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Interactive Power Systems Calculator

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1. Upper voltage pu magnitude limit vmlb Lower voltage pu magnitude limit plt Upper load limit at the bus plb Lower load limit at the bus Additional generator data i Internal generator index ireg Internal index of a controlled bus pgcost Coefficients of a real power cost polynomial Additional branch data i Internal from bus index j Internal to bus index Table 3 1 Additional Data 32 3 3 Matlab Interface Functions Matlab interface to ipsys back end algorithms provides data input output and calculation functions The input output functions read and write data from PTI23 network and cost polynomial files Any subsequent reading modification or deletion of the data is done through Matlab The included algorithms are the same ones as in ipsys and should give exactly the same results This section describes all of the interface function and similar de scriptions can be viewed from Matlab using help function The design behind the Matlab interface uses a two step approach User has access to pl functions which in turn call cpp functions This way a user has a two chances to process the input and or output data once in Matlab pl_ file and in cpp file before the data is passed to back end functions This should allow considerable data control Even without changing pl_ files a complete reconfiguration of a network can be done from Matlab allowing a wide range of experiments to be performed Whenever a back
2. or develop custom Graphical User Interface GUI The both of these approaches are used for a development of a power systems simulator and their use is described in this manual along the generic framework features The Interactive Power System Simulator IPSYS is a scripting tool used to define manipulate and analyze electrical power systems described us ing Power Technologies Inc input data format as described in 4 A user can interact with a single or multiple power systems through IPSYS shell Matlab interface MIPSYS or a single network using a GUI interface GIP SYS IPSYS libraries can be used for developing different purpose interactive programs using the C programming language Power networks are com pletely modeled as objects and can be operated on through a set of message passing methods Separate power network objects can be interfaced by set ting the power generation in one of the systems and matching load in the adjacent system Determining the level of power exchange can be done either using the IPSYS shell or C programming This User Manual describes IPSYS MIPSYS and GIPSYS functionality application Programming In terface API is described in a different manual IPSYS shell interface features can be grouped in four distinctive parts data types general type operators and functions program control and scripting and specialized power systems routines IPSYS supports real number ma trix and character string
3. t as a tab and n as a new line The following IPSYS string variable prints out as ipsys gt str Here are 2 tabs t t and a new line n ipsys gt printf str Here are 2 tabs and a new line This completes the discussion of IPSYS data types These data types are sufficient for procedural programming commonly used for scientific purposes 11 2 2 Built in Operators and Functions IPSYS interprets the usual arithmetic operators and one argument functions of different data types in an intuitive way Addition subtraction multipli cation and division can be used with real numbers matrices and a combi nation of both The only requirement is that the matrix matrix operations have compatible dimensions When a two argument operator is applied to a real number and a matrix the operator is applied element wise That is the operation is between the scalar and each of the matrix elements For example ipsys gt a 1 2 3 4 ipsys gt printf 2 3f 1 a 1 000 0 500 0 333 0 250 1 ipsys gt printf 3 3g 2 a 2 a 4 8 12 16 1 ipsys gt printf 3 3g a 2 a L 0 5 1 1 5 1 Examples of matrix matrix operations ipsys gt a 1 2 3 4 ipsys gt b 2 3 5 7 ipsys gt printf 3 3g a b 12 17 26 37 ipsys gt printf 3 3f a b b a 12 1 000 0 000 0 000 1 000 1 ipsys gt printf 3 3f a c Incorrect argument dimensions Matrix sub Matrix amp Line
