the proposed optimization approach scales favorably to large distribution networks. [ABSTRACT FROM AUTHOR]

Through spatial multiplexing and diversity, multi-input multi-output (MIMO) cognitive radio (CR) networks can markedly increase transmission rates and reliability, while controlling the interference inflicted to peer nodes and primary users (PUs) via beamforming. The present paper optimizes the design of transmit- and receive-beamformers for ad hoc CR networks when CR-to-CR channels are known, but CR-to-PU channels cannot be estimated accurately. Capitalizing on a norm-bounded channel uncertainty model, the optimal beamforming design is formulated to minimize the overall mean-square error (MSE) from all data streams, while enforcing protection of the PU system when the CR-to-PU channels are uncertain. Even though the resultant optimization problem is non-convex, algorithms with provable convergence to stationary points are developed by resorting to block coordinate ascent iterations, along with suitable convex approximation techniques. Enticingly, the novel schemes also lend themselves naturally to distributed implementations. Numerical tests are reported to corroborate the analytical findings. [ABSTRACT FROM PUBLISHER]