\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"1578",leadTitle:null,fullTitle:"Flow Cytometry - Recent Perspectives",title:"Flow Cytometry",subtitle:"Recent Perspectives",reviewType:"peer-reviewed",abstract:'"Flow Cytometry - Recent Perspectives" is a compendium of comprehensive reviews and original scientific papers. The contents illustrate the constantly evolving application of flow cytometry to a multitude of scientific fields and technologies as well as its broad use as demonstrated by the international composition of the contributing author group. The book focuses on the utilization of the technology in basic sciences and covers such diverse areas as marine and plant biology, microbiology, immunology, and biotechnology. It is hoped that it will give novices a valuable introduction to the field, but will also provide experienced flow cytometrists with novel insights and a better understanding of the subject.',isbn:null,printIsbn:"978-953-51-0626-5",pdfIsbn:"978-953-51-5303-0",doi:"10.5772/2045",price:159,priceEur:175,priceUsd:205,slug:"flow-cytometry-recent-perspectives",numberOfPages:512,isOpenForSubmission:!1,isInWos:1,hash:"fccad401cbcf998ea4de62d524abf82d",bookSignature:"Ingrid Schmid",publishedDate:"June 13th 2012",coverURL:"https://cdn.intechopen.com/books/images_new/1578.jpg",numberOfDownloads:67102,numberOfWosCitations:35,numberOfCrossrefCitations:16,numberOfDimensionsCitations:40,hasAltmetrics:1,numberOfTotalCitations:91,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 24th 2011",dateEndSecondStepPublish:"June 21st 2011",dateEndThirdStepPublish:"October 26th 2011",dateEndFourthStepPublish:"November 25th 2011",dateEndFifthStepPublish:"March 24th 2012",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6",editedByType:"Edited by",kuFlag:!1,editors:[{id:"109787",title:"M.Sc.",name:"Ingrid",middleName:null,surname:"Schmid",slug:"ingrid-schmid",fullName:"Ingrid Schmid",profilePictureURL:"https://mts.intechopen.com/storage/users/109787/images/system/109787.jpg",biography:"Ingrid Schmid, Mag. Pharm. is an Academic Research Specialist in the Department of Medicine at the University of California at Los Angeles and the technical director of the UCLA Flow Cytometry Resource which she established in 1989 in conjunction with Dr. Janis Giorgi. She received her Pharmaceutical Sciences degree from the University of Vienna, Austria. Ms. Schmid has a broad perspective on applications of various flow cytometry techniques and has developed numerous flow cytometry methods. She has published twenty-seven first-author papers, reviews, and book chapters, has co-authored an additional thirty-one publications, and has edited two books. 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Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"59783",title:"Application of Cubic Spline Interpolation Technique in Power Systems: A Review",doi:"10.5772/intechopen.74853",slug:"application-of-cubic-spline-interpolation-technique-in-power-systems-a-review",body:'With electricity becoming an inevitable part of all spheres of human life, it is imminent that the increasing demand for electricity be met. The realization of this necessity has manifested in extensive research in the field of power systems, which has brought to light the complexity of power system. The power system involves the continuous variation in connected loads and increases in power requirement which demands a corresponding increase in the generation. This is met via several means most recently characterized by the penetration of renewable energy sources. The unpredictability of these events entails the deployment of probabilistic-based load flow techniques, estimation of unknown variables and load models which includes uncertainties for their analysis.
All the analysis and research aimed at providing solutions or mitigating the above-mentioned problems require vast amounts of data which is discrete in nature or the analysis techniques involve responses that are discrete in nature. Regardless of the origin of this non-continuity of data, obtaining a continuous response is imperative because of the desired accuracy and the continuity of real-time operation.
The challenges set the stage for interpolation techniques to play an active role in mitigating the problems prevailing in the current power system studies like load forecasting, power system reactive power planning, transmission network expansion, available transfer capability (ATC) determination and market power/clearing price forecasting. Among the various interpolation techniques available, the cubic spline method has been found to be a popular method. Cubic spline also has the desirable characteristics of continuous derivatives at data points which make the design of controllers around these regions possible, and its employment has been seconded by the high accuracy obtained.
The measurements we take for analysis in everyday life are wrought with noise that may be caused by the surrounding environment. Especially in electrical measurements, such random noises may be caused by the magnetic field produced by the current, the presence of stray charges, heating caused by the flow of eddy currents, and so on. One of the possible solutions to the problems caused by noise is to take a large number of measurements. This ensures that the random noise gets canceled out on an average, and hence the integrity of the data is maintained. Other complicated methods of tackling the errors caused by noise are available, but the detailed and in-depth analysis of these methods falls beyond the scope of this chapter.
The first application of cubic splines is in finding the available transfer capability, which is an important parameter in power system operation. Ever since the advent of the deregulated power system, the computation of transfer capability has been a priority. Two quantities that require special attention in these computations are the total transfer capability (TTC) and the available transfer capability (ATC). The TTC of a power system is defined as the maximum amount of power that can be transferred over the interconnected transmission network in a reliable manner while meeting all of a specific set of defined pre- and post-contingency system conditions [1]. ATC is defined as the measure of the additional amount of power that may flow over and above the base case flows without jeopardizing power system security [1]. The system operators of the deregulated power system normally can obtain previously calculated ATC values through an open-access same-time information system. However, the need for a quick calculation of ATC poses a challenge. Some of the methods employed include DC power flow, AC power flow, optimal power flow and sensitivity. Most of the methods mentioned earlier are not sufficiently agile as far as computation speed is concerned or trades accuracy for speed. For example, DC power flow yields quick results at the expense of accuracy, whereas AC power flow compromises speed for accuracy. A solution to the abovementioned problem is found by employing curve-fitting techniques especially the cubic spline interpolation technique.
