ELE11117 Modelling and Computation for Smart Places

CW1: Smart Wind Farm with Adaptive Turbines and Maintenance Robots

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Case Study: Smart Wind Farm with Adaptive Turbines and Maintenance Robots

A cutting-edge wind farm is being developed in a coastal region known for its variable wind conditions. The farm incorporates smart turbines with adaptive technology and autonomous maintenance robots. The goal is to optimize energy production while minimizing maintenance costs and environmental impact.

The wind turbines feature an innovative adaptive technology that can dynamically adjust their efficiency based on environmental conditions and internal feedback mechanisms. This adaptation is controlled by an advanced algorithm that considers both current wind conditions and the overall system state.

Autonomous maintenance robots are deployed to inspect and repair the turbines. These robots use AI to predict potential failures and optimize their maintenance schedules. The robots’ effectiveness is influenced by the current state of the turbines and environmental factors.

The system’s performance is primarily characterized by two key variables:

1. Turbine Efficiency (𝐸): This represents the overall efficiency of the wind turbines in converting wind energy to electrical energy. It ranges from -1 to 1, where negative values represent energy consumption (e.g., during extreme conditions or maintenance).
2. Robot Productivity (𝑃): This represents the effectiveness of the maintenance robots in keeping the turbines in optimal condition. It ranges from 0 to a maximum value 𝑃_𝑚𝑎𝑥.

The relationship between these variables can be described as follows:

• The rate of change of turbine efficiency is influenced by three main factors:

1. A growth term that equal to the product of the given control parameter 𝑟 and the current efficiency 𝐸.
2. A saturation term that prevents unlimited growth, equal to −𝐸³.
3. An environmental impact term tending to decrease the rate of change of the turbine efficiency. This term is equal to the product of: (i) a given impact factor (𝛽); (ii) the turbine efficiency (𝐸); (iii) a component varying linearly as a function of 𝑃, taking values between zero (when 𝑃 = 𝑃_𝑚𝑎𝑥 ) and one (when

𝑃 = 0).

• The rate of change of robot productivity is influenced by two main factors:

1. A term representing the robots’ effectiveness in maintaining the turbines and tending to increase the robot’s productivity rate of change. This term is the product of: (i) a component varying linearly with respect to 𝑃 that takes values between zero (when 𝑃 = 𝑃_𝑚𝑎𝑥 ) and 𝑃_𝑚𝑎𝑥 (when 𝑃 = 0 ); (ii) the turbine efficiency (𝐸) scaled against its maximum value.
2. A term representing the challenges faced by robots as their productivity increases, thereby tending to decrease the rate of change of 𝑃. This term is the product of three components: (i) the given robot adaptation rate (𝛾), which represents how quickly robots can adjust to changing conditions; (ii) the current value of robot productivity (𝑃); (iii) a component varying linearly as a function of 𝐸, taking values between zero (when 𝐸 = 𝐸_𝑚𝑎𝑥 ) and one (when 𝐸 = 0).

Parameters:

• Control parameter: 𝑟 (ranges from -1 to 1)
• Environmental impact factor: 𝛽 (ranges from 0 to 1)
• Robot adaptation rate: 𝛾 (ranges from 0.01 to 0.1 per day)
• Maximum turbine efficiency: 𝐸_𝑚𝑎𝑥 = 0.6
• Maximum robot productivity: 𝑃_𝑚𝑎𝑥 = 100 units/day

The system exhibits complex behaviour as the control parameter 𝑟 varies. This may lead to critical points where the system’s behaviour changes qualitatively, potentially resulting in multiple stable operating states.

Questions:

1. Construct a mathematical model of the wind farm system from the above verbal description. Clearly state what each of the variables denotes.
2. Analyse the long-term behaviour of the model from 1. for the wind farm system for different values of the control parameter 𝑟. How does 𝑟 affect the system’s performance? Focus on the three 𝑟 values of -0.5, 0, and 0.5. For each of these

𝑟 values, provide the exact values of 𝐸 and 𝑃 after 100 time units, starting from initial conditions 𝐸(0) = 0.1 and 𝑃(0) = 50. Plot the time evolution of 𝐸 and 𝑃 for each 𝑟 value on the same graph, using different colours for each 𝑟 value.

3. For the smart wind farm system, determine the stable operating points for 𝑟 =

0.25 and 𝑟 = 0.75. Provide the exact numerical values of 𝐸 and 𝑃 for each stable point. How do these operating points change as 𝑟 is adjusted? Identify the critical value of 𝑟 where the system’s behaviour changes significantly. Create a bifurcation diagram showing how the equilibrium points for 𝐸 change as 𝑟 varies from -1 to 1.

4. Consider the wind farm system operating at a fixed control parameter 𝑟 = 0.5. The system is now subject to periodic wind speed variations, which can be modeled as a sinusoidal forcing term added to the equation for with an amplitude 𝐴 and 𝜔 the respective angular frequency.
5. (a) Simulate the system’s response to this periodic forcing for 𝐴 = 0.1 and 𝜔 =

0.5 rad/hour over a period of 100 hours. Use initial conditions 𝐸(0) = 0.5 and

𝑃(0) = 50. Plot 𝐸(𝑡) and 𝑃(𝑡) on the same graph.

