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Three orthogonalization techniques to correct errors in the computed direction cosine matrix are introduced. One of these techniques is a vectorial technique based on the fact that the three rows of a direction cosine matrix constitute an orthonormal set of vectors in a threedimensional space. The other two iterative techniques are based on the fact that the inverse and transpose of an orthogonal matrix are equal. In computing a time-varying direction cosine matrix computational errors are accompanied by the loss of the orthogonality property of the matrix. When one of these three techniques is used to restore the orthogonality of the matrix, the computational errors are also corrected. These techniques were tested experimentally and the results, given in this paper, were compared with a method used by the Honeywell Corporation.

Methods of fitting models to experimental data obtained from biological systems are reviewed. The Michaelis-Menten model is used as the working example of a familiar and well-studied biological model. Criteria for selecting models include goodness of fit, freedom from systematic errors, and simplicity. A given model is usually fitted and the optimal parameters determined by minimization of an objective function, usually the sum of the squared errors. Freedom from systematic errors is best judged graphically. The important mathematical methods of fitting models are derived from the calculus and include linear and quadratic programming. The latter leads to minimization of the sum of squares of errors. Optimal search procedures, which also perform this minimization, are surveyed. Other properties of models, such as the proper number of parameters and whether linearization is appropriate, are discussed. The specialized problems of biological models and data are considered.

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This paper deals with the problem of formulating dispatching algorithns which must trade off power system o p e r a t i n g c o s t w i t h system s e c u r i t y e f f e c t s . D e c ision theory was used t o s t r u c t u r e o p t i m i z a t i o n a l g orithms which dispatch a power systeminaccordance with d i f f e r e n t s e c u r i t y s t r a t e g i e s . The s t r a t e g i e s thems e l v e s a r e d i c t a t e d by how one w i s h e s t o t r e a t p r o b a b il i s t i c l o a d v a r i a t i o n s and contingency events. The c h i e f v a l u e o f t h i s r e s e a r c h l i e s i n p r o v i d i n g an approach t o u n i f y i n g t h e t e c h n i q u e s o f s t o c h a s t i c power flows, contingency analysis and constrained optimizat i o n . A. INTRODUCTION The b a s i c p r o b l e m d e a l t w i t h i n t h i s p a p e r can be sumnarized as: "What i s t h e b e s t way t o s t r u c t u r e and implementa power dispatching system which meets severe system security requirements?" Contingency analysis, p r o b a b i l i t y , c o n s t r a i n e d o p t i m i z a t i o n and decision theory are applied in answering this question. Contingency analysis a1 lows a dispatch operator to repeatedly pose various hypothetical "what if" events which could occur i n t h e power system, and to analyze t h e i r e f f e c t s . U s u a l l y some s o r t o f s t e a d y s t a t e model o f t h e power system, i n i t i a l i z e d t o p r e s e n t c o n d i t i o n s , i s used to provide the answers t o t h e h y p o t h e t i c a l questions. O f major importance are the questions "What i s the relative importance o f one hypothetical contingency event versus another?" and "Given that one o r more of the events occur, what canbedone about i t and what cost penalty should be applied?" The " r e l a t i v e importance" of one contingency event versus another depends upon t h e p r o b a b i l i t y t h a t t h e e v e n t w i l l occur and upon the probable effects of the event given that i t has occurred. Thus, contingency analysis and probab i l i t y t h e o r y must be combined. P r o b a b i l i t y t h e o r y i s used t o d e t e r m i n e t h e p r o b a b i l i t y o f f a i l u r e o f system components (e.g., line or generator outages) and t o det e r m i n e t h e p r o b a b i l i t y o f d i f f e r e n t l o a d d i s t r i b u t i o n s . C a l c u l a t i n g t h e p r o b a b i l i t y of various contingency events and t h e e n s u e i n g p r o b a b i l i t i e s o f t h e e f f e c t s o f t h o s e events provide a method o f comparing one event and i t s effects to another event and i t s e f f e c t s . S t i l l needed, however, i s an answer to the question "How r e l i a b l e should the system be and what i s the cost associated w i t h t h a t r e l i a b i l i t y ? " I t may be assumed t h a t : 1. a f i x e d r e l i a b i l i t y i s t o be maintained, or Engineering Committee of the IEEE Power Engineering Society for presentation at Paper F 75 427-5, recommended ...

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