In order to compute ATC, the cubic spline is employed to trace the curve of the variation of voltage magnitude and power flow with respect to the real power transfer. The ATC is then determined by the point where the limits of voltage magnitude or power flow intersect the curves. The computation of ATC takes place in two different forms—the point-to-point ATC and area-to-area ATC. The area-to-area ATC refers to the additional power that can be mobilized from the seller area to the buyer area, whereas the point-to-point ATC refers to bus-to-bus transfer (usually from a generating bus to a load bus). Another factor considered of paramount importance for the computation of ATC is the effect of contingencies like line outages. Individual consideration of line outages for a large-scale power system is an impractical approach, and hence contingency analysis is carried out using contingency ranking which helps select the critical lines.
The mathematical definition of ATC is shown in Eq. (1):
The definition of the abovementioned terms is as follows:
Transmission reliability margin (TRM): the amount of transmission capability necessary to ensure the security of the interconnected system under a reasonable range of uncertainties in the system.
Capacity benefit margin (CBM): the amount of transmission capability reserved by load-serving entities to ensure access to generation from interconnected systems to meet generation reliability requirements.
Existing transmission commitments (ETC): normal transmission flows.
Traditionally, ATC computation involves the recursive application of AC power flow with increasing power transfers, thereby tracing voltage magnitudes and MVA power flow variations with respect to real power formally called the P-V and the P-S curves as shown in Figure 1. These curves are then employed in conjunction with the limits imposed by acceptable voltage magnitudes and power flows to calculate the ATC. In the cubic spline-based approach, the cubic spline interpolation technique is used to trace the P-V and P-S curves which are then employed to calculate the ATC. This is done by first determining four known points on the curve using AC power flow and then using cubic spline interpolation to trace the curves in between those points. The four known points are denoted as Vi (Pn) and Sij (Pn), where Vi is for voltage of bus ‘I’, Sij is the MVA power flow between bus ‘i’ and ‘j’, Pn is the real power transfer and n is the index for the four points and has values of n = 1, 2, 3 and 4. The incremental steps for tracing the P-V or the P-S curve will be of 1 MW each.
Variation of voltage with real power transfer.
The first and foremost step in the process of curve tracing consists of the determination of the four points on the P-V and P-S curves. This is of paramount importance because the power system undergoes voltage collapse once the real power transfer crosses a certain limit. In the voltage collapse process, the particular bus faces a continuous drop in bus voltage once the critical point load is exceeded. Therefore, the points must be selected carefully making sure that the critical point is not exceeded.
Step 1. Perform the line contingency ranking using line-loading performance index (PIMW) and bus voltage performance (PIv). This is done to identify the critical lines. The critical lines will have a PI value greater than PIbase case.
Step 2. Find a base case by solving the AC load flow. This will act as the first point.
Step 3. Perform the simulation for line outage for one of the critical lines found in step 1.
Step 4. Specify the point or area of transfers. For the point-to-point transfer, generally a generator bus is taken as the selling bus and a load bus as the buying bus. On the other hand, for area-to-area transfer, all the generator busses in a particular area called the selling are and all the load busses in a specific buying area are considered.
Step 5. Now, the next three points are determined. The first step is to determine the fourth and final point P4. In order to do so, the sensitivity method is employed. This method is based on the limiting point of system constraints which is given in Eqs. (2)–(4)
In Eqs. (2)–(4), PTi,VL, PTi,VU and PTij,S are the estimates of P4 which are arrived at by using the linear estimates based on the lower voltage limit, the upper voltage limit and the thermal limit of each line, respectively. The lower limit of voltage VL is taken as 0.9 p.u. and the upper limit Vu is taken as 1.1 p.u. Slimitij is the thermal limit of each line. Vi0 and S0ij are the base case values computed in step (ii). The derivative terms are the reciprocal of the rate of change of voltage and MVA power flow with real power. This can be obtained by solving the AC power flow starting from the base case and making a small change in the transferred real power and noting the change in voltage and MVA transfer. The fourth known point P4 is selected as the minimum of all three values computed above. Then, based on this, P2 and P3 are chosen using the formulae given in Eqs. (5) and (6)
Step 6. Now, find the MVA and bus voltage Vi for each of the newly computed real power values—P2, P3 and P4.
Step 7. Now, use the cubic spline to trace the curve in between these points and obtain the P-V and the P-S curves.
Now, the area-to-area and point-to-point ATCs are calculated by obtaining the point where the voltage limit or the MVA limit line intersects the P-V and P-S curves, respectively.
Inference: The author of [1] has employed the proposed cubic spline-based method to compute the ATC of the Malaysian power system. The performance of the method is measured in terms of its accuracy and speed (which are the main reasons for the employment of the method). For testing the Malaysian power system, it has been simplified into a 241-bus system and further classified into three regions namely north, east, central, south and PUB. The lower and upper voltage limits are taken as 0.9 and 1.1 p.u., respectively. The results obtained have been compared with the traditional recursive AC power flow results.
It is observed from the results obtained in [1] that all the ATC values obtained are due to the MVA limitation. It is further observed that the performance of the cubic spline interpolation method is comparable to the AC power flow method in terms of accuracy. The observation of paramount importance is that while the proposed method obtains the high accuracy as found in the recursive AC power flow method, the time it requires to compute the ATC is much smaller than the AC recursive power flow method—to the extent that in some cases, the time required is up to 30 times lesser. Therefore, it may be concluded that the cubic spline method is superior to AC power flow in terms of speed and superior to DC power flow in terms of accuracy.