2. Compute and plot the phase portrait (𝐸 vs. 𝑃) for this forced system. Compare this to the phase portrait of the unforced system (𝐴 = 0) with the same initial conditions.
3. Investigate the system’s response to different forcing frequencies. Repeat the simulation for 𝜔 = 0.1, 1, and 2 rad/hour, keeping 𝐴 = 0.1. Plot the results on a single graph with four subplots, one for each frequency (including 𝜔 = 0.5 from part (a). Discuss how the frequency of forcing affects the system’s behaviour.
4. Calculate and report the time-averaged values of 𝐸 and 𝑃 for each forcing frequency over the last 50 hours of simulation. How does the forcing frequency affect these average values?
5. For the case where 𝑟 = 0.5, provide a comprehensive analysis of the system’s behaviour. Identify all possible steady-state operating conditions (equilibrium points) and classify each point as stable or unstable. For each equilibrium point

evaluate the Jacobian matrix, calculate the eigenvalues and eigenvectors, determine the system’s response to small perturbations based on these results. Create a phase portrait of the system that includes the vector field representing the system dynamics, all equilibrium points, clearly marked and labelled as stable or unstable, the eigenvectors at each equilibrium point. Based on your phase portrait and stability analysis, discuss the overall dynamics of the wind farm system under these conditions, the practical implications of your findings for the wind farm’s operation and management strategies. How the presence of multiple equilibrium points might affect the system’s behaviour and control.

7. Assess the long-term sustainability of this smart wind farm system when operating at 𝑟 = 0.6. Simulate the system for 1000 time units and provide the final values of 𝐸 and 𝑃. Plot the time evolution of E and P over this period. Are there any conditions under which the system might fail or require significant external intervention? How does the presence of multiple stable states affect the system’s resilience?
8. Compare the performance of this adaptive system to a traditional wind farm without smart features. Model the traditional system as

^𝑑𝐸 = −0.1𝐸  𝑎𝑛𝑑 ^𝑑𝑃 = −0.1𝑃.

𝑑𝑡 𝑑𝑡

Simulate both systems for 100 time units with initial conditions 𝐸 = 0.1 and 𝑃 = 50, using 𝑟 = 0.5 for the smart system. Provide the final values of 𝐸 and 𝑃 for both systems. Plot the time evolution of 𝐸 and 𝑃 for both systems on the same graph, using different colours or line styles to distinguish between the smart and traditional systems. Under what conditions would this smart system provide the most significant advantages? Consider the implications of the system’s nonlinear behaviour in your comparison.

Note: In addressing these questions, ensure that your approach includes appropriate computational and analytical methods for solving the problem and analysing the system’s behaviour. Justify your choice of methods and provide clear explanations of your analysis and findings. Your analysis should demonstrate a deep understanding of the engineering principles involved and the practical implications of your results.

When providing numerical results, round to four decimal places. All plots should be clearly labelled with appropriate titles, axes labels, and legends.

Assessment details

You need to submit a typed report (1,200-1,400 words) addressing in detail all the above questions. At least 70% of the words in your report must be dedicated to the discussion of the results. In your report, you do not have to explain the Matlab code you have used but you must discuss any mathematical equations you formed and used as part of the case study.

Along with the report you must submit the respective Matlab code you used to answer the above questions. The Matlab code should be submitted separately as .m files. Prior to submission, you need to make sure that your Matlab code produces all the results discussed in your report.

Marking criteria

Mathematical Modelling and Problem Formulation:Correct formulation of ODEs for turbine efficiency and robot productivity.Accurate representation of the wind farm system dynamics.Proper modelling of system nonlinearities and interactions between variables.Clear explanation of assumptions and limitations of the model.	20 marks
Numerical Methods and Implementation Correct implementation of numerical integration methods.Proper use of Newton-Raphson method for finding equilibrium points.Accurate implementation of eigenvalue analysis and phase portrait generation.Efficiency and robustness of the implemented algorithms.	20 marks
Data Analysis and Interpretation Thorough analysis of system response to different control parameter values.Accurate identification and interpretation of equilibrium points and their stability.Proper analysis of bifurcation behavior and long-term system dynamics.Insightful comparison between adaptive system and traditional wind farm.Critical evaluation of the system’s response to periodic forcing	20 marks
Visualization and Presentation of ResultsClear and informative plots with proper labelling and legends.Appropriate choice of graph types for different analyses.Effective use of subplots and multiple plots to convey information.	10 marks
Discussion and Engineering Insights Depth of analysis and interpretation of results.Ability to relate mathematical results to practical engineering implications.Critical evaluation of the system’s performance and potential improvements.	10 marks
Report Structure and Writing Quality	10 marks

Clear and logical structure of the report.Proper use of technical language and terminology.Adherence to word count and formatting requirements.
MATLAB Code Quality and Documentation Correct functionality of the code.Code efficiency and organization.Adequate commenting and documentation.	10 marks
Total	100 marks

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END OF ASSESSMENT BRIEF