4. 0 033 0 614 0 009 0 236 0 344 0 085 0 877 0 334 0 322 0 203 0 757 0 817 0 044 0 927 0 707 0 197 0 395 0 833 0 822 0 240 0 449 0 085 0 933 0 971 Table 2 7 lists I O functions and some miscellaneous functions Functions IsConsts IsVars IsFuncs and IsMem list the available constants variables 27 Function Required Optional Return Name Arguments Arguments Value printf 1 format string 2 one or more 1 number of chars variables printed fprintf 1 file name 3 one or more 1 number of chars 2 format string variables printed fopen 1 file name 1 success 1 2 mode string or failure 0 fclose 1 file name 1 success 1 or failure 0 IsConsts 1 number of constsnt IsVars 1 number of variables IsFuncs 1 number of functions IsMem 1 number of objects clsVars 1 variable list 1 number of deleted variables clsFuncs 1 functions list 1 number of deleted functions sleep 1 miliseconds 1 miliseconds sleeping Table 2 7 Input Output and Miscellaneous Functions used scripting functions defined and the number of objects in memory clsVars and clsFuncs can be used to clear variables or functions from IPSYS memory if they are used without any arguments all of the variables or func tions are cleared Sometimes when IPSYS CLI controls the GUI display from a loop it is needed to slow down the display of the results on the GUI network For this purpose sleep function is implemented whi
5. 1 as the load ID until a multi load options is implemented Even in the currently used format there can be multiple generators on a single bus and they should be named as described by PTI23 format in Appendix B pl_pti23Net is the Matlab function used to read both network description and cost functions It takes up to five arguments of which only the first one is required and returns the net structure The first argument is the name of the PTI23 input file the second argument is the P cost functions file the 58 third argument is the Q costs file the fourth is P costs and the fifth is the Q demand curves file Of this five parameters only the first one is required and Q and Q are to be used in the future If an argument must be entered but unused use and empty matrix For example the following will read in the data for the system in the Figure 6 1 gt gt b4 pl_pti23Net allen raw allen sup allen dmd b4 Title1 0 100 00 Tue Oct 25 16 18 59 2005 Title2 GIPSYS Title3 Ic 0 SBASE 100 bus 1x4 struct gen 1x2 struct branch 1x5 struct swing 1 pvindx 2 pqindx 2x1 double Once the system data is read in any of the algorithms listed in table 3 2 can be applied to the system As explained before Matlab passes data by value so the arguments remain unaffected and the results can be captured through the return value only For example the following runs Newton Raph
6. 3 7059 3 7059 3 70587 L MW 5 8227 5 8227 5 82273 G1 11 9104 11 9106 11 91 Go 16 7937 16 7936 16 79 L 20 0492 20 0487 20 05 Lo 18 4214 18 4219 18 42 Fi MW 3 70000 3 70000 3 70 F 2 13 0222 13 02 Table 5 1 Optimal welfare calculations Although the numbers are almost exactly the same 1 discusses the market setup as a multilateral transaction while the above is the poolco formulation Nevertheless the both formulations find the same optimal point 57 Chapter 6 Matlab Examples In this chapter a few examples are used to demonstrate the main Matlab interface features For the demonstration purposes a four bus network 6 1 with associated real power cost functions and real power demand functions is used 6 1 Matlab Input Output Matlab and ipsys use the same input file formats There are currently three different input files that can be used by Matlab to define a power systems net work The only mandatory input file is the network definition file in PT123 format The other two files provide the real power generation cost and the load demand The internal functions do limited checking of compatibility of these three files and user should pay attention that the real power limits in PTI23 file match the real power limits in defined by the cost functions Also PTI23 allows only a single load per bus To allow for a possibility of multiple loads per bus the demand file should use the default
7. 48 49 55 56 16 17 ipsys gt printf 3g t 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Note that the index matrices are actually row vectors and that indexing is one based To index contiguous matrix elements Range function can be used to generate index range For example the first two rows of a matrix can be extracted as ipsys gt a Matrix shiljak mat ipsys gt printf 6 3f a Range 1 2 Range 1 Cols a 0 000 0 000 0 010 0 000 2 150 0 000 0 040 0 000 0 070 1 100 0 030 0 000 0 060 0 020 0 000 0 250 1 IPSYS provides function for construction of constant and identity matrices Following commands calculate the inverse of matrix a ipsys gt a 1 2 3 4 5 4 3 2 1 ipsys gt i Idn 3 ipsys gt printf 3 2f i a 0 38 0 50 0 88 1 00 1 00 1 00 0 87 0 50 0 37 1 ipsys gt printf 3 2f a i a Identity transposed and constant matrices can be constructed as follows ipsys gt a Matrix 3x3 mat ipsys gt printf 3 2g a a 1 2 GI 4 5 6 7T 8 9 ipsys gt printf 3 2g Trn a 1 4 7 2 5 8 3 6 9 10 1 ipsys gt printf 3 2g Idn 3 1 0 0 0 1 O 0 0 1 Character strings in IPSYS are defined by double quotes ipsys gt a 1234 234 In this example 1234 234 is a character string and any numerical manip ulation will cause an error In addition to the ASCI character set IPSYS recognizes escaped characters
