Dissertation Defense

Dissertation Defense Compensation of Electric Arc Furnaces Based on LaGrange Minimization by: • Leonard W. White May 7, 2012

Introduction Electric Arc Furnaces (EAFs) are among the largest loads connected to the gird. They are disruptive to the grid and are generally compensated – Static Compensators (STATCOMs) being the most common equipment used. The standard compensation strategy uses the Clarke transformation to derive the compensation values. A detailed look into the operation of the compensation systems reveals a different, and more effective, compensation strategy is available. During the following presentation, we will: • Review the operation of, and problems with, the Clarke transformation as applied to EAFs. • The particular nature of EAF waveforms. • Develop the LaGrange minimization compensation strategy. • Review two operating EAFs, one with, one without compensation. • Develop and validate accurate EAF and STATCOM models. • Compare and contrast the dq0 compensation scheme with the LaGrange schemes on an identical model platform. Dissertation Defense

The Clarke Transformation: The Clarke transformation ‘converts’ a 3-phase system into an equivalent 2-phase system. It was developed in 1932 by Edith Clarke. It works by projecting 3-phase values – either voltage or current – onto a set of orthogonal axes. The d axis is aligned with the Vaaxis of the original system. The Vb values and the Vcvalues are each projected onto the d and qaxes and summed with the other projected values on that axis. The system does not have to be balanced for the transformation to be applied – as shown here: If the system is balanced, there is no residue, that is, the d and q values are all that you get. If the system is not balanced, there is a residue, called the “zero” or 0 component. Mathematically, the transformation looks like this: Dissertation Defense

The Clarke Transformation, Con’t: These elements are the sine and cosine values of the 120 angles assumed to be between the vectors. Or The coefficient is added so that the power represented by this system is invariant across the two spaces. There is an inverse transformation that reverses the operation. The transformation is well-known and finds application, especially where there is need to decouple real and reactive power. .... Which can be easily done to the transformed variables with simple operations .... Let’s take a look at the output of the Clarke transformation when the input is an unbalanced 3-phase set of currents.... Dissertation Defense

The Clarke Transformation, Con’t: Allow the inputs to be the following values: The example set is composed of currents; a voltage waveform set would look the same., i.e., there is no loss of generality. The following operations will be performed: 1. Transform the set, point-by-point, into dq0 space. 2. Plot the transformed set of values, Id ,Iq,and I0 in time domain on the same set of axes. 3. Examine the plotted values for meaning. Dissertation Defense

The Clarke Transformation, Con’t: The direct axis component is in black. Iq Id The quadrature axis component is in red. The zero component is in green. If we also have a dq0 set of voltage waveforms we know the following is true at any instant of time from instantaneous power theory: I0 At any instant of time we know the dq0 values of I and V. And for the power in the zero component: We also know the relationship between the dq values of I and V. So we can compute P and Q .... We do not know the value of .... And now, the problem: And there is no way to determine it ! Dissertation Defense

The Clarke Transformation, Con’t: The implications are the following: • We cannot compensate for zero component current using dq0. • We cannot compensate for zero component power using dq0. Power involves the product of voltage and current and, since zero component voltage is typically quite small, power is not a serious issue. Compensating for zero component current is a different problem and one of extreme interest in the operation of Electric Arc Furnaces. A brief summary: We can’t extract complete compensation information with the use of the dq0 approach. A Clarke transformation based compensation strategy will work quite well in any application where the unbalance components are small. It is less than ideal when there are significant levels of unbalance in the system. Mathematically, when the Clarke transformation is employed in a compensation scheme a Degree of Freedom has been lost. Once lost, it can not be recovered.... Aside: Two additional degrees of Freedom are lost if the angles between the waveforms is not 120. This is a relatively minor issue. We will look at this in a more detail when we compare dq0 to LaGrange compensation. Dissertation Defense

