EEEE4116 Design for a 2-Level Inverter

Advanced Control (EEEE4116)

Coursework 1

Modelling and Advanced Controller Design for a 2-Level Grid-Feeding Inverter

In this assignment you will bring together your skills of state-space equation development and controllerdesign to control a grid-tied 2-Level Converter. The design willmake use of transforming the 3-phase

behaviour of this converter into the dq frame and use the dq equivalent circuit to develop controls. If youhave not yet read the coursework summary, it is highlyrecommended you read this prior to get theunderstanding of what dq transforms are and why we are developing a control system in this way.Figure 1- Notional System Diagram: DC Source interfaced with 3-phase 2-Level Inverter interfaced to the grid. The system under investigation is a very common application when trying tolink renewable energy sourcessuch as solar panels, or energy storage systems to interface them to the national grid, or even microgridapplications where small remotecommunities rely on generatingtheir own power.We are converting DC power into 3-phase AC power to connect to the national grid. The parameters whichwill be used in the design is as follows:= 33uFω

Grid Frequency (rads-1 )= 100π rads-1f

sSwitching Frequency= 20kHzExercise 1 – System Modelling As per the coursework summary, we wish to develop our control strategy using the dq equivalent modelIshown in [ X ] that an equivalent 3-phase inverter can be modelled using the following circuits when

bserving converter dynamics in the dq domain:

gure 2- 3-Phase Inverter dq equivalent average model. Where:md: d-axis modulation index.

q: q-axis modulation index.

ω: frequency of phase voltages (rads-1 )

• Vcd / Vcq: d-axis and q-axis voltage respectively across capacitor
• Iid / Iiq: d-axis and q-axis input current respectively from inverter
• Icd / Icq: d-axis current respectively flowing into capacitor.
• Iad / Iaq: d-axis and q-axis output currents after filter.
• represents a virtual voltage source in the system (due to changing currents in inductor)

• represents a virtual current source in the system (due to changing voltages in capacitor)Hint: Note the directions of the virtual voltage and current sources. Vital to this exercise.Using Kirchhoff’s current and voltage laws on the two circuits shown in Figure 2, develop state-equations for d and qaxis voltage and currents.n our system, we will treatthe modulation indexes as inputs to our system. Using your state equations, go on toshow that the state-space equation defining the model canbe shown to be as: 𝑚 𝑚 𝑑 𝑞 Eq. 1As you may have recognised, although Iad and Iaq are variables in our dq model, this variable is not included within

the state-equation. Similar could be said about Icd, and Icq. Explain why these terms are not present in the finalstate-space equation.In addition, what sources of error doyou think could be attributed in the model, and what effect do you think thiscould have on the system?

Exercise 2 – Transfer Function Depiction Whilst state-space can describe the system with differential equations, it still does not fully replace theransfer function for model development. In fact, often a transfer unction block diagram is first developedto visualize the behaviours of a system and help formulate state-space models.In the first part of this exercise, analyse Fig 2 and construct a 代写EEEE4116  Design for a 2-Level  Inverter transfer block diagram for the above model. A

emplate has been provided below to assist you construct the block diagram. Explain in your report howyou derived the terms in the block diagram._ _ igure 3: Structure of the block diagram for the dq equivalent model. Fill in the blanks and '?' to complete the model. The boxes with ‘?’ should be transfer functions in Laplace Form. The dashed boxes are interconnectionsbetween terms to form the mathematical model of the system.constructing the model, in MATLAB enter your state-space system usingthe ss(A,B,C,D) command.Compare the eigenvalues of the state-space representation to that of the transfer functions in Fig 3, andcan you explain why there is a discrepancy between them?As shown through the state-space equations, and the transfer function block diagram, the two circuitsexhibit cross-coupling (sharing of terms in each other’s models). Why is cross coupling an issue in thecontroller design, and how could you augment the transfer blockdiagram tocompensate for cross-couplingaffects?Exercise 3 – Control Philosophy

