477 Controllability of a Small Hydro Power Plant Sotir Panovski 1 and Gordana Janevska 2 Abstract – The paper presents the analysis of controllability theory for hydroenergetic installation of the Small Hydro Power Plant "Strezevo" using mathematical algebra. The analysis is based on the linear mathematical model of this plant treated as an object of automatic control. Keywords – Controllability, Linear mathematical model, Small hydro power plant. I. INTRODUCTION Kalman introduced the notion of controllability in 1960. Controllability is an important property of a control system. This property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. At the power industrial objects the primar task, before the task of controller synthesis, is the analysis of the object's controllability. If we suppose that the mathematical model which represents the object's dynamic behavior is known, then the question is: Is it possible for every state of the system to find a define action (control vector) that brings it into the desired state? Roughly, the concept of controllability denotes the ability to move a system around its entire configuration space using only certain admissible manipulations. An example of its applicability is a water turbine. The electrical network loading changes disturb the current steady state of the turbine and this results in decreasing or increasing the turbine angular speed, which is not plausable. Therefore, the turbine controller should provide a control action which will annulate these disturbances. The above example is about the state controllability, but in most cases we are interested in the output controllability. The state controllability is usually taken to mean that it is possible, by admissible inputs, to steer the states from any initial value to any final value within some time window. A linear controllable system may be defined as a system which can be steered to any state from the zero initial state. Output controllability means the ability to manipulate the outputs of a system by admissible inputs. For a system with several outputs, it might not be possible to manipulate these outputs independently by the admissible inputs, in which case the system is not output controllable. 1 Sotir Panovski is at the Faculty of Technical Sciences, I.L.Ribar bb, 7000 Bitola, R. Macedonia E-mail: sotir.panovski@uklo.edu.mk 2 Gordana Janevska is at the Faculty of Technical Sciences, I.L.Ribar bb, 7000 Bitola, R. Macedonia E-mail: gordana.janevska@uklo.edu.mk II. MATHEMATICAL MODEL A linear mathematical model of a hydro energetic installation (Fig. 1) consists of an accumulation, a supply tunnel, a compression pipeline, a water chamber, a hydroelectric generating set and electric net is derived. 1. Accumulation 2. Supply tunnel 3. Water chamber 4. Delivery pipe 5. Turbine 6. Draft tube 7. Tail race Fig. 1. Hydro electrical power plant The block diagram of the hydro power plant as an object of automatic control is given in Fig. 2. Fig. 2 Block diagram of HPP Analyzing the dynamics of each constructive part (subsystem) of the general system, the mathematical model is developed. The model is linearised based on realistically adopted and critically studied assumptions and the following equations are obtained: • Equation of the supply tunnel ( ) r b t t v t t t v t t 1 1 2 H T C h T q T C C q Δ ⋅ + + ⋅ - Δ ⋅ + - = Δ • (1) • Equation of the surge tank c v t v v 1 1 q T q T h Δ ⋅ - Δ ⋅ = • (2) v II p I h 3 6 7 H p II II II v m h s p m 5 8 4 I I 2 1 v I H А Hydro power plant M opt ω A [m 2 ] – opening of the inlet runner Mopt [Nm] – momentum of the loading H [m] – disposal (total) head ω [s -1 ] – angular velocity of the turbine