## Services on Demand

## Article

## Indicators

## Related links

- Cited by Google
- Similars in Google

## Share

## Journal of Energy in Southern Africa

##
*On-line version* ISSN 2413-3051

*Print version* ISSN 1021-447X

### J. energy South. Afr. vol.23 n.2 Cape Town 2012

**A qualitative study of the optimal control model for an electric power generating system**

**Yidiat O. Aderinto; Mathias O. Bamigbola**

Department of Mathematics, University of Ilorin, Ilorin, Nigeria

**ABSTRACT**

The economic independence of any nation depends largely on the supply of abundant and reliable electric power and the extension of electricity services to all towns and villages in the country. In this work, the mathematical study of an electric power generating system model was presented via optimal control theory, in an attempt to maximize the power generating output and minimize the cost of generation. The factors affecting power generation at minimum cost are operating efficiencies of generators, fuel cost and transmission losses, but the most efficient generator in the system may not guarantee minimum cost as it may be located in an area where fuel cost is high. We choose the generator capacity as our control u_{i}(t), since we cannot neglect the operation limitation on the equipment because of its lifespan, the upper bound for u_{i}(t) is choosing to be 1 to represent the total capability of the machine and 0 to be the lower bound. The model is analyzed, generation loss free equilibrium and stability is established, and finally applications using real life data is presented using one generator and three generator systems respectively.

**Keywords:** mathematical model, electric power generating system, generation loss free equilibrium

**Full text available only in PDF format.**

**References**

Adejumobi I.A. (2005). Optimizing the Effectiveness and Efficiency of an Electrical System in Nigeria, Ilorin Metropolis as case study. Unpublished PhD Thesis, Department of Electrical Engineering, University of Ilorin, Ilorin, Nigeria. [ Links ]

Aderinto Y.O. and Bamibgola O.M. (2010). On Optimal Control Model of Electric power Generating System, *Global Journal of Mathematics and Statistics,* Vol. 2 No.1, p 75-85. [ Links ]

Agusto FB. (2008). On Optimal Control of Oxygen Absorption in Aquatic Systems. Unpublished PhD Thesis, Department of Mathematics, University of Ilorin, Ilorin, Nigeria. [ Links ]

Bao-Zhu G. and Tao-Tao W. (2009). Numerical Solution to Optimal Feedback Control by Dynamic Programming Approach, *Journal of Global Optimization,* p.1-15. [ Links ]

Bhunu C.P., Ganra W., Mukandavire Z., and Zimba M. (2008). Tuberculosis Transmission Model with Chemo-prophylaxes and Treatment. *Bulletin of Mathematical Biology,* Vol. 1, p. 1163-1191. [ Links ]

Billinton R. (1994). Evaluation of Reliability Worth in an Electric Power System. *Reliability Engineering and Systems Safety,* Vol. 46, p. 15-23. [ Links ]

Branimir K. and Radivo P A. (1993). Multi-objective Optimization Approach to Thermal Generating Units Maintenance Scheduling. *European Journal of Operational Research,* Vol. 84, p. 481-493. [ Links ]

Burden T., Ernstberger J. and Renee FK. (2003). Optimal Control Applied to Immunotherapy. *Discrete and Continuous Dynamical System,* Series B, p. 1-11. [ Links ]

Burghes D.N. and Graham A. (1989). An Introduction to Control Theory including Optimal Control. The Open *University,* Milton Keynes. 1989. [ Links ]

Cao J. and Wang J. (2003). Global Asymptotics Stability of a General Class of Recurrent Neural Networks with Time-Varying Delays. IEEE Transactions on Circuits and Systems-1: *Fundamental Theory and Applications,* Vol. 50 No. 1, p. 34-45. [ Links ]

Castilio-Chavez C., Feng Z., and Capurro A.F (2008). A Model for TB with Exogenous Reinfection. IMA Print, 2008. p. 1-23. [ Links ]

Craven B.D. (1995). Control and Optimization; Chapman and Hall, London 1995. [ Links ]

