Paul A. Trodden

Local Solutions of the Optimal Power Flow Problem

The existence of locally optimal solutions to the AC optimal power flow problem (OPF) has been a question of interest for decades. This paper presents examples of local optima on a variety of test networks including modified versions of common networks. We show that local optima can occur because the feasible region is disconnected and/or because of nonlinearities in the constraints. Standard local optimization techniques are shown to converge to these local optima. The voltage bounds of all the examples in this paper are between ±5% and ±10% off-nominal. The examples with local optima are available in an online archive (http://www.maths.ed.ac.uk/optenergy/LocalOpt/) and can be used to test local or global optimization techniques for OPF. Finally we use our test examples to illustrate the behavior of a recent semi-definite programming approach that aims to find the global solution of OPF.

Distributed model predictive control of linear systems with persistent disturbances

This article presents a new form of robust distributed model predictive control (MPC) for multiple dynamically decoupled subsystems, in which distributed control agents exchange plans to achieve satisfaction of coupling constraints. The new method offers greater flexibility in communications than existing robust methods, and relaxes restrictions on the order in which distributed computations are performed. The local controllers use the concept of tube MPC – in which an optimisation designs a tube for the system to follow rather than a trajectory – to achieve robust feasibility and stability despite the presence of persistent, bounded disturbances. A methodical exploration of the trades between performance and communication is provided by numerical simulations of an example scenario. It is shown that at low levels of inter-agent communication, distributed MPC can obtain a lower closed-loop cost than that obtained by a centralised implementation. A further example shows that the flexibility in communications means the new algorithm has a relatively low susceptibility to the adverse effects of delays in computation and communication.

Robust distributed model predictive control using tubes

This paper presents a new form of robust distributed model predictive control (MPC) for multiple subsystems with coupled constraints and persistent disturbances. The new method allows greater flexibility in communications than existing methods, and relaxes restrictions on the order in which distributed computations are performed. The new controller uses the concept of tube MPC, in which an optimization designs a tube for the system to follow rather than a trajectory. The contributions of this paper are the modification of tube MPC for distributed implementation and investigation of the trade between performance and communication

Optimization-Based Islanding of Power Networks Using Piecewise Linear AC Power Flow

In this paper, a flexible optimization-based framework for intentional islanding is presented. The decision is made of which transmission lines to switch in order to split the network while minimizing disruption, the amount of load shed, or grouping coherent generators. The approach uses a piecewise linear model of AC power flow, which allows the voltage and reactive power to be considered directly when designing the islands. Demonstrations on standard test networks show that solution of the problem provides islands that are balanced in real and reactive power, satisfy AC power flow laws, and have a healthy voltage profile.

Adaptive cooperation in robust distributed model predictive control

In this paper a new, adaptive cooperative form of robust distributed model predictive control is introduced. In the new algorithm, for linear, dynamically-decoupled subsystems in the presence of bounded disturbances, an optimizing subsystem determines the existence of paths in a graph representing currently-active coupling constraints. Where such paths exists, cooperation is promoted by the local agent designing a hypothetical plan for other subsystems. Robust feasibility and stabil- ity are maintained by permitting only non-coupled agents to update at each time step. By simulation, performance is shown to surpass that of using cooperation between immediately-adjacent agents, rivalling that of a ‘fully cooperative’ implementation.

MILP islanding of power networks by bus splitting

A mathematical formulation for the islanding of power networks is presented. Given an area of uncertainty in the network, the proposed approach uses mixed integer linear programming to isolate unhealthy components of the network and create islands, while maximizing load supply. Rather than disconnecting transmission lines, the new method splits the network at its nodes, which are modelled as busbars with switches between lines, generators and loads. DC power flow equations and network constraints are explicitly included in the MILP problem, resulting in balanced, steady-state feasible islands. Numerical simulations on the IEEE 14-bus test network demonstrate the effectiveness of the approach.