In this section, cubic spline is applied in modeling a very complex load—electric arc furnace (EAF). This example presents the true power of cubic splines in tracking extremely complex trajectories. The electric arc furnace’s (EAF) ability to efficiently smelt scrap iron raw materials has made it the backbone of the steel-making industries. The EAF employs high temperatures produced by low-voltage and high-current electric arcs to smelt scrap iron raw materials. The increase in productivity requirements has led to EAFs being designed for high-power applications. The operation of EAFs introduces a significant amount of harmonics, inter-harmonics and flickers in the supply system. Therefore, it is mandatory that the operators pay much attention to the power-quality considerations. In order to study these problems, it is of paramount importance to understand the nonlinear load characteristics that the EAF present to the power system. Also, in order to mitigate the power-quality issues and to further study the impact of EAFs on the power system, it is required that the EAF be modeled after obtaining its time response.
The entire operation of the EAF involves three stages namely striking, smelting and refining. In the striking process, the electric arc is built up by lowering the electrodes of all three phases, the melting process involves the melting of the material and the process ends with the stable refining. Due to the complexities involved in modeling the EAF operation in the striking and the melting processes, most of the research has been directed towards a steady-state modeling of the EAF in the refining stage of operation. The modeling of an EAF requires several parameters such as arc voltage, arc length and arc current, which are determined by the position of electrodes. Therefore, in order to accurately model the EAF, we need to know the field measurements of the electric response which involves the variation of voltage and current. The measured responses are then employed to develop an EAF conductance model using the cubic spline interpolation method.
The EAF is modeled as a function of nonlinear conductance using the cubic spline interpolation technique which is called the cubic spline interpolation model (CSIM). In this method, a set of cubic polynomials are obtained which helps understand the voltage–current characteristic of the EAF.
The steps involved are as follows:
First, a set of ‘n’ measured data points of conductance is obtained for one fundamental cycle of operation. The measurements lying in the interval [a, b] such that a = x0 < x1 < … < xn = b. For the interval between two adjacent points, a cubic function is defined as shown in Eq. (7).
where i = 0, 1, 2,…., n-1. The coefficients ai, bi, ci and di are unknown. These coefficients need to be determined based upon the following constraints:
Step 1. Each spline must pass through the given data points yi.
Gi(xi) = yi and Gn-1(xn) = yn
Step 2. Interior data points between each spline must be continuous.
Gi + 1(xi + 1) = Gi(xi + 1)
Step 3. The first and the second derivatives of the splines must be continuous across the interior data points. Therefore, the spline forms a smooth function.
G’i + 1(xi + 1) = G’i (xi + 1).
G”i + 1(xi + 1) = G”i(xi + 1)
Step 4. In addition to the conditions mentioned in steps 1–3, another boundary condition must be satisfied which concerns the derivative of the functions at the boundaries (at x0 = a and xn = b). There are two types of boundary conditions that may be required to satisfy:
Natural boundary condition: G”0(x0) = G”n-1(xn) = 0.
Clamped boundary condition: G’0(x0) = D0 and G’n-1(xn) = Dn.
where D0 and Dn are the values of the first derivatives of the unknown functions.
It is generally found that the natural boundary conditions give less accurate results than the clamped boundary conditions. Alternately, one could possibly apply a boundary condition called not-a-knot condition, which in addition to the natural boundary condition also incorporates another condition that the third derivative of the function must be continuous at x1 and xn-1.
In order to find the coefficients—ai, bi, ci and di—we follow the following steps and equations:
Step 1. Set
Step 2. Solve
and
and
where
Step 3. Set
Step 4. Set
Step 5. Set
The coefficients
Inference: The author in [2] has analyzed the performance of CSIM on an actual power system model developed in MATLAB/SIMULINK whose online diagram is also presented in [2]. The results obtained via employing the CSIM (shown in Figure 4) are then compared with the results obtained via employing two traditional methods namely (i) harmonic current injection model (HCIM), shown in Figure 2, and (ii) harmonic voltage source model (HVSM), shown in Figure 3. The results described in [2] have been obtained for three different parts—early, middle and later stages—of EAF operation during the refining period. This is done primarily because EAF does not possess a steady-state behaviour that lasts long. Since the HCIM and HVSM are unable to model dynamic behaviour, these methods are compared with the CSIM only during the latter part of the operation because of the dynamic behaviour of the EAF in the early and middle parts (Figure 4).
HVSM V-I characteristic [2].
HCIM V-I characteristic [2].
CSIM V-I characteristic (later stage) [2].
Upon employing the CSIM for EAF modeling in the early and middle stages, it is found that the performance of the EAF during these stages is highly nonlinear. The V-I curve presents a multi-valued function which makes it difficult to model the EAF. The results shown in [2] suggest that upon the comparison, the HCIM method yields the largest errors in terms of both the EAF voltage and current determined. Then, HVSM, which albeit an improvement on HCIM, still contains errors. The proposed CSIM method performs better than HCIM and HVSM and provides a better fit for the voltage and current characteristics. It is also explained in the results that the error encountered during modeling using CSIM in the early and middle stages is larger when compared to that encountered when modeling the later stages of the refining period because the number of sampling points used for the EAF model is not sufficient for modeling in the early and middle stages of refining. As an extension, it is also proposed in [2] that the cubic spline interpolation technique can also be used for modeling other nonlinear loads.
