Economic efficiency of coordinated multilateral trades in electricity markets

This paper presents economic efficiency evaluation of electricity markets operating on the basis of a coordinated multilateral trading concept. The evaluation accounts for the overall costs of power generation, network losses, and system and unit constraints. We assume a non-collusive oligopolistic competition. An iterative Cournot model is used to characterize the competitive behavior of suppliers. A supplier maximizes the profit of each of his generating units while taking rivals' generation as given. Time span is over multiple hours. This leads to a mixed integer non-linear programming problem. We use the augmented Lagrangian approach to solve iteratively for globally optimal schedules. An IEEE 24-bus, 8-supplier, and 17-customer test system is used for illustration. The results show that such a market at times of light demands exhibits little market power, and at times of large demands exhibits a great deal of market power. This contrasts with the PCMI and HHI concentration measures, which give fixed measurement values of market power. The results of two-year (730 round) market simulations show a range of deadweight efficiency loss between 0.9 and 6% compared to that of PoolCo which results in a range between 0.5 and 10% for the same test case.     

Observability for two-dimensional systems

Sufficient conditions that a two-dimensional system with output is locally observable are presented. Known results depend on time derivatives of the output and the inverse function theorem. In some cases, no information is provided by these theories, and one must study observability by other methods. We dualize the observability problem to the controllability problem, and apply the deep results of Hermes on local controllability to prove a theorem concerning local observability.Research supported by NASA Ames Research Center under Grant NAG2-189 and the Joint Services Electronics Program under ONR Contract N0014-76-C1136.Research supported by NASA Ames Research Center under Grant NAG2-203 and the Joint Services Electronics Program under ONR Contract N0014-76-C1136.     

A canonical expansion for nonlinear systems

The importance of differential geometry, in particular, Lie brackets of vector fields, in the study of nonlinear systems is well established. Under very mild assumptions, we show that a real-analytic nonlinear system has an expansion in which the coefficients are computed in terms of Lie brackets. This expansion occurs in a special coordinate system. We also explain the concept of a pure feedback system. For control design involving a nonlinear system, one approach is to put the system in its canonical expansion and approximate by that part having only feedback paths.     

Driven dynamics of time-varying linear systems

Motivated by applications to automatic control of aircraft, we consider the problem of output tracking for time-varying linear systems. The signals to be tracked are bounded on (-∞, ∞), and our goal is to compute bounded controls and bounded state trajectories that result in the desired tracking. A convolution integral representation for the desired solution is given in the literature, and we show that certain integrability assumptions can be eliminated from the associated results. Despite the fact that our system is time varying and our signals do not necessarily have Fourier transforms in the classical sense, we show that computations can be carried out using “generalized” Fourier transforms. We also consider an output tracking problem in which the driven dynamical equation is an ordinary differential equation     

Global transformations of nonlinear systems

Recent results have established necessary and sufficient conditions for a nonlinear system of the formdot{x}(t) = f(x(t))-u(t)g(x(t)). withf(0) = 0, to be locally equivalent in a neighborhood of the origin in Rⁿto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that{gldot[fdotg],...,(ad^{n-1}fldotg)}span ann-dimensional space and that{gldot[fldot g],...,(ad^{n-2}fldotg)}is an involutive set.     

Approximation of nonlinear systems having outputs

Suppose we have a nonlinear system with output $$\begin{gathered} \dot x = f(x) + g(x)u, \hfill \\ y = h(x), \hfill \\ \end{gathered}$$ an open setO of state spaceR ⁿ , and a positive integerk. We find conditions onf,g, andh so that for eachx ₀ ε O there is ann-dimensional affine linear system, which depends onx ₀ but not onu, having the property that the output time responses (starting at the statex ₀ of the original nonlinear system and this approximating linear system agree through orderk for any admissable controlu. Several possible applications of our results are examined.