8. 588 ost Figure 5 3 Production cost profile The minimal cost point found using DCOPFFlexG function Input data file 0 100 00 Tue Mar 27 16 21 28 2007 GIPSYS From Volt Volt Mw Mvar Mw Mvar 49 OF w q o 01 e 01 N HO oO WHN BB O NH oa R NO Mw gen 1 00000 2 14 17 74 134 80 0 98914 2 06 30 62 153 56 0 98918 17 74 30 62 71 64 1 00000 134 80 79 06 33 86 0 99133 71 64 79 06 112 58 1 00000 153 56 33 86 112 58 0 00000 8 74 21 73 4 55 0 38268 8 66 0 18 15 51 0 51387 21 34 0 30 2 96 3 86468 4 55 18 91 0 29 1 57948 5 59 15 60 13 98 4 83466 27 68 0 29 20 53 120 125 120 120 120 605 000 50 00 00 00 00 00 00 24 00 24 00 24 00 24 00 24 00 0 00 305 00 0 00 0 00 150 00 0 00 150 00 59 02 0 00 0 00 47 74 0 00 48 50 Mvar gen Mw load Mvar load Total I2R Mw losses Total I2X Mvar losses Generation Cost 155 262 605 000 120 000 0 081 35 275 583 025000 This is the DC optimal power flow with flexible generation minimizing pro duction cost Since it uses the DC approximation of the AC network the AC power flow SSNetNRPF should be run at this point The power flows print out shows a small difference in the cost Input data file 0 100 00 GIPSYS Tue Mar 27 16 21 28 2007 Mw 1 00000 24 00 306 91 36 41 39 41 112 40 35 1
9. Generation Cost 585 227777 End of Output 47 5 3 DC Optimal Power Flow Flexible Gen eration In the previous example generation cost was calculated for a couple of oper ating points In this example a cost dependency on the generators on buses four and six will be shown The same network 5 2 is used for this example Bus number one is the slack bus and its generation cost is accounted for but it is not shown in the graph due to dimensional limitations Both generators on buses four and six are varied between 50MW and 300MV The script file used to generate the graph data is SSNet b6 b6 raw b6 sup fopen b6 dat y for pg4 50 pg4 lt 300 pg4 pg4 10 for pg6 50 pg6 lt 300 pg6 pg6 10 SSNetGen b6 4 1 pg4 SSNetGen b6 6 1 pg6 SSNetNRPF b6 fprintf b6 dat Zg thg tig n pg4 pg6 SSNetCost b6 fprintf b6 dat n fclose b6 dat The cost graph is shown in figure 5 3 A user can easily generate such graphs to get a feel for how cost depends on a couple of generators and maybe esti mate the minimal cost operating point If there is a large number of genera tors that can be used to minimize the generation cost IPSYS DCOPFFlexG function can be used instead and for the same network it is given by ipsys gt SSNet b6 b6 raw b6 sup ipsys gt SSNetDCOPFFlexG b6 48 b6 dat 758 768 768 748 728 708 688 668 648 628 668
10. Rectifier commutating transformer resistance per bridge ohms XCR Rectifier commutating transformer reactance per bridge ohms EBASR Rectifier primary base AC volts KV TRR Rectifier transformer ratio TAPR Rectifier tap setting TPMXR Maximum rectifier tap setting TPMNR Minimum rectifier tap setting TSTPR Rectifier tap step Third record contains inverter quantities corresponding to rectifier quantities above Switch Shunt Data Ends with I 0 I MODSW VSWHI VSWLO SWREM BINIT N1 B1 N2 B2 N8 B8 I Bus number MODSW Mode O fixed 1 discrete 2 continuous VSWHI Desired voltage upper limit per unit VSWLO Desired voltage lower limit per unit SWREM Number of remote bus to control O to control own bus VDES Desired voltage setpoint per unit BINIT Initial switched shunt admittance MVAR at 1 0 per unit volts N1 Number of steps for block 1 first O is end of blocks B1 Admittance increment of block 1 in MVAR at 1 0 per unit volts N2 B2 etc as Ni Bl 72 Appendix C Power Flow Review Power flow is a steady state analytical tool of a power system The power flow within a system is completely known if real or active power P reactive power Q voltage magnitude V and voltage phase angle 0 are known at each bus 7 Only two variables are known at each bus and the goal is to solve for the other two variables The three most common bus types are load bus P Q known voltag
