UPTEC F12038
Examensarbete 30 hp
Januari 2013
Optimized Tuning of Parameters
for HVDC Dynamic Performance
Studies
Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student
Abstract
Optimized Tuning of Parameters for HVDC Dynamic
Performance Studies
Axel Andersson
HVDC (High Voltage Direct Current) is used all over the world for transmission of electric power due to lower losses compared to traditional HVAC (High Voltage Alternating Current). However, the procedure of converting AC into DC puts great demand on the control system of the converter stations. These control systems need to be tuned properly to give the HVDC system the correct dynamics to handle variations in the network load and other disturbances.
In this thesis, it was investigated if optimization algorithms can be used for tuning of the control parameters. Focus was on three parts of the control system, the Current Control Amplifier, Voltage Dependent Current Order Limiter and the Rectifier Alpha Minimum Limiter.
The Nelder & Mead Simplex method was used and several different objective functions were tested, including combinations of integral square error, integral absolute error, rise time and overshoot. Several different fault cases and scenarios were tested and results of the optimization were compared to the manually tuned control system.
It was found that the results of the optimization were comparable with the manually tuned parameters for many of the cases tested. The biggest issue encountered was that the optimization algorithm often finds a local minimum in the objective function, leading to a suboptimal solution. This issue could be solved by running the
optimization several times, using different initial values.
It is concluded that using optimization algorithms could be a useful tool for tuning of the HVDC control system.
ISSN: 1401-5757, UPTEC F12038 Examinator: Tomas Nyberg
Ämnesgranskare: Alexander Medvedev Handledare: Hector Avila, Prerna Bihani
Sammanfattning
HVDC (h¨ogsp¨and likstr¨om) anv¨ands idag som alternativ till den h¨ogsp¨anda v¨axelstr¨ommen f¨or att transportera elektrisk energi. F¨ordelen med att anv¨anda likstr¨om ist¨allet f¨or v¨axelstr¨om ¨ar fr¨amst att f¨orlusterna blir l¨agre. Dessv¨arre kr¨aver HVDC-transmissioner stora och dyra stationer som om-vandlar v¨axelstr¨ommen till likstr¨om innan den transporteras. En annan nackdel ¨ar att denna omvandling inte ¨ar helt simpel, tekniskt sett. Det kr¨avs komplicerade reglersystem f¨or att s¨akerst¨alla att man levererar r¨att sp¨anning, str¨om och effekt ut p˚a n¨atet.
Innan en HVDC-anl¨aggning byggs i verkligheten byggs en datormodell av den. Denna modell anv¨ands f¨or diverse tester och simuleringar vars syfte ¨
ar att s¨akerst¨alla systemets funktionalitet. En del av dessa tester kallas dynamic performance studies, DPS. I en DPS testas fr¨amst hur systemet beter sig vid vissa felfall och st¨orningar som skulle kunna intr¨affa vid drift. De olika parametrarna i styrsystemet st¨alls in f¨or att s¨akerst¨alla att sys-temet ˚aterh¨amtar sig tillr¨ackligt snabbt vid dessa fel. Parametrarna st¨alls in manuellt enligt ”trial and error”-princip. Antal fall som testas, samt antal parametrar som m˚aste st¨allas in, g¨or att DPS:en kan ta v¨aldigt l˚ang tid.
Ett alternativt s¨att att st¨alla in dessa parametrar ¨ar att anv¨anda op-timeringsmetoder. D˚a st¨alls parametrarna in automatiskt genom att en dator ber¨aknar fram vilka parametrar som ¨ar b¨ast. Detta g¨ors genom att optimeringsmetoden minimerar en funktion, som kallas m˚alfunktion. Vilka parametrar som datorn kommer fram till beror p˚a val av optimeringsmetod, samt hur man definierar m˚alfunktionen.
I detta arbete testas n˚agra k¨anda m˚alfunktioner, samt n˚agra egna id´eer p˚a m˚alfunktioner, f¨or att optimera tre delar av HVDC-reglersystemet. Op-timeringsmetoden som anv¨andes var Nelder & Mead Simplex-metod och
programvaran som anv¨andes var PSCAD/EMTDC.
Det visades att f¨or tv˚a av de tre delarna av reglersystemet, fann op-timeringsmetoden l¨osningar som var j¨amf¨orbara med de manuellt funna l¨osningarna. Slutsatsen blir s˚aledes att optimeringsmetoder kan vara ett bra hj¨alpmedel vid HVDC-systemstudier.
Acknowledgements
I would like to thank Hector Avila and Prerna Bihani at ABB HVDC for their support and feedback during the course of writing this thesis.
Contents
1 Introduction 1
1.1 Introduction . . . 1
1.2 Purpose and goal . . . 1
1.3 Scope of the thesis . . . 2
2 Background 2 2.1 The HVDC system . . . 2
2.1.1 AC conversion . . . 2
2.1.2 Control system . . . 4
2.1.3 Current Control Amplifier . . . 5
2.1.4 Voltage Dependent Current Order Limiter . . . 6
2.1.5 Rectifier Alpha Minimum Limiter . . . 6
2.2 Dynamic Performance Studies . . . 7
2.2.1 CCA . . . 7
2.2.2 VDCOL . . . 8
2.2.3 RAML . . . 9
2.3 Nelder-Mead Simplex Algorithm . . . 9
2.3.1 Ordering of vertices . . . 10
2.3.2 Calculation of centroid . . . 10
2.3.3 Simplex transformation . . . 11
2.3.4 Termination test . . . 13
3 Method 13 3.1 Finding an objective function . . . 13
3.1.1 CCA . . . 14
3.1.2 VDCOL . . . 16
3.1.3 RAML . . . 18
3.2 Testing the objective function . . . 19
4 Simulation setup 19 4.1 PSCAD’s Optimum Run . . . 19
4.1.1 Initial step size . . . 20
4.1.2 Normalization of parameters . . . 20
4.1.3 Tolerance . . . 20
4.2 HVDC test system . . . 20
5 Results and discussion 21 5.1 CCA . . . 21
5.1.1 Initial objective functions . . . 21
5.1.2 Objective function modification . . . 23
5.1.3 Testing the objective function . . . 25
5.2.1 Initial objective functions . . . 26
5.2.2 Modification of the objective function . . . 28
5.2.3 Testing the objective function . . . 32
5.3 RAML . . . 34
5.3.1 Initial objective function . . . 34
5.3.2 Objective function modification . . . 36
5.3.3 Testing the objective function . . . 40
6 Conclusions 42
References 44
Appendix 45
A VDCOL Comparisons 45
1
Introduction
1.1 Introduction
Today, using HVDC (High Voltage Direct Current) is the most efficient way of transporting large quantities of electric power over long distances. Two reasons for this are that direct current does not suffer from reactive losses and that transmission line costs are lower. HVDC systems are also used for connecting asynchronous AC networks and upholding stability in grids.
The downside to using HVDC are the large converter stations needed to convert the AC into DC. These stations contain a large number of compo-nents and are very costly. The procedure of converting AC into DC also puts great demand on the control system of the stations. These control systems need to be tuned properly to give the HVDC system the correct dynamics to handle variations in the network load and disturbances.
Before implementing the real HVDC system, a software model of the system is built. On this model, extensive tests are carried out to ensure the proper performance and robustness of the system during transient con-ditions. These tests are called dynamic performance studies (DPS) and are an essential part of the development of the HVDC system.