We have now derived the state-space equations for our inverter, and modelled the transfer function blockdiagram, all within the dq domain. We will now start designingthe controller for our inverter.We want our converter to be able to respond quickly to any changes in demand from the grid. Therefore, asettling time to maintain the peakvoltage will be set to 200ms. This in turn will influence the phase changeof the inverter similarly.Decide with explanation the choice of poles being used in the pole-placement algorithm and createfeedback controller K for our system. You can augment the system by feeding the output of thecontrollerdirectly into the B Matrix asfollowsFigure 4: Basic state-space block diagram to analyse performance of designed controller. In default condition, you may see the output not change at all. What can you change in the above model toevaluate the closed loop performance of the system.One you can see the dynamics of the outputs, evaluate the response, and reflect ifthe system is behaving

ccording to your design.ou will realise from the system should operate much better than that of open loop, however there is

urrently no ability to attain our desired voltages in this control scheme as we have no references for which

e controller can actuate upon. If you analyse the output of your controller ‘K’ when using the block

diagram in Fig 4, what are the outputs of K tending to, and can you explain why the system is tending tothis value.To integrate references into our model, we must introduce integral states. Identify which states you'll

reference for grid applications. Discuss in your report how these integral states are added to the statespace block diagram, and how they help achieve zero steady state error. Given the expanded state system,reperform the pole-placement with new poles, justifying your selection while ensuring desired converterperformance.In the report, discuss the adaptation to the statespace equation to incorporate integral states, the need forthem, the design of the new poles for the system. Show, with explanations how the state-space blockdiagram is augmented for the inclusion of integral states. Analyse the performance of the system, especiallyany foreseeable issues you can observe with the system? Prove to thereader that even with theadaptation, the system operates still within desired specification. Exercise 4 –  Preloading and Integrator Anti

windup Implementations

Integrator terms are very influential regarding system performance. It’s not simply that having them in our

system allows our system to achieve zero steady state error but can also heavily influence the start-up

performance of any control system, and the step response of a system when there are physics limitations.

Using the working closed loop model with integral states, place a scope on each of the integrator outputs

(that for the states as well as integral states) and the system outputs.

Figure 5: Placement of Scope on Output of integrator block

What do you observe in the relationship between the integrator outputs and the system outputs?A solution to the issues observed is to do something called IntegratorPre-Loading. You may have in the pastused the PI block within Simulink and seen the followinoption to initialize the integrator values but maynot have realised the use in this up until this exercise.Figure 6: Initialization Settings within the PI Block in Simulink You should hopefully recognise that these integratorvalues reach a steady value when all the systemderivatives equate to zero. Note these values down, and inside your integrator block, initialise theintegrator to these values.In doing this, what has changed in the system? Can you explain with reference to the state-space systemwhy the system now behaves the way it does? What assumptionsare made performing this change, andcan would anything need to be considered when apply on physical hardware?our system should be working incredibly well, incomparison to Exercise 3. However, there is still one issuethat still needs to be resolved regarding the integrator. You probably will not see the issue if you havincorporated the integrator pre-loading successfully, so we need to do a step test to highlight this issuehe following would typically be quite irregular for a grid tied inverter, but we are doing this to push the

system into new behaviour. Use a ‘Step’ block on one of your references and apply a sizeable step input at half your simulation time (enough time to allow the system to stabilize before the step reference isapplied). What can you observe happen with the system inputs, and why is this an issue? (If you don’t seeany issues, apply larger steps, and analyse all states and inputsuntil a possible issue may arise)We need to limit the effects of this issue, which raises the issue if integratoranti-windup. If a controller hasan integrator, integrator anti-windup means that the integrator is turned off, when the when the systemhits limit values (or “Saturates”). Can you explain why we would want to turn off the integrator when oursystem reaches these conditions?n Figure 7 is the typical structure of an integrator anti-windup. Analyse the structure, adapt your system toincorporate the Anti-windup scheme below. Clearly describe your methodology and show you understandwhat is going on when you incorporate into your system.Figure 7: Integrator Anti-windup Structure 0Exercise 5 – Real Model Testing

The final part to this coursework is to apply our controller model to an actual circuit to show what has beendesigned would work. Here it is not expected that you build upa whole inverter simulation, and you will beprovided with the backbone of the inverter model as shown in Fig 8.To the simulation, you may choose two pieces of software, MATLAB Simulink, or PLECS. Please ensure youdownload the respective file for the application you wish touse. PLECS simulates power systems faster, butMATLAB is easier to implement the observer design through script. Both pieces of software will produce

the same results for the same system.Figure 8: Simulink Model of Three-Phase Inverter which can be downloaded on Moodle. You task here is to adapt the model to incorporate the controller, into the switching model. Oneimplemented, stress test your controller, note any differences between the average model you made forExercises 1-4, and try and explain why suchdifferences are observed. You may notice that Iiq is a non-zerovalue when the system reaches steady state. Can you explain why Iiq is the value to which it settles at?Stress testing can involve load testing using the connected resistors and apply the switch at a givenmoment in time. Can change the reference values.What happens to the system is you set the resistors to be a very low value (approx. 100Ω)? Can you explainwhy the converter behaves in this way?Exercise 6 – Optimal Controllers Development The methods of controller design covered in this coursework has been a very popular industrial method ofcontroller design for many years, and you have in fact used similar techniques when designing PI controllersin your previous studies. However, as computers have become faster, and algorithms refined, techniques ofcontrolleroptimisation have been rapidly employed in industrial applications.In this exercise, we will briefly look at the Linear Quadratic Regulator (LQR). The mathematics of the controlis relatively complex and out of the scope for this module, however it is important to understand how youcan use software tools such as MATLAB to develop optimal controllers for state-space systems.What is LQR Control? The Linear Quadratic Regulator is an Optimal Control method, which looks at how to drive a system from itscurrent states to the required reference states at a minimal cost, or in other words, to achieve our systemreferences using the least amount of energy.To achieve this, the following cost function is evaluated:

0Where the feedback control law, as normal is defined by Eq. 3:𝑢 = −𝐾𝑥Therefore, the algorithm looks for a value for a controller K, which not only makes the closed loop systemstable, but minimizes the overall control effort J.The way we cantune this system is by selection of the Q and R matrices. With reference to Eq. 2 you cansee that the Q matrix weights the states ‘x’, and the R matrix weights the inputs ‘u’.The Q and R matrices are each diagonal matrices, whose dimensions are equivalent to the number ofstates, and inputs respectively. For example,for our system we would have the following matrices, if Q andEach diagonal element in the Q matrix corresponds to the state of that row. For example, the non-zerovalue on the 2nd row of Q, associates a weighting to the 2nd states of x, in this case Iiq.Likewise, for R, the non-zero value on the 1st row of R corresponds to the 1st input, in this case md.Tuning the controller can be done by understanding these rules:

• Q Matrix Tuningo The smaller the weighting in given states, the more controller effort provided to this state. Inother words, the smaller the value, the faster the bandwidth for that givenstate.

• R Matrix Tuning

o The larger the weighting given to the inputs, the more restricted the respective input.▪ So, a smaller value for R, the faster an input can react to system changes.▪ A larger value for R, the more restricted, and thus slower an input would react.Eq. 2Eq. 3Eq. 4For this exercise, you are going to develop an LQR controller, making use of the lqr() function withinMATLAB. The control architecture will not need any change to thatfrom Exercise 4 & 5.Read through the documentation of the lqr() function in MATLAB and ensure you know how to use thefunction.In your report, describe the process of designing the LQR controller, noting that we still desire our system tostabilise at our references within 200ms. Show a good design philosophy and describe the process in whichyou designed the controller to meet the specification. You may also want to think about the followingquestions in your report when writing about your design:Whatperformance improvement can you observe between the controller designed using LQR and poleplacement in our closed loop system?What are the fundamental differences between LQR and other control techniques such as pole-placement,and what advantages canuse LQR optimisation, and whichscenarios might it be a preferable method ofdesignWhat are the trade-offs involved when choosing the weightings in Q and R? Can you show the impact ofchanging each of the matrices and relate it back to theory?Given the name of this controller is a Linear Quadratic Regulator, do you believe these controllers wouldwork well in non-linear environment, and if not, are there some work arounds you could propose?Writing your report The style of the report should be a technical report.You may use the headings for each exercise to breakdown the subheadings in your report, but the style of the report you submit for your assignment should bein the style as if you were creating a professional document colleagues may look to understand your design.Do not write your report as if answering an exam, i.e Ex5 a) Ex5 b). Doing this you will lose marks inpresentation for your reportEach exercise has a few questions which you are expected to answer, however if there are other elementsof the design which you wish to discuss and analyse further basedon your studies on the course or extrareading, you are highly encouraged to write up on this. This is how you will achieve the top marks in therubric.

Coursework Support Sessions Each week there is a coursework support session on teams. This is your opportunity to ask any questions to the coursework, from use of MATLAB, Simulink, or general theory on controller design. You’rehighly encouraged to attend each of the seminars each week and come prepared with questions.The support sessions take the following structure each week:

• First 10-15 minutes will involve a small presentation or demonstration on different aspects of thecontroller design, or small software tutorial.

• Remaining 45 minutes will be a group Q&A. Please do have questions ready as this part of thesession is directed by you.

• If there is time remaining 1-to-1 help sessions can take place. It will be first come first serve. If I amunable to get to people requested, I will attempt to meet you another time in the week when I amfree.For any coursework questions, please contact Dr. David Dewar (David.Dewar1@nottingham.ac.uk).

Submission Requirements: This report should be no longer than 20 pages (title pages and contents page are not included)Please do not include anything which might identify you as the writer of the report. All reports are to bemarked anonymously.Therefore, please DO NOT INCLUDE any names, student ID’s or emails in the document.