The Clarke Transformation, Con’t: It does not matter if the system is a 3-wire or a 4-wire system. • In one case the unbalance current circulates ... • In the other it returns through a common connection ... The point is that the unbalance current is simply not accessible for compensation by use of the Clarke transformation. Next, we will take a detailed look at EAF waveforms …. And, as we will see, these waveforms do not match many of the fundamental constraints that were applied in the development of the Clarke transformation …. Dissertation Defense

EAF Waveforms: Kobe-Weiland Copper Accurate data from EAFs is difficult to obtain. The subject EAF is used to melt copper for recycle; it is rated at 4 MW, with a dedicated 88 kV transmission line used to supply the EAF only. The input voltage to the EAF transformer is 12.47gndY/7.2 kV. Real-time metering data were obtained from the EAF, using a Fluke 443. Eight separate data sets, each containing: Two (2) full cycles with a resolution of 2.4 electrical: • Four (4) voltages – Each phase plus neutral to ground. • Three (3) currents – Each phase. Data captured in this way is extremely reliable and provides hard information about actual EAF operations. The following is a sample of captured EAF data: Dissertation Defense

EAF Waveforms, Con’t: Voltage Waveforms Current Waveforms Data presented is from data set No. 8. All captured data sets are ‘similar.’ When the current waveforms are summed we see that there is a considerable amount of unbalance current …. Dissertation Defense

EAF Waveforms, Con’t: The degree of current unbalance can be seen by looking at the individual ABC waveforms plus the sum plotted on a common axis set: We’ll see this plot later .... Again, all captured data sets are ‘similar.’ Dissertation Defense

EAF Waveforms, Con’t: The overall relationship of arc voltage vs. arc current can be seen in the following plots …. These plots are similar to lissajous figures and clearly show that the v-I relationship is complex. The multiple crossings indicate the presents of harmonic content. Overall, the relationship can go from lead-to-lag and vice-versa several times in a cycle. “lead” and “lag” are steady state concepts; the idea here is that the voltage:current relationship changes over a single cycle. So we see that, in general: Dissertation Defense

EAF Waveforms, Con’t: • The waveforms are non-sinusoidal. i.e., they are not identical from cycle-to-cycle • The waveforms are non-periodic. • The waveforms have no obvious symmetry. • The peak values of the waveforms are different from phase-to-phase. Current waveforms • The peak values of the waveforms are different from cycle-to-cycle. • There are multiple zero-crossing points within one cycle. • The waveforms do not originate from a system in the “steady-state” as this is usually defined. • The voltage/current waveforms vary widely in their lead/lag relationship. Conclusion : • Any analysis that depends on assumptions contrary to the above will be flawed. • Any compensation scheme based upon assumptions contrary to the above will be not provide the expected corrections. Dissertation Defense

LaGrange Minimization: Consider another approach to the problem of instantaneous power: Partition currents into two parts: (1) A part that contributes to only ‘Active’ power. (2) A part that contributes to only ‘Passive’ power. The concept is to minimize the active currents while guaranteeing that the passive currents do not contribute to active power over all three phases. That is: Minimize: Subject to: Dissertation Defense

LaGrange Minimization, Con’t: Introduce a LaGrange operator, , and rewrite the current and constraint equations as a combined set: The object is to determine  such that: First, expand F to obtain: Dissertation Defense

LaGrange Minimization, Con’t: In matrix format: Solving the set for  gives: From which the passive currents are: And the active currents are: Dissertation Defense

LaGrange Minimization, Con’t: Some comments: 1. No assumptions about the nature of the input waveforms were made by the derivation method. 2. The solution is unconstrained mathematically, but there may be constraints imposed by the electrical topology and Kirchhoff’s Laws. 3. There is a second derivation that involves the neutral conductor for application in wye connected systems where there is neutral control. Next, confirm the validity of the method with the two methods of computing instantaneous power. Power under this method is defined as: For reference, the other two methods of expressing the same power are: The following plot shows all three for the same 3-, V-I, input data set Dissertation Defense

