DIFFERENTIAL EQUATIONS USED TO ANALYSE AND OBSERVE EFFECT OF IMPACT LOADS ON WIND TURBINE AS IT AFFECT THE ENTIRE SYSTEM

INCLUDING THE GRID NETWORK

1ADOKO SAMBWA 2OKOM ANTHONY C. 3IDIAGBONYA EDDIE A. 4EGBON OSAZEE J. 5EZE NDUBUISI D. 6OSARENKHOE EZEKIEL

Energy Consultants (Renewable Energy) 1Asa-Lambda Technology Ltd. C/o: Crossing Over Ventures,

9, Akpakpava Road, Benin City, NIGERIA. Email:adokosam@yahoo.com

Abstract A moving mass of air that strikes a wind turbine delivers a dynamic or impact load or force. Problems resulting from such forces to wind turbine may be analyzed or observed using differential equation, under strict compliances on the basis of certain accepted assumptions wind turbine structure needs to behave elastically, and no dissipation of energy should take place at the point of impact, and finally, the effect of loads impact should be directly proportional to the magnitude of the applied force, then, on the basis of the principle of conservation of energy, it may be assumed that the wind force at the instant of impact with the wind turbine, its kinetic energy is completely transformed into the internal strain energy of the resisting system (wind turbine). At this instant, while the transformation from kinetic energy to mechanical energy is taking place, there are visible signs of vibrations being occurred within the wind turbine which may have negative affects the general performances of the wind turbine and consequently, hindering the purpose in which it is designed for. However, since vibrations, deflections, and stability are of primary interest in this work, other very fundamental issues as stresses will not be discussed in this paper. It is clear that recent events towards the development of commercial wind turbines have triggered special interest in wind energy applications. Considerable interest has been manifested in the ability of wind turbine to be classified as a viable and cheap energy source, compared with the fossil fuels based energy sources. Despite all these glorious acknowledgements, there are some buckets of unsolved problems in wind energy sector. Few among these obstacles are centered on the unpredicted nature of wind. This paper used differential equation in observing the behaviour of wind turbine as it is influenced by the wind forces across it. Keywords: Motion, Impact Loads, Differential Equations and Application.

1. INTRODUCTION

Considering the environment which the wind turbine plays its fundamental role, engineering practices consider that for a given construction project, the selection of structural and machine components should be based on some characteristics. Those are: strength, stiffness, and stability. For a wind turbine, we opine that the members making up the unit are assumed to be in stable equilibrium. Although, not all structural arrangement are necessary stabled. Physically looking at the wind turbine, it is obvious that it stands on one end, and might conclude that the stress at the base would be equal to the total weight of the wind turbine divided by its cross-sectional area.

However, the equilibrium of the wind turbine is precarious. With the smallest imperfection in the turbine or the wind turbulence, the turbine might lose its equilibrium, sometimes fall down. This illustration may initiate a sense of feeling of the importance in wind turbine’s design ad implementation.

This paper will explore these phenomenal analytically while observing certain behaviour of the wind turbine under severe climate conditions. But the problem becomes more complicated because different phenomena contribute to the capacity of the wind turbine structure, depending on its length, and the weight of its rotor and cases [1] Various cases of structural instability problems which may exist are beyond the scope of this paper.

The chief concern of this paper will be the analysis of equations that describe the behaviour of the wind turbine that is the turbine under an influence of a force as a function of time as it affects the production of electricity, putting in mind the structural instability, the buckling of the turbine at the worse case scenario. These are crucially important problems of engineering design.

Figure 2: A damaged Wind Turbine by high Wind Speed. CourtesyWikipedia.

Figure 3: Wind turbulence Courtesy Wikipedia

2. THE IMPACT LOADS AS IT AFFECTS TRANSMISSION Impact load or force causes the wind turbine structure to swing. The acceleration of the structure is proportional to the vector sum of all forces acting on it. The wind turbine being a one degree of freedom experiences rotational effects [2] In furtherance of the above, consider the wind turbine structure with a torsional modulus of stiffness K at the base. The behaviour of the turbine’s structure is subjected to a vertical force P (mg) and a horizontal force F. Keeping P constant, and F varies continuously, the simple question then arises; how will this structure, behaves if, F = o ? This is a special situation and it corresponds to the investigation of dynamic stability, self buckling, and much other behaviour which may be exhibited under such condition. Figure 4: Damaged Wind turbines by impact loads. In the list of boundary conditions, the wind turbine’s structure is assumed to experience only rotational effect as it cannot bend i.e., its ability of having one degree of freedom. Let us assumed = rotation, the restoring moment will be K , then with F = , the upsetting moment will be: P sin = P … for <<< 1. Hence, if K > P : the system is stable K < P : the system is unstable. Buf if K = Pl , the equilibrium is neither stable nor unstable, but is neutral.

