SYSTEMATIC SENSITIVITY ANALYSIS
USING DESCRIBING FUNCTIONS

Fred C. Schweppe and James Gruhl

M.I.T. Energy Lab, Cambridge, Mass. 02139
1. INTRODUCTION

This paper discusses simple, straightforward procedures for performing systematic sensitivity analysis on the input-output behavior of large mathematical models that are implemented as digital computer programs. The techniques were developed as part of an EPRI-funded study undertaken at the MIT Energy Laboratory, Model Assessment Group on the validation of the Baughman-Joskow "Regionalized Electricity Model" (REM); see, for example, reference 1. REM is used only as an example to give focus to the general ideas, and no conclusions or results about REM itself are given (these can be found in other documents such as in reference 2).

REM is a sophisticated computer program that simulates the dynamic behavior of portions of the U.S. energy supply/demand market with particular emphasis on the electric sector. For the sake of developing a simplified mathematical representation of REM de fine:

The elements of a and y can be generalized to cover series of discrete, or even continous, functions in time. The concepts of this paper, however, were developed and tested using the simple constant a and y as defined in Figure 1. The y are outputs in 1997 which is the terminal year for the base case simulation.

a

α

Once the inputs & and outputs y have been defined, any large computer model can be viewed simply as a nonlinear function f which translates the a into the y:

y = f (a)

(1.1)

a: vector of exogenous input parameters

y: vector of model outputs in 1997

This point of view is valid independent of whether the model is dynamic or static, a simulation or an optimization, deterministic or stochastic, etc.

To address the issue of sensitivity analysis of a model, define:

: base case input

Yo = f (a): base case output

Aa: input perturbation (e.g., uncertainty)

a = a +▲a: perturbed input

y(Aα) = f(α +Aα): perturbed output

8y(Aα) = y(Aα) - Yo = f(αo +Aα) - f(α)

= output perturbation (e.g., uncertainty).

A vast amount of material can be gathered for model sensitivity analysis if the following question can be answered:

Given a characterization of Ag
what are properties of dy(4a)?

(1.2)

In fact one might de fine the whole process of sensitivity analysis understanding, testing, and evaluation of the properties of dysa).

as the

The very nature of dy (Aa) suggests that it might be developed by a differentiating process, and in fact a Taylor series expansion of (1.1) yields

If the higher-order terms in (1.3) could be ignored, the Jacobian itself could provide the answer to (1.2). Unfortunately, for many cases of concern, the nonlinearities represented by the higher order terms are critical. For the REM example the Jacobian J was a 13 by 15 matrix and it was numerically estimated three different ways:

(1) using a 1% perturbation in a, which is partially displayed in Table 1,

(2)

using a 10% perturbation in a, partially displayed in Table 2, and finally

(3) using a 20% perturbation in a displayed in Table 3.

The fact that these three different numerical estimates were radically different in many of their entries demonstrated that the model's behavior was significantly nonlinear, thus question (1.2) could not be adequately answered by simple linear characterizations.

Two approaches for dealing with this inherent nonlinear nature of the model were developed:

(1) Criterion Sensitivity Analysis, and

(2) Describing Functions.

This paper emphasizes the Describing Function approach but the Criterion Sensitivity Analysis concept is briefly reviewed in the next section.

2. CRITERION SENSITIVITY ANALYSIS

Some of the basic ideas underlying the Criterion Sensitivity Analysis approach are reviewed here in a simplified fashion. More details can be found in Ref. 3.

Again consider the question posed in (1.2). Assume that the characterization of the input perturbation (uncertainty) Aa is expressed as

CAPITAL

COSTS

INSTALLED CAPACITY IN 1997

COAL GAS
NUCLEAR 1.5083 -0.0004

OIL.

LWR INT.COM. 0.0012-1.4855 -3.7630

COAL -0.7900 -0.0004 -0.0005 0.8967 -0.0038

OIL -0.0136 -0.0004 -0.0021 -0.0025 0.0390 INT.COM.-0.0009 -0.0004 -0.0005 -0.0099 -0.0209

Table 1 Capital Cost and Capacity Expansion Portions of Normalized Gradients from 1% Parameter Perturbations in REM

Table 2 Capital Cost and Capacity Expansion Portions of Normalized Gradients from 10% Perturbations

Table 3 Complete Table of Normalized Gradients from 20% Parameter Perturbation Results

INSTALLED

CAPACITY