LaGrange Minimization, Con’t: Black Pen - Widest Yellow Pen – Middle Red Pen - Narrow instantaneous power is the same at all points of the data set The implication is that if we remove the passive current from the input current waveform the overall input power will remain unchanged. Dissertation Defense

LaGrange Minimization, Con’t: In summary: Complex power under LaGrange is the same as for either the ABC or dq0 power theories. LaGrange minimization requires no real power. For a balanced set of current waveforms LaGrange minimization provides equivalent compensation to dq0. For a un-balanced set of current waveforms LaGrange minimization provide optimal compensation as based on the derivation method. The next step is to devise a validation method and compare LaGrange to dq0 …. Dissertation Defense

Validation of LaGrange Minimization: The original concept was to locate an EAF with STATCOM compensation and to perform the following steps: • Duplicate the physical STATCOM in PSCad • Duplicate the STATCOM control system in PSCad. • Collect voltage and current data from the EAF. • Apply the model using the dq0 compensation technique. • Apply the model using the LaGrange compensation technique. • Compare and contrast the two techniques …. AmeriSteel in Charlotte, NC The following was accomplished: • A steel plant was located with a 33 MVA EAF and a 20 MVA STATCOM. • A non-disclosure agreement was executed between the company and NCSU. • Site visits were made; unprecedeted access was given to all EAF and STATCOM documentation. And then a problem developed: The plant Owner does not have the compensation scheme or control system documentation. Further, this information was not available from the STATCOM manufacturer (ABB). Dissertation Defense

Validation of LaGrange Minimization, Con’t: In the face of this problem, a new validation scheme was devised: • Duplicate the physical STATCOM in PSCad. • Devise a control scheme that would work with both dq0 and LaGrange. • Use an EAF model. • Apply the model using the dq0 compensation technique. • Apply the model using the LaGrange compensation technique. • Compare and contrast the two techniques …. This was accomplished …. Some comments: • The original idea was to use a ‘canned’ STATCOM model but there were several problems. No information about the internal workings of ‘canned’ models. The models use the Clarke transformation as a part of their internal control scheme. Aside: For LaGrange to be valid the dq0 transformation must be avoided anywhere in the scheme…. • The PSCad EAF model has serious problems that keep it from meeting the validation requirements of the work. More a bit later …. Dissertation Defense

The EAF and EAF power supply system: The following diagram shows the overall arrangement of the EAF in relation to the utility system and compensating STATCOM: Utility Point of Delivery (POD) EAF & EAF transformer Utility transmission system Filter @ carrier frequency Utility Power Factor correction capacitor(s) STATCOM /w DC link Starting with the STATCOM model.... Dissertation Defense

A NPC STATCOM model So – a STATCOM model and an EAF model were written from the basic concepts. Both were then validated. First the STATCOM model …. The starting point was the electric utility point of delivery. Harmonic content information was obtained - at a time when the EAF was not in operation. the highlighted harmonics are those that were at lease 0.1% of the total magnitude. these input harmonics were scaled to match the delivery point voltage .... ... and then duplicated in the utility source model .... Dissertation Defense

A NPC STATCOM model, Con’t: The result was the final PSCad model of the utility source: Switching is included so that the model can either provide a pure sine wave – for initial testing – and the actual more complex waveform for final model validation. Overall, the system looks like this: ... The next step is the STATCOM itself .... Dissertation Defense

A NPC STATCOM model, Con’t: The inverter section of the STATCOM is a standard Neutral Point Clamped topology.... Component values are taken from the documentation for the actual plant .... Gates are controlled by use of a standard two-tier, phase-locked, Pulse Width Modulation scheme with the carrier at 1,500 Hz. Same as AmeriSteel STATCOM Dissertation Defense

A NPC STATCOM model, Con’t: Output of the carrier generator looks like this: The relations that are used to generate the gating signals is: This is at reduced frequency (600 Hz) to better display the waveforms Using this carrier with a simulated reference waveform .... Dissertation Defense