In the DPS, several different cases and configurations are tested to ensure the system complies to specifications. These tests range from minor voltage drops to cable breaks. Several variables need to be taken into account when rating these tests, including recovery time, phase margin and overshoot. The parameters of the control system are tuned until the system characteristics satisfy the specification.
The parameters of the control system have up to this time been tuned by hand, using trial and error. The amount of tests and parameters that need to be taken into account has made the DPS a very time consuming process.
1.2 Purpose and goal
The purpose of this thesis is to find a method to automatize the dynamic performance studies using optimization algorithms in order to save time and resources. Using optimization algorithms could also help find solutions with better performance and robustness than the manually found solutions. The solution the algorithm finds optimal depends on the function which it minimizes. This function is called the objective function.
The goal of this thesis is to find and implement an optimization algo-rithm and an objective function, so that the algoalgo-rithm is quicker and gives better results than manual trial and error. The algorithm and objective function will be implemented in PSCAD/EMTDC, a software for power system simulations.
1.3 Scope of the thesis
The HVDC control system contain a huge number of different controllers and functions with corresponding parameters to be tuned. This thesis focuses on the tuning of three core components of the HVDC control system, the Current Control Amplifier (CCA), Voltage Dependent Current Order Lim-iter (VDCOL) and Rectifier Alpha Minimum LimLim-iter (RAML), all essential for the performance and stability of the system.
2
Background
The goal of this chapter is to explain the necessary concepts needed to understand the problems, methods and results of this thesis.
2.1 The HVDC system
This section describes the basic functionality of the HVDC system. The sys-tem described, and used for this thesis, is the conventional line-commutated current-source converter type.
2.1.1 AC conversion
The principle used to convert AC into DC in an HVDC system is the same principle used in electronics. In electronics, diodes are used for rectifying the AC voltage. In HVDC systems, thyristors are used. Thyristors are es-sentially diodes, which conduct in the forward direction but block in the backward direction. However, they have one important feature diodes lack. Thyristors have an input that controls when the thyristor conducts. How-ever, when the thyristor has switched on and is conducting, it is not possible to switch it off, it will conduct until the voltage across it crosses zero.
Figure 2.1 shows a three phase rectifier bridge using thyristors. It is similar to the classic rectifier bridge used in electronics where diodes are used.
Assume the thyristors conduct at all times in the positive region, effec-tively making them behave like diodes. The output will be a direct voltage as shown in figure 2.2.
Figure 2.2: Voltage output from rectifier bridge
Now assume that a delay is introduced, so that the thyristors conduct a fraction later than they do in the diode case, see figure 2.3.
Figure 2.3: Voltage output from rectifier bridge using firing angle α This delay is called α or the firing angle. It can be seen that the DC voltage is reduced compared to the case where α = 0.
It can be shown [1] that the DC-voltage is given by Udc =
3√2
π Uaccos(α) (2.1)
where Uac is the phase-to-phase RMS voltage. It can be seen that the DC
voltage can be controlled by α which can be varied between 0 and 180 degrees. This corresponds to a change in the DC voltage from 3√π2Uac to
−3√2
Figure 2.4: Voltage output from rectifier bridge using firing angle α, showing the overlap angle µ
In reality, the thyristors are not ideal. There will be some overlap be-tween the already conducting thyristor and the triggered thyristor, leading to the case in figure 2.4. This is called the overlap angle µ.
The rest of the period is called the extinction angle or commutation margin, γ. This leads to the following well known HVDC expression:
α + µ + γ = 180 (2.2)
2.1.2 Control system
The basic HVDC system consists of two connected converter stations, called the rectifier and the inverter.
Figure 2.5: Basic HVDC system
Figure 2.5 shows a basic HVDC system consisting of a rectifier and an inverter connecting two AC networks. The DC power at the rectifier PdcR
is given by
PdcR= UdcRIdc (2.3)
The DC current Idc is given by the voltage drop across the DC line divided
by the resistance R which inserted into equation 2.3 gives PdcR= UdcR
UdcR− UdcI
where UdcR and UdcI are the DC-voltages at the rectifier and the inverter
respectively. The power at the inverter is calculated in the same manner. It is clear that the power transmitted depends on the voltage in the rectifier and the inverter, which is controlled by varying α.
The requested DC power is compared to the actual DC voltage Udc and
the required current order Iorder is calculated. This current order is sent to
the current control amplifier (CCA) which can be seen in figure 2.6.
Figure 2.6: Basic HVDC control system
The CCA is a PI controller designed for a stable and responsive current control. The CCA calculates the necessary firing angle αordin order to keep
the DC current Idc at the requested level. The firing angle αord is sent to
the firing control (FC) and control pulse generator (CPG) which translate αord into firing pulses that are sent to the thyristors.
2.1.3 Current Control Amplifier
The Current Control Amplifier (CCA) is a slightly modified PI controller. Its main objective is to keep the current at a desired level. It is tuned to match the dynamics of the system to give a fast yet stable response. The transfer function can be written as [2]
GCCA(s) = G
1 + KpTis
Tis
. (2.5)
G is called the gain, Kp is called the proportional factor and Ti is the time
constant. These are design variables and are tuned to give the system the proper characteristics.
2.1.4 Voltage Dependent Current Order Limiter
The Voltage Dependent Current Order Limiter (VDCOL) is a protective function located before the CCA. Its objective is to limit the current order when the DC voltage decreases. This is to avoid instability during AC disturbances in the inverter network. It also provides safe restarts after fault clearances. The characteristics for the VDCOL can be seen in figure 2.7.
Figure 2.7: VDCOL function
IO ABS MIN and IO ABS MAX set the global minimum and maximum for the current order respectively. UD LOW , UD HIGH and IO LIM de-cides the location and steepness of the slope.
2.1.5 Rectifier Alpha Minimum Limiter
If the rectifier AC voltage decreases, the firing angle will decrease in order to make up for this loss. If the voltage suddenly goes back to normal, it can cause spikes in the DC current. The Rectifier Alpha Minimum Limiter (RAML) function is used to prevent these spikes by detecting disturbances and increasing the minimum allowed firing angle αmin.
The RAML has two different functions for handling three phase and sin-gle phase faults. Three phase faults are detected via the RAML REF param-eter and single phase faults are detected via the CRAML REFparamparam-eter. If a fault is detected, αminis increased to an angle specified via the DL LEVEL
and CDL LEVEL parameters. When the fault is cleared, αmin will slowly
decrease to its original value. The rate at which it decreases is controlled by the RAML DECR parameter.
2.2 Dynamic Performance Studies
The Dynamic Performance Studies (DPS) are carried out to ensure that the HVDC system meets the system specifications with regard to performance and stability during transient conditions. The parameters of the different functions of the control system are tuned until these specifications are met. This section explains how the DPS are carried out for the CCA, VDCOL and RAML.
2.2.1 CCA
The CCA is tuned by step responses. The parameters tuned are G and Kp.
The time constant Ti is normally not changed [3].
A value ∆I is added or subtracted to the current order of the rectifier. The value of ∆I is usually in the magnitude of 0.1 p.u. This change in the current order causes a step in the direct current. When the current has settled, the step ∆I is removed and the current returns to its normal operating point. Figure 2.8 shows a typical current step used when tuning the CCA.
Figure 2.8: Current step for tuning of the CCA
The performance of the CCA is determined by the rise-time and over-shoot. Rise-time is the time it takes for the current to reach 90 % of the reference step. The rise-time of both the positive and negative steps need to be taken into account. To differentiate between the cases, the rise-time and overshoot of the negative steps will be denoted as fall-time and undershoot, respectively. The desire is to minimize rise-time, fall-time, overshoot and undershoot.