The problem of maintaining a good power factor (greater than 0.85) is a challenge faced by most of the industries. Some of these bulk consumers are penalized for operation under a low power factor. The application of cubic splines is presented to topic address this challenge and provide a commendable solution. Rolling mills and electric arc furnaces constitute very large loads in the power system. When the problem of power factor is considered, these loads (particularly electric arc furnaces) may be termed as a necessary evil because of their extremely low power factor. The variation in the arc length during operation results in the introduction of severe and rapid fluctuations in the reactive power and the voltage, and when the short circuit occurs, the power factor drops to values as low as 0.1. The large impact current and reactive power generated result in significant waste of energy and may also cause the power system to lose its stability. This may cause decadence in the quality of load and endanger the users. The solution to the abovementioned problem is cited in a process called reactive power consumption. Currently, the reactive power compensation is achieved dynamically by the placement of a dynamic reactive power compensation device at access points of such interference loads. These devices are usually represented as a fixed capacitor and thyristor-controlled reactor (FCT). The FCT enables smooth control of the reactive power and also has the desirable feature of maintaining its voltage unchanged. In addition, the FCT can effectively suppress voltage fluctuation and solve the voltage distortion and flicker problem and improve the power quality. The continuous and smooth variation in the reactive power is obtained by the variation of the thyristor conduction angle. For the FCT to effectively carry out its task, it is imperative that it calculates the control angle quickly and accurately in a real-time environment. However, the existence of a nonlinear relation between the control angle and the reactor amplification factor makes the real-time calculation a cumbersome process.
The remainder of this section reviews the calculation and the corresponding control of the control angle via employing cubic spline interpolation technique. The calculation of the control angle, α, is based on the reactor amplification factor for each phase. Cubic spline interpolation is employed because it provides a low-order polynomial interpolation polynomial and also increases the smoothness of the interpolation function.
The basic thyristor-controlled reactor (TCR) consists of a pair of anti-parallel thyristors in series with an inductor as shown in Figure 5. The thyristor delay angle varies between 90° and 180°. As a result, the fundamental current is completely reactive. An increase in the delay angle leads to a decrease in the fundamental reactive current, which is equivalent to increasing the reactance or reducing the susceptance and hence results in a decrease in the reactive power. Hence, the TCR can be seen as being equivalent to a variable susceptance which can be controlled using the delay angle ‘α’. This is because the AC voltage remains constant but the value of the fundamental current changes, which in turn results in the variation in reactive power. The equivalent fundamental susceptance of the TCR is given in Eq. (15)
Thyristor controlled reactor.
The relationship between the delay angle, α, and the amplification factor of the equivalent impedance is shown in Eq. (16)
As can be seen, there exists a nonlinear relation between the delay angle and the amplification factor. The application of cubic splines makes the control of delay angle with reactive power variation easier, quicker and smoother. It must be observed that the inductance current in the TCR does not depend on the inductance but is rather governed by the thyristor conduction angle. The accurate control of the thyristor angle makes the accurate control of inductor current and hence reactive power possible.
The problem is formulated as follows. Eq. (9) suggests that in order to control the delay angle using the amplification factor ‘k’, we need to find a solution to the nonlinear equation. Albeit possible, it is a cumbersome task to solve nonlinear equations, not to mention the considerable amount of time the solving takes which make real-time application and control a tedious task. Moreover, of paramount importance is the accuracy of the controlling action since it determines the reactive power compensation. The cubic spline interpolation is employed as a means to calculate the control angle ‘α’ having known the amplification factor ‘k’. Cubic spline is used as the method of interpolation because of the advantages it provides in terms of simplicity of calculation, numerical stability and smoothness of the interpolated curve.
As previously stated, the existence of a nonlinear relation between the control angle and the amplification factor makes the real-time computation a lengthy process. In order to achieve the aim using cubic spline interpolation, the control angle is regarded as the dependent variable and the amplification factor ‘k’ as the independent variable. According to [3], the author generated 158 data points using Eq. (2). The aim is now to generate 157 interpolating cubic polynomials that fit in between these points. The form of these cubic polynomials is shown in Eq. (17)
In Eq. (10), i = 1, 2, …., 157 and ki ≤ k ≤ ki + =1.
The fitting coefficients a0, a1, a2 and a3 need to be computed for each interval which gives a unique cubic polynomial for each interval. When carrying out the control procedure first, the amplification is determined, and then the corresponding cubic polynomial is used to arrive at the required conduction angle value.
Inference: The results shown in [3] suggest that the conduction angle determined by using cubic splines shows significant match with the values obtained by simulation. Also, the use of cubic splines yields quicker results—a trait which would be beneficial for real-time applications. Hence, it can be said that the cubic spline interpolation method is able to solve the problem of quick computation of the conduction angle with accuracies relevant to engineering applications.
In this section, the application of cubic splines is of paramount importance since it marks the limit of a power system—the maximum amount of power that can be supplied before the continuous uncontrolled drop of voltage. The impact of voltage instability on the power system has been so damaging that significant research has been conducted in this direction. The phenomenon of voltage instability is characterized by a sudden and uncontrollable drop in voltage as a response to a disturbance that has occurred on the power system. This disturbance could be anything from the variation of load to loss of a line or lightning strike, and so on. Most of the studies have been focused on the steady-state aspect of voltage stability. In order to determine the proximity of the system to voltage collapse, we need to estimate or find the voltage stability margin (VSM). Many methods have been proposed in the past to estimate the steady-state voltage stability margin. One of the popular methods is the continuous power flow method. A major disadvantage of the continuous power flow is that it requires a considerable amount of time and hence it cannot be employed in real-time applications.