Multi-Vehicle Cooperative Search using Distributed Model Predictive Control

This paper applies a form of Distributed Model Predictive Control (DMPC) to the problem of multi-vehicle search of an area. The objective of the team of vehicles is to thoroughly search an area of unknown content, collecting rewards while avoiding collision and duplication of effort. The DMPC algorithm employed, which offers guaranteed feasibility and flexible communications, has recently been extended to a cooperative form, where agents consider a greater portion of the global objective. It is shown by simulation that use of this cooperative cost leads to an improvement in system-wide performance over that of a simple greedy implementation. This paper applies a recently-developed form 1 of Distributed Model Predictive Control 2 (DMPC) to the problem of multi-vehicle cooperative search, where a team of vehicles navigate through and search an unknown environment, while avoiding collision and duplication of efforts. Key features of the developed method are (i) path planning decisions are made autonomously and on-line, based on the dynamics model, by vehicles, (ii) collision avoidance is guaranteed at all times, and (iii) the control algorithm permits flexible communications between vehicles. It is shown that, by using a cost function that considers the team objective rather than a simple greedy objective, cooperation between efforts is encouraged and better system-wide performance may be achieved. Classical search theory is well established for static environments with a single agent. The problem of multi-vehicle search requires path-planning decisions to be made for each vehicle involved; ideally, the vehicles should search the unknown area thoroughly and as quickly as possible, coordinating to avoid duplication of efforts, while avoiding collision with each other. One way of ensuring complete coverage of the area is to employ a pre-specified exhaustive search method, such as Zamboni search; 3 however, such a method makes no allowance for uncertainties in the environment, such as new threats or obstacles. Similarly, most results in the problem of coverage for robotics, 4 for applications such as floor cleaning, harvesting, or mine hunting, do not consider search environments where path planning might need to be adaptive; moreover, no consideration is given to the dynamic or kinematic constraints of the vehicle. In an unknown environment, as vehicles progess, knowledge of the area increases and plans may be required to change, either in response to new information about the environment or changes in the intentions of others. Research in this area includes a team of vehicles moving through the search area with a uniform longitudinal front, and only lateral relative motion between vehicles, with limited look-ahead, 5 and swarms of vehicles arranging themselves into a formation with maximal sensing capability yet minimal inter-vehicle communication. 6 Some authors propose decomposition of the state space into searchable cells: in [7] and [8] the space is discretized so that vehicles have a finite number of heading choices, whereas in [9] and [10] waypoints are generated for the vehicle to follow using a low-level controller. In this paper, the vehicle dynamics model and kinematic constraints are included in the search problem, and Model Predictive Control 11, 12 (MPC) is used as the control method. The vehicles plan finite-length paths through the search area, which is known in extent and decomposed into a cellular grid; thus, this method is comparable to a limited look-ahead approach, but with vehicle dynamics considered. Sensors on the vehicles search cells as the paths are followed, with the aim of collecting rewards associated with cells

A One-Step Approach to Computing a Polytopic Robust Positively Invariant Set

A procedure and theoretical results are presented for the problem of determining a minimal robust positively invariant (RPI) set for a linear discrete-time system subject to unknown, bounded disturbances. The procedure computes, via the solving of a single LP, a polytopic RPI set that is minimal with respect to the family of RPI sets generated from a finite number of inequalities with pre-defined normal vectors.

Distributed predictive control with minimization of mutual disturbances

Abstract In this paper, a distributed model predictive control scheme is proposed for linear, time-invariant dynamically coupled systems. Uniquely, controllers optimize state and input constraint sets, and exchange information about these–rather than planned state and control trajectories–in order to coordinate actions and reduce the effects of the mutual disturbances induced via dynamic coupling. Mutual disturbance rejection is by means of the tube-based model predictive control approach, with tubes optimized and terminal sets reconfigured on-line in response to the changing disturbance sets. Feasibility and exponential stability are guaranteed under provided sufficient conditions on non-increase of the constraint set parameters.

Persistently exciting tube MPC

This paper presents a new approach to deal with the dual problem of system identification and regulation. The main feature consists in breaking the control input to the system into a regulator part and a persistently exciting part. The former is used to regulate the plant using a robust MPC formulation, in which the latter is treated as a bounded additive disturbance. The identification process is executed by a simple recursive least squares algorithm. In order to guarantee sufficient excitation for the identification, an additional non-convex constraint is enforced over the persistently exciting part.