Stability of the CCA is ensured by inspection of the Nyquist curve. The Nyquist curve is drawn by performing a frequency sweep of the system. The main factor taken into account is the phase margin.
The frequency sweep needed to draw the Nyquist curve usually takes a lot of time. To save time, experts use a rule of thumb to make sure the phase margin is acceptable. This rule sets maximum limits on G· Kp and
2.2.2 VDCOL
The VDCOL is tuned by performing voltage drops in the inverter AC net-work. The drops can vary in magnitude and time. Both single phase and three phase faults are tested. Figure 2.9 shows a typical case for tuning of the VDCOL. Shown is the recovery of the DC power after a three phase fault in the inverter AC network. The fault occurs at 0.1 s and is cleared at 0.2 s.
Figure 2.9: DC power recovery after a three phase fault in the inverter AC network
To measure the performance of the VDCOL, the recovery time of the DC power is monitored. The recovery time is calculated from when the voltage drop is cleared, until the DC power has reached 90 % of its pre-fault value. The maximum recovery time is stated in the system specification.
Stability is of outmost importance when tuning the VDCOL. To ensure stability, several variables are monitored. The recovery of the DC power should be controlled and not have severe overshoot or dip after recovery, see figure 2.9.
Figure 2.10: DC voltage recovery after a single phase fault in the inverter AC network
Furthermore, the DC voltage should not spike during recovery, although some overshoot is generally acceptable (usually about 10 % above pre-fault value [4]). Figure 2.10 shows a DC voltage recovery after a single phase fault which would be regarded as acceptable.
The extinction angle γ of the inverter is monitored closely when tuning the VDCOL. It should not drop too low during recovery. Usually a few degrees below its stationary value is acceptable [5]
If γ drops too low, commutation failures could occur. This is because of the physical properties of the thyristors. When commutation failures occur, the ability to control the firing of the thyristors is lost for a short period of time.
Figure 2.11: Extinction angle during recovery after single phase fault in the inverter AC network
Figure 2.11 shows the extinction angle after a single phase AC fault. After the fault is cleared at 0.2 s, γ decreases in a slow and stable manner which is what is aimed for.
For tuning of the VDCOL, the parameters UD HIGH, UD LOW, TC UP REC and TC UP INV are used. UD HIGH and UD LOW are ex-plained in section 2.1.4. The TC UP REC and TC UP INV parameters are part of a low pass filter acting on Ud prior to the VDCOL function for the
rectifier and inverter, respectively.
2.2.3 RAML
The RAML function is tuned by applying voltage drops in the rectifier AC network. Faults of different magnitudes and durations are tested. Both single phase and three phase faults are tested.
The performance and stability is measured in the same way as for the VDCOL, with the exception of the extinction angle, which is not monitored for the RAML.
The parameters tuned in the RAML function are RAML DECR, CRAML REF, RAML REF, CDL LEVEL and DL LEVEL.
2.3 Nelder-Mead Simplex Algorithm
The Nelder-Mead Simplex Algorithm is an algorithm first published in 1965 by J. A. Nelder and R. Mead [6]. The goal of the algorithm is to mini-mize a function of n variables, usually called the objective function (OF). It accomplishes this by forming a simplex which iteratively changes shape and location in order to locate the minimum of the OF. It should not be confused with the Simplex Algorithm of Dantzig, an algorithm for linear programming.
An n-simplex is defined as an n-dimensional polytope, which is the con-vex hull of n + 1 vertices. For example, a simplex in 1 dimension is a line segment, a simplex in 2 dimensions is a triangle and so on. For each iteration the algorithm replaces the vertex with the highest OF value with a vertex of lower OF value. This is performed until a minimum of the OF is found.
The algorithm starts with a user defined simplex of any size. Each iteration of the algorithm include the following steps:
• Ordering of vertices • Calculation of centroid • Simplex transformation • Termination test
The algorithm will continue this loop until the termination criteria have been fulfilled
2.3.1 Ordering of vertices
In this step, the algorithm orders the vertices according to the objective function value at these points so that OF (x1)≥ OF (x2)≥ · · · ≥ OF (xn+1).
2.3.2 Calculation of centroid
The centroid on the opposite side of the vertex with the worst OF value is calculated. The centroid is calculated as
c = 1 n n+1 ∑ i=2 xi
Figure 2.12 shows a simplex in two dimensions with centroid c. In the figure, OF (x1)≥ OF (x2)≥ OF (x3)
2.3.3 Simplex transformation
This step contain different operations, depending on the OF value at the specific points. It starts with the reflection operation.
Reflection The reflection point xr and corresponding OF value is
calcu-lated. The reflection point can be expressed as
xr= c + α(c− x1)
where α is a design constant. In most implementations of the algorithm, α = 1.
Upon reflection, there exist three outcomes that lead to different actions: • OF (xr) > OF (x2): Here, the reflection point is worse than the second
worst vertex. If this is the case, contraction is performed.
• OF (x2) ≥ OF (xr) ≥ OF (xn+1): Here, the reflection point is better
than, or equal to, the second worst vertex, but not better than the best vertex. If this happens, x1 is replaced by xr and the transformation
is complete.
• OF (xn+1) > OF (xr): Here, the reflection point is better than the
best vertex, i e. a new objective function minimum is found. If this happens, expansion is performed.
Figure 2.13: Simplex using reflection point xr, dashed line showing the
orig-inal simplex
Expansion The expansion point is expressed as
xe= c + γ(xr− c)
where γ is a design constant defined by the user. In most implementations, γ = 2.
Figure 2.14: Simplex showing expansion point xe, dashed line showing the
original and reflected simplex
Figure 2.14 shows the simplex after expansion. If OF (xn+1) > OF (xe),
i.e. the expansion point is better than the current best point, x1 is replaced with xe and the transformation is complete. Otherwise, x1 is replaced by
xr and the transformation is complete.
Contraction Contraction is performed using the better of the two points
x1 and xr. The contraction point is defined as (assuming xr is the better
point)
xc= c + β(xr− c)
where β is a constant defined by the user. In most implementations, β = 12
(a) (b)
Figure 2.15: Simplex, performing the contraction operation using (a) xr and
(b) x1 with the dashed line showing the original simplex If OF (xc) is better than the current worst vertex, it replaces it and
the transformation is complete. Otherwise, the reduction operation is per-formed.
Reduction During the reduction operation, n new vertices are calculated as
xi= xn+1+ δ(xi− xn+1)
for i = 1 . . . n. This simplex is then accepted and transformation is com-pleted. Figure 2.16 shows the reduction operation.
Figure 2.16: Simplex showing reduction operation around best point, x3, dashed line shows original simplex
2.3.4 Termination test
When a new simplex has been formed, some termination criterion is tested for stopping the algorithm. Without it, the algorithm would continue until it is stopped manually. Several different termination criteria exist [7]. For example, the algorithm could terminate when the simplex has shrunk to a certain size, or the objective function values of the vertices are close enough, or the number of iteration has reached a certain limit, or a combination of criteria.
The PSCAD implementation uses the objective function termination cri-teria [8]. It terminates when the difference in objective function values be-tween iterations becomes less than a value specified by the user.