An alternative approach is to use the time-synchronized voltage and current phasor measurements obtained from phasor measurement units (PMUs). In the presented method, the author proposed a combination of a coupled single-port Thevenin equivalent model and cubic spline extrapolation in order to find the point of voltage collapse or the voltage stability margin (VSM). The concept is based on the fact that the voltage collapse point of the load impedance equals the Thevenin equivalent impedance.
Thevenins’ equivalent voltage and equivalent impedance as seen from load bus ‘i’ as given in Figure 6 can be written as shown in Eq. (18)
Coupled single port Thevenins’ equivalent.
The definitions for the abovementioned variables can be found in [4].
The load impedance ZLi of bus ‘i’ can be arrived by using Eq. (19)
The load bus voltage VLi and the load bus current ILi are obtained through the PMU measurements.
Figure 7 shows the variation of Zli and Zth as a function of the load parameter λ. As can be seen, the Zli equals the Zthi at the point of maximum critical loading, and this point gives the maximum value of the load parameter—λmax. This maximum value of load parameter can be arrived at by equating an approximate function that extrapolates the Zli versus λ curve to the point that it meets the Zthi line.
Variation of Thevenin and load impedance with λ [4].
The choice of cubic spline extrapolation is justified by the author as the superior fitting to the impedance trajectory as evidenced by extensive simulation results. The cubic spline extrapolation proceeds by developing different cubic polynomials for the interval between measurements based on certain constraints.
For ‘m’ PMU-measured data points obtained, there will be ‘m-1’ intervals present, each of which is represented by Eq. (20)
The inputs for extrapolation are taken as three sets of |ZLi| and λ and the point of |Zli| where λ is to be extrapolated and found. The λ thus found corresponds to the point where Zli equals Zthi and hence is equal to λmaxi. Each bus in the considered system will have its own corresponding λmaxs. The λmax = λsys for the entire system is found by taking the minimum of all λmaxi obtained for each of the individual load busses. This value of λsys corresponds to the value of λmaxi for the weakest load bus in the system. The proximity of the load bus to the voltage collapse point is given in Eq. (21)
After the computation of VSM, the author of [4] proceeds to find an index which helps in determining whether the load increase results in the violation of the reactive power limits at the generator busses. The author begins with the simple power equation and arrives at the equation of a surface given by Eq. (22)
where PL, load real power; QL, load reactive power; QG, generator reactive power.
The surface defined lies in the (PL, QL, QG) space. Upon cutting the surface with the constant power factor planes, we get the PL–QG curves which are similar to PV curves or λV curves. The existence of an approximate quadratic relation between λ and V is extended to find a quadratic relation between λ and QG. This approximate quadratic relation is modeled and given in Eq. (23)
In Eq. (16), αi, βi and γi are parameters that are different for each generator bus and need to be determined. The parameters can be determined using three sets of PMU readings. Then, using the computed values of the parameters, an estimate for the extreme of QGi is arrived using the condition that at the extreme limit dλ/dQGi = 0 holds. Enforcing this condition gives us the following value as the estimate shown in Eq. (24)
The index i stands for all the generator busses of the system. At each load step, it is ensured that the estimated value of generator reactive power—Q Giex—remains within the bounds of the reactive power limits that are specified for a generator bus.
The algorithm for proceeding with the described method in order to compute the VSM is as follows:
Step 1. Obtain three sets of PMU measurements and use them to compute three sets of |ZLi|, λ and QGi. Also, we possess the knowledge of the admittance matrix.
Step 2. Compute the parameters αi, βi and γi for each of the generator bus of the system.
Step 3. Compute the estimate of the extreme of the reactive power for each generator
Step 4. If there is no violation of the reactive power limits of any generator, then proceed to step 6. Otherwise, proceed to step 5.
Step 5. Change the bus type from PV to PQ bus for the bus whose reactive power limit has been violated.
Step 6. Compute the impedance matrix as shown in [4].
Step 7. Compute the Thevenin equivalent impedance.
Step 8. Estimate the value of λmaxi by using the cubic spline extrapolation technique for each load bus of the system.
Step 9. Find the value of λsys which is the minimum of all values computed in step 8.
Step 10. Determine the VSM.
Inference: The author of [4] has performed test runs of the proposed algorithm on the following test system—IEEE 30 bus system, IEEE 118 bus system and IEEE 300 bus system. Further, the obtained results have been compared with two other previously available methods. The obtained results indicate the superior performance of the proposed cubic spline technique when compared to the other methods. Upon comparison of the percentage error, it may be observed from [4] that the cubic spline method is almost 10 times more accurate than the other methods.
In this final section, the application of cubic spline technique has been illustrated in the field of deregulated electricity market. In this market, market power issues predominantly spoil the basic idea of maintaining equilibrium within the market players.
Market power is the ability of showing one’s monopolistic nature on the price of the commodities in the market. This has become a challenging issue in the context of the present electricity market and will become more challenging and play a significant role when private generation companies start participating in buying/selling the power [5]. Due to the increase in demand and the regulatory policies, private parties have started investing in the power sector, especially in the renewable energy sources. Thus, it is inevitable for the independent system operator (ISO) to estimate market power for taking crucial decisions [6].