A NPC STATCOM model, Con’t: An early test of the gating signals – at reduced carrier frequency for better visualization – looked like this: For all three phases, the overall gating looks like this: A master control line is included to allow the STATCOM to be turned on/off at specific instants in time.... Dissertation Defense

A NPC STATCOM model, Con’t: The overall current regulator looks like this: This is the value of the series inductance This scales the actual voltage back to the reference voltage There’s one of these for each of the three phases .... Dissertation Defense

STATCOM model validation: In order to validate the model it is necessary to direct the STATCOM to deliver positive and negative reactive power at the rating of the STATCOM.... Starting with the dq0 matrix equation, solved for currents .... And forcing P to be zero, gives: Which is incorporated into the following compensation strategy: Desired reactive power from the STATCOM Desired phase currents that will accomplish this goal .... Dissertation Defense

STATCOM model validation: Directing the model to deliver -20 MVA then switch to +20 MVA produces: Power waveforms: Note the accuracy of the directed output …. Voltage & current waveforms : Dissertation Defense

STATCOM model validation: And, the other direction, from +20 MVA to -20 MVA: Power waveforms: Voltage & current waveforms : Dissertation Defense

STATCOM model validation: The STATCOM model has been validated .... In summary: • The STATCOM model produces negative reactive power at its design rating .... • The STATCOM model produces positive reactive power at its design rating .... • The STATCOM model will smoothly transition between the two extremes in either direction .... Next, onward to the EAF arc model .... Dissertation Defense

The EAF model: The existing PSCad EAF model is unsuitable for the work because: Originally used for flicker studies. It does not provide the unbalance that we need to demonstrate the advantages of the LaGrange method .... • It’s a 3-phase model .... • It provides equal loading on all phases .... • It is undocumented .... • The PSCad EAF model created as a part of this work overcomes all these disadvantages. • The original model is based on the following non-linear differential equation set: The base publication references an harmonic domain solution but this was not actually used in the solution of the set. • These equations were solved by several different methods in MatLab and then, once the final solution method was chosen, in PSCad .... Dissertation Defense

The EAF model, Con’t: The MatLab solution: • The MatLab solution attempts to duplicate the work of the original source publication with a method other than a harmonic domain solution .... • The following solution methods were used: • Euler’s method • Runge-Kutta, 4th order • Modified Euler Method (Heun’s Method) • In the publication, the model is driven by a sinusoidal current source .... Note that this is not the way a “real” arc functions. More later .... • The Runge-Kutta and the Modified Euler method both provided results that identically matched the referenced publication .... Reasons: Fast .... Sufficient accuracy – a 2nd order solution .... Fully discrete .... Does not require interpolation of values at each step .... • The Euler method did not provide accurate results – as expected .... • The Modified Euler method was selected as the best solution method .... Dissertation Defense

The EAF model, Con’t: The results from the MatLab solution: These results identically match those from the source publication .... • Again, this solution is driven by a pure sine wave .... • Onward to the PSCad solution .... Dissertation Defense

The EAF model, Con’t: • The PSCad solution drives the equation set with the actual current through the arc. The arc current is determined by application of the circuit constraints .... • This is more difficult. A flow chart of the final solution looks like this: • Provisions are made for the introduction of Gaussian noise. • Gauss noise will be useful for future work, especially with flicker analysis .... • The results of the PSCad solution look like this: Dissertation Defense

The EAF model, Con’t: The results from the PSCad solution: • Note the differences in the Voltage, Current, and Arc Radius waveforms.... • The V vs. I relationship remains similar to the original plot .... Dissertation Defense

The EAF model, Con’t: A 3-phase validation was performed .... For this validation, 3 copies the following circuit were used. The arcs are wye-connected. Each source is phase shifted by 120 from the other: The line currents looked like this: Comments: The Kobe data scale has been reduced so that the size is closer to the model graph .... No Gauss noise has been added to the model.... The model has been submitted to IECON 2012 for publication as “A single phase PSCad electric arc furnace model.” • Which is very close to the original on-site data captured from the working 4 MW Electric Arc Furnace at Kobe .... • The EAF model is ready to applied to the dq0 to LaGrange comparison, but first .... Dissertation Defense