3
Method
This section describes the methodology used for this thesis. The methodol-ogy can be divided into two steps:
• Finding an objective function • Testing the objective function
3.1 Finding an objective function
To make an algorithm optimize the performance of the system, the impor-tant factors that define the performance need to be represented mathemat-ically. These factors then form the objective function and minimization is
performed on this function. The minimum of the objective function corre-sponds to what the user has defined as the optimal system.
The most important feature of the objective function is that its minimum corresponds to what normally is considered optimal for an HVDC system. To verify that this is the case, the solution found by the algorithm is com-pared to what an expert HVDC designer would consider optimal. These criteria can be found in section 2.2.
It is also important that the objective function is smooth and contain few local minima, as otherwise the algorithm could converge to non-optimal solutions. To evaluate the smoothness, several different start values are tested. If the objective function converges to the same minimum, using several different start values, the likelihood increases that it is the global minimum.
The process of finding a good objective function takes an evolutionary path. A similar approach is taken in [9]. The idea is to start with an objective function and evaluate how it performs. The objective function is then modified if needed.
For evaluating the different objective functions, the Nelder-Mead Sim-plex algorithm is used. It has been used previously for similar problems with success [9] [10]. It also seems to be the best alternative for multi-variable optimization in PSCAD. The alternative in PSCAD is the Hooke-Jeeves al-gorithm, proposed by R. Hooke and T. A. Jeeves in 1961 [11]. However, the Hooke-Jeeves algorithm tend to converge more slowly than the Nelder-Mead Simplex algorithm, due to its need to evaluate more objective function values per iteration.
An alternative approach is to use some external application, such as MATLAB for example, for the actual optimization. The application would then receive the objective function value from PSCAD, evaluate the new parameter values, and send these values back to PSCAD each run.
Another approach is to write a new optimization module in PSCAD. This makes it possible for the user to choose algorithm freely.
Given the time span and scope of this thesis, it was decided to go with the PSCAD built-in optimization module.
3.1.1 CCA
The CCA is one of the most important functions to design to get the proper dynamics for the system. It is also the function that is most often described in papers on HVDC optimization.
Objective functions widely used for similar problems is the Integral Square Error (ISE)
OF (Id) =
∫
and the Integral Absolute Error (IAE) OF (Id) =
∫
|Iord− Id|dt
Both these functions integrate the error between the reference and actual value of the direct current. It is easy to understand why such functions can be used to tune a step response. Slow step responses would render a big error in the early part of the step, too quick step responses will have a big overshoot. Both these scenarios will render a big integral value. Hence, the minima of the functions will be at some trade-off between slow solutions and quick solutions with much overshoot.
Another objective function that has been tested is a very intuitive func-tion consisting of only the recovery time and overshoot. They are simply added together to form the objective function. The idea is that minimiz-ing this will also optimize the system. This will also lead to some form of trade-off between the two parameters.
Along with performance, stability needs to be taken into account. As described in section 2.2.1, when tuning the CCA, the phase margin has to be taken into account. The way this is handled in this thesis is by limiting G· Kp and Kp as per the rule of thumb.
The need of limiting G· Kp and Kp brings a problem, The Nelder-Mead
Simplex algorithm has no formal way of handling constraints. A way to solve this problem is to punish solutions where the limits are violated. This can be done by adding a piecewise function to the objective function. This function adds a weight whenever the limits are violated. In the case of the CCA, it adds a weight whenever G· Kp or Kp are greater than their maximum values as specified by the user. An equation for such a function is OFconstraints(G, Kp) = C if G· Kp > (G · Kp)max or Kp > Kpmax 0 otherwise
where C is a constant. For this function to have the needed effect, C has to be significantly larger than the objective function. This creates a big step in the objective function when the parameters are outside of their limits, making sure the minimum of the objective function lies inside of the limits. Adding the constraints for stability, the complete expressions for the initial objective functions for tuning of the CCA are the following:
Integral square error
OF (Id, G, Kp) =
∫
where OFconstraints(G, Kp) = C if G· Kp > (G · Kp)max or Kp > Kpmax 0 otherwise
Integral absolute error
OF (Id, G, Kp) = ∫ |Iord− Id|dt + OFconstraints(G, Kp) (3.2) where OFconstraints(G, Kp) = C if G· Kp > (G · Kp)max or Kp > Kpmax 0 otherwise
Recovery time and overshoot
OF (Id, G, Kp) = Wrecovery(Tf(Id) + Tr(Id) + Wovershoot(Yu(Id) + Yo(Id)) + OFconstraints(G, Kp) (3.3) where OFconstraints(G, Kp) = C if G· Kp > (G · Kp)max or Kp > Kpmax 0 otherwise
The functions Tf(Id) and Tr(Id) represent the time it takes for the current
to reach 90 % of the negative and positive step respectively. The functions Yu(Id) and Yo(Id) represent the undershoot and overshoot of the negative
and positive step, respectively. Wrecovery and Wovershootare weights. The
ra-tio of the weights decide how much the two terms contribute to the objective function.
3.1.2 VDCOL
The focus when tuning the VDCOL is a bit different compared to that of the CCA. The point of the VDCOL is to provide stability. An objective function focusing mainly on performance would not be appropriate.
A way of optimizing the VDCOL is described in paper [9]. The way it is done in this paper is by using ISE to optimize the DC current recovery. The reference is a user-defined ramp function. This solution has problems with instability. The way the instabilities are handled in this paper is by adding a piecewise function to the objective function that adds a weight when these instabilities occur.
The approach taken in this thesis is to put stability first, to make sure instabilities do not happen in the first place. Stability is handled by mini-mizing the overshoot in Udand undershoot in γ as described in section 2.2.2.
To perform this, modified ISE functions are used. For the Ud function
OFUd(Ud) = ∫ Yo(Ud)dt (3.4) where Yo(Ud) = { (Udlimit− Ud) 2 if U d≥ Udlimit 0 if Ud< Udlimit
and for the γ function
OFγ(γ) = ∫ Yu(γ)dt where Yu(γ) = { 0 if γ ≥ γlimit (γlimit− γ)2 if γ < γlimit
γlimit and Udlimit are user defined constants.
In this thesis the performance of the VDCOL is measured by the DC power recovery, see section 2.2.2. To optimize the power recovery, integral errors are used. By using integral errors, a quick recovery is ensured while punishing overshoot. Both ISE and IAE are tested. For power reference, a normal reference step is used. By minimizing the overshoot in Ud and
undershoot in γ, stability is upheld during the step.
Adding both performance and stability to the objective function, the expressions for the initial objective functions used for tuning of the VDCOL are the following:
Integral square error
OF (Pd, Ud, γ) = WP ∫ (Pref− Pd)2dt + WU ∫ Yo(Ud)dt + Wγ ∫ Yu(γ)dt (3.5) where Yo(Ud) = { (Udlimit− Ud) 2 if U d≥ Udlimit 0 if Ud< Udlimit and Yu(γ) = { 0 if γ ≥ γlimit (γlimit− γ)2 if γ < γlimit
WP, WU and Wγare weights that decide how much the corresponding terms
contribute to the objective function.
Integral absolute error
OF (Pd, Ud, γ) = WP ∫ |Pref − Pd|dt + WU ∫ Yo(Ud)dt + Wγ ∫ Yu(γ)dt (3.6) where Yo(Ud) = { (Udlimit− Ud) 2 if U d≥ Udlimit 0 if Ud< Udlimit and Yu(γ) = { 0 if γ ≥ γlimit (γlimit− γ)2 if γ < γlimit
WP, WU and Wγare weights that decide how much the corresponding terms
contribute to the objective function.