11. be additional information needed for a particular algorithm This additional data can include various cost 30 Field z Value Figure 3 2 Bus 6 Data Structure functions but user can freely keep any appropriate information within the data structures Currently in addition to the bus record discussed above this software uses generator and branch records from the input PTI input file and cost function files Since this is work in development there are possibilities to define data not used by any algorithms yet This will be discussed in the function calling section Table 3 1 lists elements not read from the PTI input file but used in currently implemented algorithms While ipsys scripting language used special functions to manipulate network data Matlab allows direct access to all of the network structures As with all Matlab programming vectorizing should be used as much as possible to improve the performance 31 Additional network data swing Internal index of the swing bus pvIndx Internal indexes of the PV buses pglndx Internal indexes of the PQ buses Additional bus data i Internal numbering index qmax Maximum available VAR at the bus qmin Minimum available VAR at the bus pg Real generation at the bus qg Reactive generation at the bus vaub Upper voltage phase limit in degrees valb Lower voltage phase limit in degrees vmub
12. ipsys a lt d lt ae D lt a D lt ae Dc OIA OAC ac a ac e ac ae AIA ac a ac ae a ae a dc a IAA ICA A I RAK Interactive Power Systems Simulator Carnegie Mellon University Aug 25 2007 08 49 46 dd ipsys gt quit prompt After started IPSYS is accepting commands as described in the rest of this manual The following sections introduce IPSYS data types operators com mands and includes the reference information 2 1 Data Types IPSYS uses real number matrix and character string variables as well as a number of predefined constants The following is the list of predefined constants Constant Symbol Constant Value M_E e M_LOG2E logo e M_LOG10E logio e M_LN2 In 2 M LN10 In 10 M_PI T M_PL2 z MPLA z M1 PI l M 2 PI 2 M2 SQRTPI M_SQRT2 V2 M_SQRT1 2 v RAND_MAX RAND_MAX Table 2 1 Common mathematical constants Real variables can be introduced through an interactive command or script and follows the standard real number format ipsys gt a ipsys gt b 2 5 cos M_PI A matrix can be entered from the command line ipsys gt a 1 2 3 4 5 7 9 7 41 or from a file ipsys gt a Matrix 3x3 mat where 3 X 3 mat file contains oF H Ww NON W BN 0 or a constant matrix can be defined as ipsys gt a Matrix 3 3 M_PI ipsys gt printf 6 3f a a 3 142 3 142 3 142 3 142 3 142 3 142 3 142 3 142 3 142 If a matrix is entered from the command l
13. sensitivity matrix can be obtained using an appropriate function All of these functions 24 return a sensitivity matrix and two index vectors defining the sensitivities For example ipsys gt dpgdvl i jl SSNetDPgDV1 b6 ipsys gt printf g t g n i 1 1 j 1 1 1 2 OP Vi for the users benefit Indexes are used for original bus and should not be used for indexing internal data structures All of the sensitivity functions take a Net variable as single argument and can be reliably used only after calling SSNetMakeSens first means that dpgdvl q The only purpose of displaying the indexes is The next group of calculates Locational Marginal Prices LMP while opti mizing either the flexible supply demand or both For example SSNetLMPG all 3 1 0 1 finds nodal LMPs of network named all when line 1 gt 3 is con gested using 0 1 increment and flexible supply The first vector entry is the node to which the price reference node is connected The second argu ment is the price reference node and the third argument is the transmission constraint increment used to calculate Lagrangian multipliers It is very im portant that the line is either congested or not for the both values of the transmission limit That is a line should be either congested for both val ues of the transmission limit or for none In the later case all the LMP values should be equal Similarly SSNetLMPSD all 3 1 0 1 c
14. setup allows for in teresting and flexible visual demonstrations of how network might evolve For example if loads are modified and power flow is run from within a loop voltage level changes can be observed visually on the GUI display using la bels Pause function can be used to slow down the execution to be able to observe the output To preserve data consistency during execution GUI window is disabled while CLI is executing a command and similarly CLI window is disabled while GUI command is executing Since the entire internal representation of the network GUI and CLI is updated with every change network can be continuously and programmat ically changed and manipulated through functions and scripts The disad vantage of this design approach is that the network can be and is frequently during editing in an invalid state for example slack bus is missing exis tence of isolated buses etc This is not a problem as long as an operation is not performed that requires a well defined network Various commercial packages address this by separating editing and running with Edit and Run 38 mode This is a good solution but limits what can be done using scripts Accidentally these packages also have limited scripting capabilities 39 Chapter 5 IPSYS Examples This chapter shows some examples how some of the major IPSYS features introduced in the previous chapter can be used The first section illustrates linear algebra capabilities s