The EAF model, Con’t: A RSCad solution: • This is not a part of the present work. It was completed by Saman Babaei as an independent validation that the solution method can be programmed in RSCad and used on the Real Time Digital Simulator (RTDS) . • As an aside, the source for this simulation is somewhat stiffer than the source that was used for the PSCad model .... • Note the differences in the Voltage and current waveforms .... • The V vs. I relationship remains similar to the original plot .... Dissertation Defense

The dq0 to LaGrange Comparison: • To compare the two compensation methods the STATCOM must be programmed to remove reactive power using, (1) the dq0 approach and then, (2) the LaGrange approach .... • The equations for the dq0 approach are the following: • With • Which gives: • Which reduces to the following implementation: Note the ‘0’ component !!!! This is the standard dq0 compensation approach .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • The matrix equation for the LaGrange passive current is the following: • Which expands to: • Which reduces to the following implementation for passive currents: Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: This is the LaGrange compensation approach .... • Both these circuits were implemented in the same PSCad model, with a switching arrangement to select between the two .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • The remainder of graphs are presented with the following timing: • The model starts at time t=0. • The STATCOM starts at time t=0.2 S with dq0 compensation • The STATCOM switches at time t=1.20 S from dq0 to LaGrange compensation. • First, a test to see how the model deals with a fixed 20 MVA reactive load: The ripple gradually vanishes as the dc offset decays .... STATCOM turns on Switch from dq0 to LaGrange This confirms that both compensation schemes are operating properly .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • Next, a test showing the EAF load .... • This graph shows power delivered by the utility: A smoothed plot will show the changes more clearly .... STATCOM turns on Switch from dq0 to LaGrange • The real power from the utility increases with LaGrange compensation. • The reactive power from the utility decreases with LaGrange compensation. Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • Smoothed power from the utility: • Amount of the real power increase: 500 kW (1.6%) STATCOM turns on Switch from dq0 to LaGrange This clearly shows the advantage of the LaGrange compensation scheme .... • Amount of the reactive power decrease: 1 MVA (11%) • Taking a detailed look at the power increase .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • A detail of the power increase: The power is not only higher but it also appears to be smoother .... To put this into perspective, the increase translates into approximately 27 tons/day of additional steel production.... • The implication is that LaGrange compensation will make the arc more stable as compared to dq0 compensation .... STATCOM turns on Switch from dq0 to LaGrange • Taking a look at the utility line voltage .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • A detail of the Utility line voltage (Phase A, RMS): The line voltage appears to be a bit smoother under LaGrange control .... • The line voltage actually falls a bit when LaGrange is applied. This is due to the relatively low X/R ratio of the entire system – around 3:1 – and is a reflection of the increase in real power delivery. • The increased ‘smoothness’ of the line voltage implies that flicker will probably be lower under LaGrange control .... STATCOM turns on Switch from dq0 to LaGrange • Taking a look at the input current to the system .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • A detail of the Utility line current fundamental: The line current appears to be smoother under LaGrange control .... STATCOM turns on Switch from dq0 to LaGrange • A look at the harmonic content will reveal a slight decrease in 3rd harmonic content and no significant increases elsewhere .... • Taking a look at the power to the arc itself .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: • A detail of the powers to the arc: • The real power to the arc increases by the same amount as the power delivered by the utility .... STATCOM turns on Switch from dq0 to LaGrange • The reactive power to the arc increases; this additional reactive power is delivered by the STATCOM .... • taking a look at the STATCOM .... Dissertation Defense

The dq0 to LaGrange Comparison, Con’t: Note that STATCOM is delivering rated MVA…. • A detail of the STATCOM output: Reactive power increase matches increased reactive power to the arc .... Average real power is zero for both schemes .... STATCOM turns on Switch from dq0 to LaGrange • taking a look at the STATCOM DC Bus .... Dissertation Defense

Dissertation Defense