3.1.3 RAML
Due to the similarities between the tuning of the VDCOL and the RAML, see section 2.2.2 and 2.2.3, the same discussion for optimizing it can be made.
For performance, ISE and IAE are used on the DC power, with a step function as reference. For keeping stability during the recovery, the over-shoot in UDC will be monitored the same way as for the VDCOL.
For tuning of the RAML, the following objective functions are tested:
Integral square error
OF (Pd, Ud) = WP ∫ (Pref − Pd)2dt + WU ∫ Yo(Ud)dt (3.7) where Yo(Ud) = { (Udlimit− Ud) 2 if U d≥ Udlimit 0 if Ud< Udlimit
WP and WU are weights that decide how much the corresponding terms
Integral absolute error OF (Pd, Ud) = WP ∫ |Pref − Pd|dt + WU ∫ Yo(Ud)dt (3.8) where Yo(Ud) = { (Udlimit− Ud) 2 if U d≥ Udlimit 0 if Ud< Udlimit
WP and WU are weights that decide how much the corresponding terms
contribute to the objective function.
3.2 Testing the objective function
The previous steps were about finding a good objective function to optimize the system. In this step, the objective function is tested and the results are compared to the reference solution, tuned by experts.
For the CCA, the standard current step, used throughout the objective function development, is used to compare the two solutions
For the VDCOL and RAML, six critical cases are used to compare the optimized parameters with the reference parameters. These cases are:
• Single phase fault, 10% remaining voltage, 100ms • Single phase fault, 70% remaining voltage, 100ms • Single phase fault, 10% remaining voltage, 300ms • Single phase fault, 70% remaining voltage, 300ms • Three phase fault, 10% remaining voltage, 100ms • Three phase fault, 70% remaining voltage, 100ms
For the VDCOL, these faults are applied to the inverter AC network and for the RAML, they are applied to the rectifier AC network.
To compare the solutions, both performance and stability will be taken into account, using the criteria discussed in section 2.2
4
Simulation setup
4.1 PSCAD’s Optimum Run
Optimum Run is a module available in PSCAD which gives the user the possibility to use optimization algorithms to optimize a set of parameters.
PSCAD 4.2.1 Professional, which was used for this paper, includes two algo-rithms for multi-variable optimization: the Nelder-Mead Simplex algorithm and Hooke-Jeeve’s algorithm.
4.1.1 Initial step size
The initial step size of the algorithms is decided by the user via the Initial Step Size variable. A bigger initial step size means that the algorithm will search a wider area, which could lead to a higher probability that the global minimum is found, but it will also lead to slower convergence and increases the possibility of running into unstable solutions.
It was found experimentally that an initial step size of 10-25 % of the initial parameter values seemed to give a good trade-off among these at-tributes.
4.1.2 Normalization of parameters
The Optimum Run module applies the same initial step size to all parame-ters. This will lead to imbalance among the different parameters due to their different values. Some parameters will have a larger relative step size com-pared to that of other parameters. With the big difference in magnitudes between parameters, this can be quite significant.
To balance this out, it is necessary to normalize the parameters. This can be done by initiating normalized parameters in the module, so that the optimization algorithm sees the parameter as having the same magnitude. Before the HVDC model receives the parameters, they are multiplied with appropriate factors to give them their true values.
4.1.3 Tolerance
The termination criterion for the optimization algorithm is set via the tol-erance variable. The objective function value is compared to the objective function value of the previous iteration. If the difference between these values becomes less than the tolerance, the algorithm terminates [8].
Using a large tolerance can lead to the optimization terminating pre-maturely, even when it hasn’t found a minimum. Using a small Tolerance can lead to unnecessary fine tuning of the parameters, which increases the number of iterations. It was found experimentally that a tolerance of about 10−4 times the expected value of the objective function provided a good trade off.
4.2 HVDC test system
The HVDC model used in this thesis is a back-to-back system with a short circuit ratio of about 3 in both the rectifier and inverter.
5
Results and discussion
5.1 CCA
5.1.1 Initial objective functions
Integral Square Error Figure 5.1 shows the step response of the op-timized CCA using the Integral Square Error objective function, equation 3.1. Here, G = 128.5 and Kp = 0.7. The optimized system is very quick with recovery times for the negative and positive step being 8ms and 10ms respectively. The optimal solution possesses a bit of overshoot. The max-imum overshoot was measured to be 0.0216 p.u. or 27 % which is above what normally is acceptable (15-20 %)
Figure 5.1: Step response with CCA tuned using the ISE objective function
Table 5.1: Test runs using ISE objective function Ginitial Kpinitial Gf inal Kpf inal Runs Obj. function
15 1 127.5 0.71 177 0.6704· 10−4 30 2 115.9 0.78 219 0.6740· 10−4 50 1.5 55.0 1.64 62 0.7120· 10−4 80 1 128.5 0.70 151 0.6703· 10−4 120 0.5 125.3 0.72 88 0.6704· 10−4 170 0.4 147.2 0.61 132 0.6710· 10−4 200 0.3 204.9 0.44 73 0.6940· 10−4
Table 5.1 shows the different test runs used when evaluating the perfor-mance of the ISE objective function. Given that 3 runs with starting values far apart converged to practically the same minimum, it can be concluded that it likely is the global minimum. The number of runs until convergence varied from 62 to 219. Some runs converged to a local minimum very close to the initial guess while some runs converged very far from the initial guess.
Integral Absolute Error Figure 5.2 shows the step response of the opti-mized CCA using the Integral Absolute Error objective function (equation 3.1). The values of G and Kp were found to be 300.1 and 0.3 respectively. The solution found is very quick where the recovery times are about 6 and
9 ms. However, the solution has even worse overshoot than the ISE solu-tion. It also has some oscillations after the negative step. The maximum overshoot was measured to 0.046 p.u or 57.5 % of the total step.
Figure 5.2: Step response with CCA tuned using the IAE objective function
Table 5.2: Test runs using IAE objective function Ginitial Kpinitial Gf inal Kpf inal Runs Obj. function
15 1 194.0 0.44 98 0.2267· 10−2 30 2 187.7 0.47 180 0.2258· 10−2 80 1 190.7 0.47 78 0.2261· 10−2 150 0.6 164.6 0.54 140 0.2269· 10−2 200 0.3 204.8 0.43 48 0.2259· 10−2 250 0.2 253.5 0.33 53 0.2199· 10−2 300 0.2 300.1 0.30 78 0.2188· 10−2
Table 5.2 shows the test runs used to evaluate the IAE objective function. It had problems with local minima, even more so than the ISE case, with only one start guess converging to the minimum objective function value. The runs often converged to solutions very near the initial guess. The number of runs until convergence ranged from 48 to 180.
Recovery time and overshoot Figure 5.3 shows the step response of the optimized CCA using the objective function consisting of the sum of the recovery time and overshoot, equation 3.3. The algorithm found the objective functions minimum to be at G = 144.2 and Kp = 0.62. The resulting step response has recovery times of 7 and 9 ms and a maximum overshoot of 0.022 p.u or 27.5 % of the total step which is comparable to the ISE case but a lot lower than the IAE case.
Table 5.3 shows the test runs used to evaluate the recovery time and overshoot objective function. All runs converged to different minima. The number of runs until convergence ranged from 35 to 119.