Ehsani A., Ranjbar A. M. and Fotuhi-Firuzabad M. (2007). Optimal and Reliable Scheduling of Competitive Electricity Markets. *The Arabian Journal for Science and Engineering,* Vol. 32, p. 281-300. [ Links ]

Fister K.R., Lenhart S. and Mcnalty J.S (1998). Optimizing Chemotherapy in an HIV Model. *Electronic Journal of Differential Equations,* Vol. No. 32, p.1- 12. [ Links ]

Fister K.R. and Panettta J.C. (2000). Optimal Control Applied to Cell-Cycle-Specific Cancer Chemotherapy. *SIAM Journal of Applied Mathematics* Vol. 60 No. 3, p. 1059-1072. [ Links ]

Fister R.K. and Lenhart S. (2004). Optimal Control of a Competitive System with Age structure. *Journal of Mathematical Analysis and Approaches,* Vol.1, p. 526-537. [ Links ]

Hosking R.J., Joyce D.C. and Turner J.C. (2003). First Steps in Numerical Analysis (Second edition). Hodder Headline Group, London, p. 128-142. [ Links ]

Jain M.K. (1983). Numerical Solution of Differential Equations (Second Edition). Wiley Eastern Limited, Delhi, 1983 p. 150-172. [ Links ]

Kathirgamanathan P. and Neitzart T. (2008). Optimal Control Paramet Estimation in Aluminium Extrusion for Given Product Characteristics. Proceedings of the *World Congress in Engineering, Vol. II,* London, p. 1-6. [ Links ]

Khorasani S. and Adibi A. (2003). Analytical Solution of Linear Ordinary Differential Equations by Differential Transfer. *Journal of Differential Equations,* Vol. 2003 No. 79, pp. 1-18. [ Links ]

Kirschner D. (1996). Using Mathematics to Understand HIV Immune Dynamics. *Notices of AMS,* Vol. 43 No. 2, p. 191.-202. [ Links ]

Lee K.Y., Ortiz J.L., Mohtadi M.A. and Park Y.M (1998).Optimal Operation of Large Scale Power Systems, IEEE *Transactions on Power Systems,* Vol. 3 No. 2, p. 413-420. [ Links ]

Lee K.Y, Ortiz J.L, Park YM. and Pond L.G. (1986). An Optimization Technique for Power Operation. *IEEE Transaction on Power Systems,* Vol.1 No. 2 p. 53-159. [ Links ]

Manafa M.N.A. (1978). Electricity Development in Nigeria (1896 - 1972). University of Lagos Press, Lagos. [ Links ]

Matilde PL,, Jose A.M. and Sanchez R.M.C.(2009). Exact Solutions in First-Order Differential Equations with Periodic Inputs. Electronic *Journal of Differential Equations.* p. 141-148. [ Links ]

Naevadal E. (2003). Solving Time Optimal Controls with a Spreadsheet. *Journal of Economic Education,* Vol. 4 p. 99-122. [ Links ]

Poppe H. and Kautz K. (1998). A Procedure to Solve Optimal Control Problems Numerically by Parametrization via Runge - Kutta Methods. *Journal of Mathematical Programming and Operations Research.* Taylor and Francis, p. 391-431. [ Links ]

Pingping Z. (2009). Analytic Solutions of a First Order Functional Differential Equation. *Electronic Journal of Differential Equations,* Vol. No. 51, p. 1-8. [ Links ]

Sabatini M. (1990). Global Asymptotic Stability of Critical Points in the Plane. *Journal of Dynamical Systems and Ordinary Differential Equation,* Vol. 48 No 2. p. 97-103. [ Links ]

Salley B. (2007). Mathematical Study of AIDS Infection Dynamics. A paper presented at the 5^{th} International Workshop on Contemporary Problems in Mathematical Physics and Applications, Cotonu, Republic of Benin, p 1-15. [ Links ]

Received 15 January 2011

Revised 29 November 2011