The Herfindahl-Hirschman index (HHI) is used to measure the market concentration that will reflect the number of players in the market and also the inequality in their market shares. The HHI is defined as the sum of the squares of market shares of all the players as given in Eq. (25)
where N is the number of players and Si is the ith player market share in percentage [7].
It measures or indicates the proportional deviation of the price at the firm’s profit-maximizing output from the firm’s marginal cost at that output. It is defined as shown in Eq. (26)
where LIi is the Lerner index for a given firm i, ρi and mci are the price and marginal cost, respectively, and
This index reflects the impact of load variation on market power and geographic difference of market due to network constraints. Must run share (MRS) represents the effect of load variation and nodal must run share (NMRS) represents the geographical difference of market powers. The equation for NMRS is shown in Eq. (27)
where
Steps involved in estimation of market power using NMRS [5, 8].
Step 1. Define the number of generators and their active power limits.
Step 2. Determine
Step 3. Calculate distribution matrix [M−1]
Step 4. Calculate NMRS of generator 1 on load 1.
Step 5. Repeat step 4 for calculating NMRS for the remaining generators on each load.
Step 6. Repeat steps 4 and 5 for various cases.
Step 7. End.
Step 8. Calculate the NMRS for all the generators using the conventional method as discussed in steps–7.
Step 9. Plot NMRS of a generator against the maximum generation of other generators.
Step 10. Connect two points at a time using cubic spline interpolation technique (piecewise polynomial) using MATLAB built-in function, that is, for example, a = spline (b, c, de) and ff = spline (b, c) where ‘a’ gives the interpolated values which correspond to the query points in de [9].
The application of cubic’s spline interpolation for calculating the respective market power of generation companies on all busses for any given load or operating condition is implemented on a sample IEEE 14 bus test system. This system consists of four generator busses and an additional slack generator at bus number 1 in addition to the nine load busses. The interconnection of the system is accomplished with 20 transmission lines.
In Figure 8, the operating condition is such that the total load is increased by 10% from the base case. The fast and accurate calculation of the market power at any given operating point of the generator may serve or help the principal purposes of various bodies like the GENCOs, DISCOs and ISOs. The knowledge of market power for any given operating condition will enable these above bodies to take suitable actions to change their own operating points in order to fulfill their objectives. Figure 8 represents the interpolation plot for the case when load is increased by 10%.
Normal case versus increase in load by 10%.
The interpolation is done between the five known operating points which are the maximum generation levels of Gen3, starting from 60 MW in steps of 10 MW. For these operating points, the NMRS values are calculated. As we increase the maximum generation level of a particular generator, it affects the NMRS of its neighboring generators. In this case, we have monitored the NMRS of Gen2 on bus 3 with the increase in the working range of Gen3 to demonstrate this fact. As a result, the base case plot for Gen2 tends to drop or go down as we keep increasing the generation level values for Gen3. As the load is increased by 10%, the NMRS plot shifts up. Thus, it represents an important observation that as the load goes up at a certain time of day in a particular region, then the GENCOs which lie in such a subsystem will have an increased market power. Due to the piecewise polynomials which are attained due to the application of cubic’s spline interpolation between the operating points, GENCOs, DISCOs and ISOs can easily take suitable actions to not let anyone take undue advantage of the varying market power due to changes in operating conditions. Thus, it helps in attaining a zero market power. This work has been extended under various system conditions and an elaborate study is made in [10].
Inference: Market power reflects the amount of influence that a company has on the system in which it operates but in power systems Market power is the ability to maintain prices above the competitive levels for a significant period of time. Hence, it is of utmost importance to find the market power of the system under normal and abnormal condition that a system has to face.
In this chapter, a complete literature review is made on how cubic spline interpolation technique has been widely used in power systems application. ATC calculation, electric arc furnace modeling, static var compensation, voltage stability margin and market power estimation are some of the areas where cubic spline interpolation techniques are extensively used. The independent system operator (ISO), utilities and consumers can utilize this tool to predict the behaviour of the system/generation companies/utilities and bring back the system towards economic/system stability.
The need for efficient and practical synthesis of biologically active molecules remains one of the greatest intellectual challenges with which chemists are faced in the twenty-first century.
Organic synthesis is a compound-creating activity often focused on biologically active molecules and occupies a central role in any pharmaceutical development endeavor. The field of organic synthesis has made phenomenal advances in the past 50 years, yet chemists still struggle to design synthetic routes that will enable them to obtain sufficient quantities of complex molecules for biological and medicinal studies. The diversity of β-carboline compounds offers a great advantage for being developed into new drugs because of their unique and complex structures, developed through old and underexplored species evolution.
The drive to develop methodology allowing improved access to such compounds has arisen after the demonstration of the useful physical and chemical properties possessed by this class of compounds such as improved lipophilicity and decrease in oxidative metabolism.
Naturally occurring compounds have always played a vital role in medicine and, in particular, β-carboline has progressively become real players in recent drug discovery. The β-carboline moiety represents core structure of several natural compounds and pharmaceutical agents. Compounds containing this subunit are pervasively present in plants, marine organisms, insects, mammalian including human tissues and body fluids in the form of alkaloids or hormones [1, 2, 3, 4, 5, 6, 7]. Several β-carboline-based compounds of natural or synthetic origin are ascribed with different pharmacological properties [8] which include antimalarial [9, 10], antineoplastic [11, 12], anticonvulsive [13], hypnotic and anxiolytic [14], antiviral [15], antimicrobial [16], as well as topoisomerase-II inhibitors [17, 18] and cGMP inhibitors [19] (Figure 1).