Figure 5.3: Step response with CCA tuned using the recovery time and over-shoot objective function
Table 5.3: Test runs using recovery time and overshoot objective function Ginitial Kpinitial Gf inal Kpf inal Runs Obj. function
15 1 33.1 2.19 76 0.5976· 10−1 25 2 33.3 2.20 84 0.5980· 10−1 50 1.5 42.6 1.96 35 0.6027· 10−1 80 1 82.6 1.07 81 0.5510· 10−1 120 0.5 136.8 0.66 81 0.5500· 10−1 170 0.4 144.2 0.62 60 0.5487· 10−1 200 0.4 135.2 0.63 119 0.5507· 10−1
5.1.2 Objective function modification
It is clear from looking at these step responses that these objective func-tions do not produce good results. The most important feature of a good objective function, that it finds the best solution for the problem, is missing. The resulting step responses have too much overshoot. They are also very sensitive to the initial start guesses and have a tendency to converge to local minima.
The overshoot needs to be reduced. The only objective function of the three that can perform this task explicitly is the recovery and overshoot objective function via the weight constants. Increasing the weight of the overshoot should reduce the overshoot. This was experimented with, with poor results. Changing the weight did indeed reduce the overshoot, but it still had the problems of convergence to local minima. Using weights also has another disadvantage. The weights will be very system dependent. Different systems have different characteristics and would need different weights to find the optimal solution. Because of this, the user would have to find new weights for every new system tested. This would lead to an optimization process in itself and would take extra time.
An alternative approach that was tested was to remove the use of over-shoot in the objective function and use it as a constraint instead, the way G· Kp and Kp is constrained. Using the objective function consisting of the recovery time, the user can find the quickest solution with a specified
amount of overshoot. To test this, the overshoot was set to be under an arbitrary value, in this case 0.016 p.u or 20 % of the total step, and the recovery time was optimized. Figure 5.4 and Table 5.4 show the results of this test.
Figure 5.4: Step response with CCA tuned using recovery time, overshoot constrained to 20% of step
Table 5.4: Test runs using recovery time objective function, overshoot con-strained to 20% of step
Ginitial Kpinitial Gf inal Kpf inal Runs Obj. function
10 2 50.0 1.52 34 31.65· 10−3 15 1.75 49.9 1.50 48 31.65· 10−3 20 2 50.0 1.60 34 31.70· 10−3 25 1.75 50.1 1.54 42 31.65· 10−3 30 1.5 49.8 1.72 48 31.70· 10−3 35 1.75 49.8 1.53 35 31.70· 10−3 40 1.5 50.2 1.55 40 31.65· 10−3
Using this setup, all the initial guesses found the same minimum which makes it very likely it is the global minimum. This minimum corresponds to a recovery time of 0.03165 s for the positive and negative step combined. Some runs converged to 0.0317 s, which is a difference of 50 microseconds which also is the time step of the simulated system. This difference is so small it is regarded as negligible. The value of G ranges from 49.8 to 50.2 and Kp ranges from 1.5 to 1.72 which has to be regarded as fairly narrow. The amount of runs until convergence showed consistency and averaged around 40 runs. The initial guess does not seem to have much influence on the convergence speed, with the starting guess furthest from the final value being the quickest with 34 runs.
The obvious downside to this approach with limiting the overshoot is that the user must know what value to set it to. Because of the different characteristics of systems, this could be difficult. Usually however, the de-signer performing the DPS has a good idea what this value should be. If the system is totally unknown, it would be possible to try different values of the overshoot, and see how the recovery time changes. At some point, a small
decrease in overshoot will lead to a big increase in recovery time and vice versa. Between these points there is a range where the trade-off between recovery time and overshoot is good, and the designer could use this range for the CCA.
5.1.3 Testing the objective function
To test this objective function and proposed method, the recovery time was optimized while the overshoot was constrained. This was done for several values of the overshoot, ranging from 10 % to 25 %. The results can be seen in figure 5.5 and table 5.5. Recovery time here is the combined recovery time of the positive and the negative step. Maximum overshoot is the maximum overshoot of either the positive or negative step.
Figure 5.5: Recovery time as a function of overshoot
It can be seen that the recovery time decreases fairly linearly in the interval of about 10 % to 17 % overshoot. From 17 % and up it flattens out some and then declines quickly again. There is no obvious range where the trade-off between recovery and overshoot stands out as being particularly good. Because of the linear characteristics, it is difficult from this test, to make any intelligent choice of G and Kp. The reference solution (tuned manually by experts) has G = 28 and Kp = 2, which according to the table, has an overshoot of around 16 %.
Table 5.5: Test runs using recovery time objective function, overshoot con-strained
Max Overshoot (%) Recovery time (ms) G Kp
10 76.25 7 2.1 11.25 72.65 8 2.2 12.5 66.25 10 2.2 13.75 56.45 14 2.2 15 49.95 21 2.2 16.25 40.50 27 2.2 17.5 35.05 35 2.1 18.75 32.85 43 2.0 20 31.65 50 1.6 21.25 29.45 57 1.5 22.5 23.75 73 1.2 23.75 19.70 81 1.1 25 19.00 94 1.0 5.2 VDCOL
It was found experimentally that weights of WP = 1, WU = 10−4 and
Wγ = 1 provided a good trade-off between performance and stability and
was used along with Udlimit = 1.1 and γlimit = 16 throughout this section.
5.2.1 Initial objective functions
Integral square error Figure 5.6 shows the recovery after a voltage drop in the inverter AC network using the VDCOL parameters obtained by using the Integral Square Error objective function (equation 3.5). The DC power recovers in about 90 ms which is well under the required recovery time of 120 ms. The DC power has a bit of a dip after recovery which is probably due to its quick recovery time. The DC voltage is below its limit of 1.1 p.u at all times. The extinction angle is kept above 16 degrees.
Table 5.6 shows the different test runs used to evaluate the ISE objec-tive function for tuning of the VDCOL parameters. The parameters are presented in order TC UP RE, TC UP INV, UD HIGH, UD LOW. As can be seen, all runs converge to different minima. The number of runs until convergence ranged from 100 to 123.
Figure 5.6: AC fault recovery with VDCOL tuned using ISE objective func-tion
Table 5.6: Test runs using ISE objective function
Initial values Final values Runs Obj. function
0.05, 0.065, 0.8, 0.2 0.036, 0.079, 0.75, 0.15 119 0.4300· 10−1 0.04, 0.05, 0.9, 0.3 0.036, 0.061, 0.73, 0.32 115 0.4748· 10−1 0.02, 0.03, 0.7, 0.15 0.024, 0.045, 0.78, 0.16 123 0.4320· 10−1 0.06, 0.08, 0.6, 0.25 0.048, 0.078, 0.66, 0.34 100 0.4760· 10−1
Integral absolue error Figure 5.7 shows the recovery after an AC voltage drop in the inverter network using the VDCOL parameters obtained using the Integral Absolute Error objective function (equation 3.6). The DC power recovery time is about 80 ms and holds steady above 90 % with only a minor dip. Udand γ stay within their limits during the recovery, which is essential.
Table 5.7 shows the different runs used to evaluate performance of the IAE objective function for tuning of the VDCOL. The number of runs ranged from 74 to 216 which is rather wide compared to the ISE range.