Figure 5: Artist impression of wind under severe climate condition

3. THE PROCESS Let denotes the angular displacement, measured counter clockwise from the equilibrium position. The weight of the nacelle and rotor is mg (g: gravity acceleration). It causes a restoring force mgsin tangent to the curve of the motion [3]. Invoking Newton’s second law which stipulates that, at each instant this force is balanced by the force of acceleration mℓ , where ℓ is the acceleration [4]. So, the resultant of these two forces is zero, and this gives the mathematical representation as follows: mℓ + mg sin = o … … … (2) Dividing the above expression by mℓ, yield;

+

K =

+ K sin = o … … … (3)

For << 1, Sin = Fig. 6: System dynamic representation Let v = : the angular velocity, then

= vl =

½ v2 = Kcos + c … … … (4) Let multiply equation (4) by mℓ 2 ½ m (ℓv)2 – mℓ2 Kcos = mℓ 2 C … … … (5) Since = v represents the angular velocity, ℓ is the velocity. The first term in equation (5) is the Kinetic energy, the second term (including the minus sign), is the potential energy of the wind turbine structure, and mℓ2 C is the total energy. This may be one the reasons wind turbine is not recommended for roof mounts! From (5)

“ℓ2 – mℓ2 cos m ℓ 2 C / ℓ 2

– mg cos = mc = P cos + mc

P: must remain constant K: is the modulus of elasticity of the turbine’s materials. When varies, the angular velocity v’ is determined.

F= mgsin : is the resultant of all forces towards the turbine P = mg : comprises the weight of the rotor and nacelle. Visibly, the turbine structure will perform oscillations along an arc of a circle of radius ℓ. The impact of F causes the turbine to be in a state of unstable equilibrium, that is any slightest load impact will influence the wind turbine to initiate a motion [5]. In this analysis, we will equally try to predict the dynamic displacements, time history, and modal analysis. But the latter may require an intensive analysis and such cannot be elaborated in the present work. 4. THE MOTION Equation (4) represents the motion of the wind turbine under impact loads. ½ 2 = K cos d

= … … … (6)

In the normal working conditions, the motion of the turbine structure may be restricted to a relatively small aptitude with a small . This implies that: Sin = => I I <<< 1. Hence,

+ = 0

Initial condition dictates that = and

… … … (7)

For amplitudes beyond the above approximated ones, equation (3) can be inverted as follow:

Integrating the above equation over one complete cycle, twice the half – cycle, and 4 times the quarter cycle, yields:

T = 4

T = 4

T = 4

Where F (K, ) is legend’s elliptic function

F (K, ) =

T = 2

5. INVESTIGATIONS / SIMULATIONS What will be the behaviour of the turbine for a period of ℓ (m) swing at initial angle of 5, 10, 15 and 20 degrees? This answer will portray series of mathematical conclusion of this work.

Swing Length

Initial Angel : degree

Period T=

2

Pulsation

W = 2

Motion

Angular Velocity

Cos

0.5 10 1.404 4.472 9.9400 0.0875 1 10 1.98 3.171 9.9403 0.620 1.5 10 2.43 2.58 9.9401 0.05012 2 10 2.8 2.24 9.96 0.0486 2.5 10 3.14 2 9.939 0.039

Table 1: Simulation at 10 degree initial angle, using different swing length of structure under impact load

Swing Length

Initial Angel : degree

Period T=

2

Pulsation

W = 2

Motion

Angular Velocity

Cos

0.5 15 1.404 4.472 14.9105 0.1317 1 15 1.985 3.163 14.9100 0.094 1.5 15 2.432 2.582 14.9100 0.076 2 15 2.8 2.236 14.9100 0.0645 2.5 15 3.14 2 14.909 0.0587

Table 2: Simulation at 15 degree initial angle, using different swing length of structure under impact load

Swing Length

Initial Angel : degree

Period T=

2

Pulsation

W = 2

Motion

Angular Velocity

Cos

0.5 20 1.404 4.472 19.880 0.169 1 20 1.985 3.163 19.880 0.120 1.5 20 2.432 2.582 19.885 0.889 2 20 2.8 2.24 19.881 0.0830 2.5 20 3.14 2 19.879 0.0757

Table 3: Simulation at 15 degree initial angle, using different swing length of structure under impact load