Bioactive β-carboline-based compounds.
Further, the significance of β-carboline-based compound is underscored by the way that two of the β-carboline-based mixes Tadalafil and Abecarnil (Figure 2) are clinically utilized for erectile brokenness and CNS issue, individually [20, 21, 22].
β-Carboline-based drugs.
Many bioactive β-carboline-based natural products have been found to be an important source of drugs and drug leads. Most of the natural products of interest to the pharmaceutical industry are secondary metabolites and several such β-carbolines, derived from marine invertebrates, have been in clinical trials as experimental anti-cancer drugs. The significant favorable position offered by utilizing these metabolites as valuable formats, is that they are as such exceedingly dynamic and specific. Being created ordinarily to secure a specific living being, they have been exposed to evolutive pressure for a few a huge number of years and have been chosen to achieve ideal action and to perform particular capacities.
Synthesis of medicinally important β-carboline-based natural products is challengeous in synthetic organic chemistry. Current research activities while primarily with the academic laboratories, have generated convincing evidence that these natural products have an exceedingly bright future in discovery of life saving drugs [23] included antibacterial, analgesic, anti-inflammatory, antimalarial, anticancer, antiparasitic and antiviral agents [24]. Although large numbers of novel β-carboline compounds have been isolated from plants, marine organisms, insects, mammalian including human tissues.
Furthermore, huge numbers of these substances have articulated natural action, without a doubt, not many have been advertised as pharmaceutical products. Some of the compounds have also been valuable as “lead” compounds, which have led to derivatives of them being marketed [25, 26].
In addition, the biological diversity of many of the β-carboline compounds still partially unknown. A considerable lot of them have indicated fascinating bioactivities both in vitro and in vivo measures, although just couple of molecules have been up to this point brought into facilities and onto the pharmaceutical market. Be that as it may, precedents are realized where cutting-edge clinical or preclinical preliminaries, did by utilizing common β-carboline items have prompted promising outcomes in the investigation of new prescriptions a variety of diseases including cancer and infective pathologies. Synthetic organic chemistry is able to produce sufficient amounts for a broad biological application and to provide access to synthetic analogs for structure-activity relationships (SAR) studies.
In particular, alkaloids establish one of the biggest classes of natural products and are synthesized by terrestrial and marine organisms on every transformative dimension and a standout amongst the most encouraging being indole alkaloids. Indole alkaloids, their action, synthesis, and potential use in medicines have been as of now inspected in a few articles [27, 28, 29]. Marine indole alkaloids speak to a rich gathering of characteristic natural compounds and can possibly turned out to be new medicinal chemistry leads for different psychiatric disorders, just as to give better bits of knowledge into the comprehension of serotonin receptor work. These atoms are sensible synthetic targets, which further improve their incentive as conceivable medicinal chemistry studies; be that as it may, hardly any, have been set up as a feature of manufactured or therapeutic science thinks about intended to produce advanced leads.
In this class, β-carbolines that consist of a pyridine ring that is fused to an indole skeleton and biological activity of their derivatives is also well established [30].
Also, substance blend might be utilized to illuminate normal procedures at the atomic dimension through biomimetic approaches, to affirm the structures of natural compounds which are typically settled depending just on spectral information, or to develop new synthetic methods for tackling the challenge of the complex chemical templates designed by nature. Significant endeavors are identified with the structure of particles that in nature are created by metabolic changes happening with high return and rate, and furthermore with high regio-, diastereo- and enantio-particularity.
During the last two decades, β-carboline-based natural products have been the focus of many investigations [31]. The β-carboline is a core-unit of several natural compounds and pharmaceutical agents. Compounds containing this core-unit are pervasively present in plants, marine animals, insects, mammalian including human tissues and body fluids in the form of alkaloids or hormones. Several β-carboline-based compounds of natural or synthetic origin are ascribed with different pharmacological properties which include antimalarial, antineoplastic, anticonvulsive, hypnotic and anxiolytic, antiviral, antimicrobial, as well as topoisomerase-II inhibitors and cGMP inhibitors.
The Pictet-Spengler reaction since its discovery in 1911 has been the key step of the synthetic strategies formulated for obtaining either substituted or fused β-carbolines [32]. The utility of Pictet-Spengler reaction is immense as it allows the option to either construct the tetrahydro-β-carboline (THBC) core first with appropriate substitution which could be extended after cyclization or to install the different substitutions which undergo cascade reactions during cyclization to afford the new THBC derivatives. These THBCs can then be oxidized to generate the desired β-carboline-derivative. However due to major significance associated with this heterocyclic moiety, alternate strategies for generating new β-carbolines are desired. In this context one of the possible strategies could be generation of a β-carboline core that bears a functional group at a suitable position that could be synthetically designed for producing substituted or fused β-carbolines. The presence of an electrophilic site in the form of formyl group in close proximity of the indole NH which is a nucleophilic site makes it an attractive template for the synthesis of substituted and 1–9 annulated β-carbolines. Alternatively, intramolecular cyclization could also be achieved with the N-2 to generate 1–2 annulated β-carbolines.
The synthesis of 1-formyl-9H-β-carboline was firstly reported by Gatta and Misiti [33] while carrying out the studies toward SeO2 mediated oxidation of variously substituted THBCs. During the synthesis of carboline he unexpectedly obtained the 1-formyl-9H-β-carboline instead of the expected 1-methyl,1-phenyl-3-(methoxycarbonyl)-1,4-dihydro-4-oxo-β-carboline when the reaction of the diastereomeric mixture of 1-methyl,1-phenyl THBC was carried out with SeO2 in dioxane. Probably the reaction was preceded through the oxidation of the benzylic moiety affording the benzaldehyde, followed by the aromatization of C-ring and finally the oxidation of the C-1-methyl to the formyl group (Figure 3).