Figure 5.7: AC fault recovery with VDCOL tuned using IAE objective func-tion
Table 5.7: Test runs using IAE objective function
Initial values Final values Runs Obj. function
0.05, 0.065, 0.8, 0.2 0.034, 0.079, 0.83, 0.14 74 0.6925· 10−1 0.04, 0.05, 0.9, 0.3 0.029, 0.060, 0.89, 0.32 125 0.6200· 10−1 0.02, 0.03, 0.7, 0.15 0.034, 0.080, 0.84, 0.14 88 0.6923· 10−1 0.06, 0.08, 0.6, 0.25 0.051, 0.092, 0.70, 0.34 216 0.6205· 10−1
5.2.2 Modification of the objective function
Comparing the two objective functions it can be seen that the DC power recovery is very similar in the two cases. The IAE recovery rises a bit slower compared to the one of ISE. Ironically, this makes the recovery time faster (the time it takes for the power to reach 90 % of pre-fault value). The IAE recovery is smoother, it does not dip as much as the ISE one. However, the differences are very small.
The fact that the objective functions are calculated from the instant of fault clearing, means that these objective functions will tend to favor solu-tions that are quick to get rid of this error. This has one big disadvantage; it goes against the purpose of the VDCOL. The point of the VDCOL is to delay and control the recovery for a smoother and safer restart.
A way to solve this problem would be to not calculate the error in the DC power where its not necessary, in this case during the ramp up. This
is controlled via the lower bound of the integral. Changing this value was tested and it appeared to delay the restart.
Figure 5.8: AC fault recovery with VDCOL tuned using ISE objective func-tion, lower integral bound 0.3 s
Figure 5.8 shows the same fault case as earlier, but with the integral calculated from time 0.3 s and onwards. The ISE objective function was used. Compared to the other case (figure 5.6) where the lower bound was at 0.2 s, just at the fault clearing, it can be seen that the recovery is delayed, and the dip is essentially gone.
Because the integral is calculated from 0.3 s and onwards, the algorithm will search for solutions that have a small error in this range. It is obvious that solutions with a slow recovery will have a large error in this range because they have not recovered at 0.3 s. According to the test earlier, solutions with a quick recovery will have a dip (or possible overshoot) in this range, and will be prevented as well.
Figure 5.9 shows the same fault case using IAE, here the lower bound is also 0.3 s. The results are almost identical to the ISE case which suggests that there is not much difference between them when using the lower integral bound modification. Because of this, only the ISE solution is investigated from here on.
From these tests, it appears that the user can essentially choose the re-covery speed of the system by changing the lower integral bound. Obviously, setting it too low would lead to the unwanted case of too quick solutions, as described earlier. Setting it too high could lead to several different
sce-Figure 5.9: AC fault recovery with VDCOL tuned using IAE objective func-tion, lower integral bound 0.3 s
narios because too many solutions would be considered optimal, i.e. have a zero error within the integral range. The limits of Ud and γ also affect the
recovery time. Tightening these limits would reduce the possible recovery time, but increasing stability.
When tuning the VDCOL, several different test cases have to be tested. Optimizing only one fault case and using these parameters on a different case, it is very unlikely that it would provide the same optimal result. Opti-mizing with regard to all necessary cases is therefore necessary to properly optimize the system as a whole.
A way to run several faults is to simply place them one after another. Figure 5.10 shows such a case consisting of two faults run in succession.
Figure 5.10: Two faults run in succession, single phase and three phase The first fault is a single phase fault, the same that has been used so
far. The second is a three phase fault in the inverter AC network with 10 % remaining voltage and duration 0.1 s. The two faults are similar in the sense that the tuning is performed the same way. This makes it probable that the objective function obtained thus far translates well to a three phase fault case. To optimize both fault cases at the same time, the objective function is calculated for both cases individually. After each run, the objective functions of the two cases are added together and this combined objective function is subject to minimization.
Figure 5.11: Single phase fault recovery with VDCOL tuned using two faults, objective function ISE with lower integral bound 0.3 s
Figure 5.11 shows the single phase fault recovery using the parameters obtained after performing optimization on two faults. It can be seen that the recovery time is slightly reduced compared to tuning only the single fault case, which is expected. However, the reduction is not that big. There is no dip in the DC power but there is a slight halt in the recovery just after it reaches 90 %. Since it does not dip below the 90 % mark, the recovery time is still kept low at just over 100 ms. Ud and γ are a bit more damped
than the single fault case which is expected with the recovery time being a bit slower.
Figure 5.12 shows the three phase fault using the same set of parameters. The recovery of the DC power is very smooth and does not dip below 90 % after it reaches it. The recovery time is about 130 ms. Ud and γ both reach
their limits during recovery.
Figure 5.12: Three phase fault recovery with VDCOL tuned using two faults, objective function ISE with lower integral bound 2.3 s
Table 5.8: Test runs using two faults, ISE objective function with increased lower integral bound
Initial values Final values Runs Obj. function
0.05, 0.065, 0.8, 0.2 0.042, 0.074, 0.84, 0.16 78 0.3396· 10−2 0.04, 0.05, 0.9, 0.3 0.032, 0.059, 0.82, 0.36 67 0.3975· 10−2 0.02, 0.03, 0.7, 0.15 0.041, 0.075, 0.85, 0.14 85 0.3274· 10−2 0.06, 0.08, 0.6, 0.25 0.051, 0.088, 0.72, 0.34 88 0.3766· 10−2 Two starting guesses converged to the same set of parameters. The number of runs ranged from 67 to 85 which shows good consistency.
5.2.3 Testing the objective function
To test the modified objective function, the lower integral bound was set to the desired recovery time, in this case 120 ms after fault clearing. All six critical cases were used, which include:
• Single phase fault inverter AC network, 10% remaining voltage, 100ms • Single phase fault inverter AC network, 70% remaining voltage, 100ms • Single phase fault inverter AC network, 10% remaining voltage, 300ms • Single phase fault inverter AC network, 70% remaining voltage, 300ms
• Three phase fault inverter AC network, 10% remaining voltage, 100ms • Three phase fault inverter AC network, 70% remaining voltage, 100ms The faults were placed in succession, 5 seconds apart. The objective function for all the faults were added together and optimization was carried out on this combined objective function. Default start values were used. The first run, it was found that fault with 10 % remaining voltage and duration 300 ms had severe overshoot in Ud. A few different start values were tested
but the overshoot persisted. To be able to perform the optimization, it was decided to remove the limit of Ud for this particular fault.
Table 5.9: VDCOL parameters before and after optimization, compared to reference values
Parameters
Start 0.05 0.06 0.8 0.25
Final 0.0372 0.0648 0.862 0.260
Reference 0.034 0.049 0.91 0.35
Table 5.9 shows the VDCOL parameters before and after the optimiza-tion, it also includes the reference parameters which were obtained by ex-perts.
The optimization completed in 139 runs. The objective function value using the initial parameters was 0.884· 10−2 compared to 0.324· 10−2 for the optimized parameters.
Single phase fault to ground, inverter side, 10 % remaining voltage, duration 100 ms The complete characteristics of this fault, using the three sets of parameters can be seen in appendix A.1. It can be seen that the optimization has greatly improved the recovery speed of this fault, compared to the initial values. It has reduced the recovery time from 150ms to about 100. The recovery time of the reference solution is around 120 ms which is in line with the desired speed. None of the solutions have any problems with overshoot in Ud or undershoot in γ.
Single phase fault to ground, inverter side, 70 % remaining volt-age, duration 100 ms The characteristics of this fault can be seen in appendix A.2. The recovery speed of this fault is reduced by about 10 ms after the optimization, making it about as quick as the reference solution. All solutions recover within 120 ms. Ud and γ recover without issues for all
cases.
Single phase fault to ground, inverter side, 10 % remaining voltage, duration 300 ms Appendix A.3 shows the characteristics of this fault.