6. INTERPRETATION OF RESULTS While modern wind power turbine is becoming reliable, our results show that it is not possible to eliminate structural failures and abnormal conditions. Using differential equations, it is possible to achieve different results in different direction that is, stability with respect to displacements in the x – direction but instability in the y – direction. This is a saddle point. The wind turbine structure is not stable because stability must be achieved in all directions and dimensions. The procedures so established in this work evolve in a deterministic pattern, i.e. simulations made will allow the prediction of the state of the system, even moderately far into the future. But, the wind turbine dynamic behaviour is not linear [7]. Nonlinearity simply means that the measured values of the properties of a system in a later state may not depend on the measured (observation) done earlier. “Frequently, the problem of modeling real-world systems with mathematical equations begins with a linear model. But when finer details or more accurate results are desired, additional nonlinear terms must be added [8]. Adopting this technique, we have observed with keen interest that the behaviour of the wind turbine structure subjected to impact load depends not only on the fundamental laws of Newtonian mechanics that govern equilibrium of the forces but also on the physical characteristics of the materials of which the turbine is fabricated. Although, there existe some efforts to improve or develop a vibration – resistant and fracture – resistant” cast material for offshore wind turbines. The research project being carried by “Siempelkamp Gieberei GmbH” aims to reduce the weight of the nacelle. This may have multiple positive impact. Our interest has been focused on the possibilities that the mechanical stability of the system will be guaranteed under such arrangement. Equally, viewing the work being carried out by two German companies, “Vensus Energie System GmbH & Co. KG, Sanarbrucken” towards producing the first ever gearless 2.5 MW wind turbine [9]. This will reduce tower head weight. The weight reduction will definitely improve the problem associated with impact load and guaranty some level of stability. 7. APPLICATIONS / CONCLUSION Differential equations arise in many engineering and other applications as mathematical models of various physical and other systems. The fundamental goals in differential equations and their applications is to find all solutions of given equations and investigate their properties [6].

It is no doubt that differential equation has tremendous engineering benefit, because of its involvement in many physical laws and relations which may be represented mathematically. It allows a physical system to be modeled mathematically and then the solution will have a physical interpretation. This has been the fastest way of obtaining a first idea of the nature and purpose of differential equations and their applications. Very often, it seems as if differential equations deals with pressing problems involving a very large and complex system about which two little is know. The problem, as it is the case here, for stability of wind turbine might be serious and urgent, but difficult to define with the utmost precision. It is therefore necessary to predict what might happen thereby suggesting what could be done about the situation at hand. In these circumstances, the prediction or the study of the behaviour of the turbine might be done by modeling the wind turbine using appropriate equations. Then, the model will be fitted into an appropriate mathematical structure that embodies techniques to convert input information into predictions. Basically, the differential equation is a tool which applies by the systematic observation and analysis of systems, structures to improve measures relating to prevention, mitigation, preparedness and optimal response to situations. However, this descriptive view seems not to have a wide acceptance from the scientists as being confirmed by fewness of research in this respect. We believe that the dynamic analysis of the behaviour of a wind turbine’s impact, with longer – term horizons. In the thru perceptive, mitigating or preventing if possible the effect of loads impacts may guaranty the stability of the generation and the transmission grid. The realization that many disasters caused by load impacts could be prevented, as well as the recognition of their devastating effects on the generation, transmission grid, and the economics. Figure 8: The effect of load impact (courtesy: Internet pages) The prevention and mitigation as fundamental elements of disaster management has been underpinned by scientific knowledge. Differential equations with all its techniques and involving methods have increased the scientist’s ability to forecast the probability of structure collapse occurring during a particular time frame.

Differential equations’ applications have made it possible to predict the amount of energy being transmitted to a wind turbine while the dynamic motions of the wind turbine are being observed, thus, proving the gear box and rotors design. Modeling techniques using differential equations have made long – and short –term forecasting of eventualities, thus saving many lives. In summary, differential equations are equally suitable for the learning techniques as tools to assist in increasing knowledge and ability to understand complex systems and processes in an ever-wider range of scales in space and time. [7]. 8. REFERENCE: 1. Adoko et al [2007]: “Differential equations used to observe the dynamic

behaviour of wind turbine” International Solar Energy Society (ISES) Congress, Beijing China 2007

2. J. Turner et al [1956], “stiffness and deflection of complex structures”, extracted

in Wikipedia. 3. Feynman, R.P et al [1963], lectures on physics. Vol. 1 Addison - Wesley 4. Kleppner. D. et al [1973], an introduction to mechanics, McGraw - Hill 5. Erwin Kreyszig [1988] Advanced Engineering Mathematics, Sixth Edition. John

Wiley & Son, NY 1988 6. Isaac Newton [1687], “Philosophiae Naturalis Pricipia Mathematica” which

contains the Newton’s Law of Motion. 7. Adoko et al: ibid 8. S. Neil Rasbad [2007], “Chaotic Dynamics of nonlinear systems” Wiley

professional 2007. 9. www.worldrenewableenergy.com