Synthesis of 1-formyl-9H-β-carboline.
Later Gatta and co-workers [34] reported an improved synthesis of methyl 1-formyl-9-H-pyrido [3,4-b] indole-3-carboxylate from 1-methyl-3-methoxycarbonyl-β-carboline via oxidation with SeO2 in dioxane. These workers further reported the application of 1-formyl-9H-β-carboline for the synthesis of canthin-6-one [35]. They extended the synthetic utility of for the generation of pyrimido-[3,4,5-lm]-pyrido-[3,4-b]-indole derivatives in the synthesis of different derivatives of this carboline moiety.
Suzuki et al. [36] reported the total synthesis of various naturally occurring 4,8-dioxygenated β-carboline alkaloids (Figure 4). The synthetic route involved two methodologies (i) an improved Fischer indolization for affording 7-oxygenated indole via protecting the phenolic group with a tosyl group and (ii) construction of a 4-methoxy-β-carboline skeleton by the C-3 selective cyclization of the C-2 substituent of the indole. Then, 4-methoxy-β-carboline was converted into 1-nitrile derivative with diethylphosphoryl cyanide (DEPC) via N-oxide by a modified Reissert-Henze reaction.
Total synthesis of naturally occurring 4,8-dioxygenated β-carboline alkaloids.
Takasu et al. [37] also reported the synthesis of different β-carboline-based compounds including the natural products Kumujancine, MVC (4-methoxy vinyl β-carboline), Creatine and their corresponding salts. They followed the synthetic strategies which involved the Pictet-Spengler reaction of tryptamine hydrochloride with ethyl glyoxylate in ethanol, followed by acylation with acetyl chloride which furnished THBC in 44% yields (Figure 5).
Synthesis of tosyl salt of β-carboline-based compounds.
Condie and Bergman [38] reported the condensation of 1-formyl-9H-β-carboline with ethyl azidoacetate which produced a non-isolable intermediate which immediately underwent intramolecular cyclization via the attack of nitrogen of indole subunit at the ester functionality. The resulting 5-azidocanthin-6-one was further transformed to 5-aminocanthin-6-one via catalytic reduction (Figure 6).
Synthesis of 5-aminocanthin-6-one via intramolecular cyclization process.
Suzuki et al. [39] reported the synthesis of canthin-6-one derivative from 1-formyl-9H-β-carboline and its 4-methoxy derivative. In addition many researchers are continuous trying to do more research in this field. Because the β-carboline gives more interest to natural product chemist and it is a huge scope for researchers (Figure 7).
Synthesis of 5-aminocanthin-6-one.
The Morita-Baylis-Hillman (MBH) reaction have also been used by Singh et al. [40] for 1-formyl-9H-β-carbolines (38) with various activated alkenes led to the formation of expected MBH product (40) as well as unnatural canthin-6-one derivatives (41). It was discovered that exclusive formation of either product 40 or 41 could be achieved by modulating the amount of DABCO used in the reaction as well as the reaction time (Figure 8).
Morita-Baylis-Hillman reaction of 1-formyl-9H-β-carbolines.
In an extension of this study, they disclosed the potential of substituted 1-formyl-9H-β-carboline for achieving the synthesis of indolizinoindole derivatives as depicted in Figure 9. The N-alkylated derivatives (42) were subjected to MBH reaction with various acrylates and cycloalkenones in the presence of DABCO or DMAP to afford the MBH adducts (43) which were transformed into indolizinoindole derivatives 45 (R1 = CO2Me) via reaction with PBr3. The reaction was preceded through the formation of allyl bromide 44.
Synthesis of indolizinoindole derivatives of 1-formyl-9H-β-carbolines.
A Claisen rearrangement have also been used for the synthesis of different β-carbolines by using of allyl alcohol in the presence of p-toluenesulfonic acid, which upon heating at 200°C for 30 min resulted the final product in 84% yield [41]. Alternatively, 4-amino-β-carboline synthesized by Fischer indole synthesis reaction when the hydrazine was used as a reactant, which is postulated to occur via initial hydrazone formation, followed by isomerization and loss of ammonia (Figure 10).
Claisen rearrangement for the synthesis of different β-carbolines.
Another oxidant for changing over tetrahydro-β-carbolines to the completely fragrant framework is elemental sulfur, which is usually utilized when utilization of palladium or platinum is not feasible. For example, in Still’s synthesis of eudistomins52, aromatic esters 53 were produced by heating 52 with sulfur in xylenes at reflux condition [42] (Figure 11).
Oxidation of tetrahydro-β-carbolines.
For the synthesis of 4-alkoxy-β-carbolines 61, Oxidation of tetrahydro-β-carbolines 57 with 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) has also found one of the best way of synthetic method [43] (Figure 12).
Synthesis of 4-alkoxy-β-carbolines.
Using different reaction conditions and reports, it is evident from the past years in medicinal chemistry filed that a wide range of synthetic methods have been reported for the generation of β-carboline moiety and its analogs. However with the new strategy developed for the synthesis of the β-carboline substrate this chapter demonstrated the extensive utility of this prototype design and synthesis of new β-carboline analogs. We believe that this substrate has great potential in medicinal chemistry division and would be more beneficial for pharmaceutical industry.
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