It can be seen that all solutions have problems with this fault. None of the solutions recover within 120 ms, although the optimized solution is the quickest at about 150 with the other two being slightly slower. All solutions have overshoot in Ud. The reference solution has the least initial spike in Ud
but possesses some oscillations. The extinction angle dips dangerously low, in both the optimized and the reference case.
Single phase fault to ground, inverter side, 70 % remaining voltage, duration 300 ms The characteristics of this fault can be seen in appendix A.4. Both the initial solution and the optimized solution recover within 120 ms. The reference solution experiences some oscillations which makes it unable to recover within the desired 120 ms. None of the solutions have any problems with Ud or γ.
Three phase fault to ground, inverter side, 10 % remaining voltage, duration 100 ms Appendix A.5 shows the characteristics of this fault. The optimized parameters greatly improve the recovery speed of the system, reducing the recovery time from about 200ms to about 130ms. By increasing this recovery speed it also increases the overshoot in Ud. The reference
solution is quicker than the initial solution but not as quick as the optimized.
Three phase fault to ground, inverter side, 70 % remaining voltage, duration 100 ms Appendix A.6 shows the characteristics of this fault for the three cases. The initial solution has a recovery time of about 100ms. Both the reference and the optimized solution has a recovery time of about 60 ms which is a big reduction. None of the solutions had problems with overshoot in Ud or undershoot in γ.
5.3 RAML
It was found experimentally that weights of WP = 1 and WU = 10−4
pro-vided a good trade-off between performance and stability and was used along with Udlimit= 1.1 throughout this section.
5.3.1 Initial objective function
Integral Square Error Figure 5.13 shows a single phase fault in the rectifier AC network, duration 0.1 seconds and remaining voltage 10 %. The RAML parameters used were obtained with the integral square er-ror objective function (equation 3.7). The parameters optimized here were CRAML REF and CDL LEVEL. The optimal values obtained were 0.94 and 34. It can be seen that the DC power recovery starts with a fast rise immediately after the fault is released, which also can be seen in Ud, then
flattens out. The recovery time is about 130 ms which is slower than the 120 ms required.
Figure 5.13: AC fault recovery with RAML tuned using ISE objective func-tion
Table 5.10: Test runs using ISE objective function Initial values Final values Runs Objective function
0.9, 60 0.94, 34 45 0.1888· 10−1
0.9, 35 0.91, 34 25 0.1889· 10−1
0.6, 60 0.58, 35 45 0.1923· 10−1
0.6, 35 0.62, 35 37 0.1923· 10−1
Table 5.10 shows the simulations used for evaluating the integral square error objective function). It can be seen that the CDL LEVEL parameter converges to nearly the same value in every run, but CRAML REF barely changes from its starting value, which likely means that CDL LEVEL is more crucial for the end result than CRAML REF. The rate of convergence is good with the number of runs ranging from 25-45.
Integral Absolute Error Figure 5.14 shows the fault recovery using the parameters obtained by using the integral absolute error objective function (equation 3.8). It is nearly identical to the ISE case. The recovery time is the same, around 150 ms, which does not meet the desired recovery time.
Table 5.11 shows the simulations used to evaluate the IAE objective func-tion. As evident by the little difference in the objective function, all simula-tions give roughly the same performance. Like the ISE case, CDL LEVEL converges to a tight range of values, while CRAML REF does not, this im-plies that the result does not depend heavily on the value of CRAML REF.
Figure 5.14: AC fault recovery with RAML tuned using IAE objective func-tion
Table 5.11: Test runs using IAE objective function Initial values Final values Runs Objective function
0.9, 60 0.93, 36 51 0.5712· 10−1
0.9, 35 0.95, 36 43 0.5713· 10−1
0.6, 60 0.49, 34 60 0.5755· 10−1
0.6, 35 0.62, 35 40 0.5781· 10−1
5.3.2 Objective function modification
It is obvious that the two objective functions tested do not meet the system requirements. They are too slow which, like the VDCOL case, could depend on the integral favoring solutions that remove the big error at the instant after fault clearing.
The same argument can be made for the RAML as for the VDCOL. Both the VDCOL and RAML are included for a slow safe restart of the system. To delay the start, the same method that worked for the VDCOL was tested. The lower bound of the DC power error integral was increased by 100 ms.
Figure 5.15 shows the fault recovery using the parameters obtained using the ISE objective function with the lower bound of the integral set to 0.3 s, which corresponds to 100 ms after fault clearing. It can be seen that the spike after fault clearing is greatly reduced. The overshoot in Ud is reduced
by over 50 %. The DC power recovers to 90 % in around 120 ms with only a minor dip after.
Table 5.12 shows the simulations used to evaluate the modified objective function. Again CRAML REF does not change much from its initial value, and CDL LEVEL converges to values fairly close. The differences in the
Figure 5.15: AC fault recovery with RAML tuned using ISE, lower integral bound 0.3 s
Table 5.12: Test runs using ISE objective function, lower integral bound 0.3 s
Initial values Final values Runs Objective function
0.9, 60 0.96, 46 27 0.1242· 10−2
0.9, 35 0.92, 42 23 0.1133· 10−2
0.6, 60 0.70, 46 24 0.1297· 10−2
0.6, 35 0.64, 46 25 0.1300· 10−2
final values most likely depend on the oscillation of the DC power during recovery which creates local minima in the objective function. What stands out the most is the quick and consistent convergence. The number of runs varies from 23 to 27 runs.
This test shows the efficiency of increasing the lower bound of the inte-gral calculating the error in the DC power. It is even more evident in the RAML than the VDCOL. The test also shows that an optimization using this objective function gives a result very close to what is desired, for this type of fault.
When tuning the RAML, both single phase and three phase faults are tested. The objective function developed so far was tested using single phase faults. Three phase faults are tuned similarly, so the same objective function that worked for the single phase case was tested to start with. The parameters used to tune the three phase faults were RAML DECR and DL LEVEL.
Figure 5.16 shows the recovery of a three phase fault in the rectifier AC network. The parameters used were obtained using the ISE objective function with lower integration bound 0.3 s. The DC power recovers in a reliable manner with a recovery time of 140 ms, which is 20 ms above the
desired time.
Figure 5.16: AC fault recovery with RAML tuned using ISE objective func-tion, lower integral bound 0.3 s
Table 5.13: Test runs using ISE objective function, lower integral bound 0.3 s
Initial values Final values Runs Objective function
0.85, 35 0.87, 44 33 0.2217· 10−2
0.60, 60 0.66, 66 95 0.6411· 10−2
0.60, 35 0.65, 45 33 0.2194· 10−2
Table 5.13 shows the simulation used to evaluate the ISE objective func-tion with lower integral bound 0.3 s using a three phase fault. One of the simulations found a solution with very poor performance with an objec-tive function three times as big as the other two which ended up roughly the same. For these two solutions the DL LEVEL values are close but not RAML REF which is similar to the single phase case.
This test shows that the objective function that gives good results in the single phase case also gives a good result in the three phase case.
Just like for the VDCOL, it is essential that the RAML is tuned so that it works for all the necessary fault cases. Optimization with regard to all these cases is therefore necessary. The same approach as in the VDCOL case was tested. A three phase and a single phase fault were placed in succession. Close enough for a quick run time, but far enough to reach steady state before the other fault goes active, see figure 5.17. The objective function of the two faults were calculated individually and added together for optimization. For this test, RAML DECR, RAML REF, CRAML REF, DL LEVEL and CDL LEVEL were tuned simultaneously.
