Sensitivity analysis of nonsmooth power control systems with an example of wind turbines

highlights

1. Generalized Derivatives Contribute to Nonsmooth and/or Discontinious Studies of Nonlinear Control Systems.

2. New Sensitivity Theory Applied to Nonsmooth-Time-Varying Systems of Differential-Algebraic-Equations.

3. Nonsmooth Sensitivity Analysis of Power Control Systems.

4. Wind Turbine Power Control System is Analyzed Given Nonsmooth-Time-Varying Input Wind Speed Profiles.

5. New Controls and Sub-Controls Need to be Introduced for Wind Turbines Exposed to High Wind Speeds.

Abstract

This paper focuses on dynamic sensitivities of nonsmooth power control systems. Recent advances in nonsmooth sensitivity theory are applicable to large-scale practical problems modeled as nonsmooth semi-explicit differential-algebraic equations (DAEs). This theory is applied here to a wind turbine control system with nonsmooth input parameter profiles. First-order sensitivity information is characterized from an auxiliary, nonsmooth sensitivity system which includes discontinuities because of nonsmooth wind speed profiles. Dynamic simulations of sensitivities are provided which give new insights on design and implementation (e.g., the system’s control limits).

Introduction

Studying aspects of power/energy systems, such as their modeling, stability, parametric characterization and sensitivity, has been a typical process for advancing control structure/strategies for such systems (see the layout of the books [1], [2]). The literature of power systems and their control has been transforming towards time-domain analysis due to the highly nonlinear nature of power systems. This is especially true in the emerging power systems such as wind and solar [3], [4]. Most power systems can be formulated as a DAE system in the following semi-explicit form:

$$\begin{array}{ccc}\hfill \dot{\mathbf{x}}(t,\mathbf{p})& =& \mathbf{h}(t,\mathbf{p},\mathbf{x}(t,\mathbf{p}),\mathbf{y}(t,\mathbf{p})),\mathbf{x}(t_0,\mathbf{p})=\mathbf{c},\hfill \\ \hfill \mathbf{0}& =& \mathbf{g}(t,\mathbf{p},\mathbf{x}(t,\mathbf{p}),\mathbf{y}(t,\mathbf{p}\left)\right),\hfill \end{array}$$

where $\mathbf{x}$ are the differential state variables, $\mathbf{y}$ are the algebraic state variables, $\mathbf{p}$ are the system’s parameters and $\mathbf{c}$ are the initial conditions of the differential state variables. The parameters of the system are critical for any control system as they can be used to describe many cases: uncontrolled input, stochastic input, design-fixed-valued input and a generated control signal, among others. Sensitivities of $\mathbf{x}$ and $\mathbf{y}$ w.r.t. $\mathbf{p}$ are therefore of critical importance to modeling improvements or control design/strategy.

Aside from the challenges associated with studying DAEs that are nonlinear and/or high in dimensions, the presence of nonsmoothness in $\mathbf{h}$ and/or $\mathbf{g}$ can cause failure in traditional stability, sensitivity and simulation methods. As a result, many works in the literature avoid nonsmoothness wherever it may appear. One simple way for that is to approximate, for example, a nonmooth or discontinuous profile with a smooth or continuous one because of a lack of computationally-relevant nonsmooth/discontinuous theory. While this may be appealing, it is often the case that errors or misrepresentations from such approximations are not traceable nor quantifiable. Moreover, smoothing approximations often require user-defined constants that can be arbitrary in nature and can increase the complexity in theoretical/numerical treatments in addition to introducing numerical error.

In many power/energy systems, especially the emerging ones that are less predictable and under significant evolution, modeling parametric profiles with nonsmooth or discontinuous profiles can be very informative because it often represents an extreme case scenario, which in return gives an idea about the extreme response. For example, it can be assuring if an extremely sudden drop in an input signal or parameter does not cause failures in the system, such as exceeding the mechanical constraints or the physical considerations. Examples of this can be found in General Electric technical report [5] and the National Renewable National Lab tests [6] where in the simulation sections for faults and response, the signals and scenarios used, such as extreme voltage/impedance fault, are modeled by nonsmooth-time-varying profiles. In many cases they are assumed to be extreme (near or full discontinuous) profiles. In many power control systems, we find control signals that are switching between on-off and start a linear signal from dead (zero) input. In order to have more rigorous analysis for the design and scenarios, one would prefer having a rigorous simulation tool that takes these control signals (nonsmooth or discontinuous) and provides how the state variables change w.r.t. controls. In many situations it is important to test how the state variables change w.r.t. any time-varying input profiles, so for example, we can make sure the state variables that physically have to be continuous are compatible with the designed controls.

Emerging power control systems such as Wind Turbine Generators (WTGs) are very complex and highly nonlinear. They are also often relatively high in their dimensions (10+), which is greater than many traditional power systems. It is the case that WTG control system design and implementation in large scale faces many challenges due to the fact that the major parametric input for the system, the wind speed $v_{wind},$ is uncontrolled and often considered stochastic in nature. This has triggered few sensitivity-based studies [7], [8], [9], [10], which through simulations and basic analyses concluded how the system seems sensitive w.r.t. some uncontrolled parameters, including $v_{wind}$ and the power grid resistance $R$ and reactance $X$. However, these studies only provided numerical sensitivities for fixed-valued $v_{wind}$ in steady-state, and did not provide a general sensitivity analysis that can be applied if, for example, $v_{wind}$ is modeled by a time-varying nonsmooth profile.

We turn our attention to recent developments in nonsmooth (i.e., continuous but nondifferentiable) semi-explicit DAEs theory [11], [12], [13], which are applicable to (1) with nonsmooth right-hand side (RHS) functions, and thus are suitable for present purposes. Successful application of this theory to practical problems can be found in process systems engineering [14], [15], pharmaceutical manufacturing [16], [17], [18] and systems biology [19], [20]. The reader is encouraged to refer to [21] for a detailed account of this area of work. The main contribution of this paper is the application of new nonsmooth sensitivity theory to power control systems of the form (1). In particular, we provide a demonstrative example of the theory applied to the emerging, highly complex and nonlinear, WTG control system. The simulation case chosen in this paper is for a nonsmooth $v_{wind}$ profile that is transitioning from a lower level to an upper level. Dynamic sensitivities are obtained which provide new insights about critical behaviors within the system. Thanks to the computationally-relevant sensitivity information of state variables w.r.t. nonsmooth $v_{wind},$ some observations in [4], [22] regarding the system’s control limits are expanded here.

The paper structure is as follows: Section 2 provides background on nonsmooth DAEs, including sensitivity theory, necessary for the analysis. Section 3 presents the nonsmooth WTG control system considered, including an analysis of its parametric sensitivities in Section 3.1, a discussion of the resulting numerical simulations in Section 3.2, and a comparison of results to a smoothing approach in Section 3.3. Section 4 concludes the paper and discusses future work.