15. use namespaces a con vention is introduced where Matlab files in this package start with pl_ C mex interface files with cpp_ and regular C functions not used in Matlab are the same as before Results using Matlab and ipsys should be exactly the same since they are using the same code 29 3 2 Matlab Interface Data Matlab maps data read from a PTI23 file into a structure consisting of arrays of network objects That is a net object has an array of buses generators branches etc and each element of an array contains all of the data defined in the PTI23 file for that particular element For example a six bus network is defined in b6 p23 input file and its main structure in Matlab is given as while for example array of buses is given as The convention used for Value 0 100 00 Wed Feb 9 21 12 42 2005 GIPSYS Figure 3 1 Six Bus Data Structure naming structure elements is that all the elements read in from the input file are capitalized and any additional data is named with small letters User is free to change already existing elements or add new data Description of the capitalized elements can be found in Appendix B Small letter variable naming should be obvious and is listed in the following tables for network elements actually used in this software The mandatory data that the cpp functions expect to be present in the network structure is the data read from the PTI input file There can also
16. variables All of these data types can be entered interactively with a script or in the case of matrices from a file Power system networks are modeled as separate objects that can be read in from a file manipulated modified simulated optimized and the resulting system can be stored in an external file Multiple power system networks are stored at the global scope and as such can be accessed from any built in or script ing function General purpose operators and functions include usual arith metic and logical operators transcendental and linear algebra functions Any other functions can be added to this set provided that the function pa rameters operate on available data types and the result is also an available data type Program flow control structures provide enough functionality for a complete procedural programming language This includes logical tests looping and input and output functionality User functions defined as ipsys scripts or C functions are supported with input and output arguments passed by value From users perspective there is no difference between ip sys and C functions Due to the object oriented power system modeling data and functional encapsulation are provided by the power system objects Power systems functions include AC power flow implementation of different optimal power flow algorithms as well as a programmatic way to manipulate the networks being simulated The optimization routines us commercial l
17. with out any errors scripting functions are compiled into execution trees Script ing functions can call built in functions and built in functions could easily call scripting functions Scripting function can be defined anywhere within IPSYS session but the recommended method is by using include statement A function is defined similar to Matlab functions The following example il lustrates all of the function definition and calling features LinSolve function is defined in LinSolve ipsi file as function x y LinSolve a b if Rows b Rows a printf Incompatible dimensions n return 0 if Rank a Cols a printf Rank deficient matrix a n return 0 x b a and included in IPSYS workspace with include LinSolve ipsi Now LinSolve function can be called in three different ways ipsys gt include LinSolve ipsi ipsys gt a rand 3 3 ipsys gt b rand 3 2 ipsys gt sol LinSolve a b ipsys gt printf 5 2f sol sol 0 73 0 83 1 94 0 48 1 80 0 51 ipsys gt printf 5 2f LinSolve a b 19 0 73 0 83 1 94 0 48 1 80 0 51 ipsys gt sol non LinSolve a b ipsys gt printf 5 2f n non 0 00 LinSolve is defined to return two variables Calling function can use both variables just one of them or none Return variables are assigned to vari ables in calling function s memory space from left to right In addition if only one variabl
18. 0 0 00000 18 87 30 98 0 31 120 00 BB WN N 0 98120 24 00 0 00 0 00 37 50 3 40520 20 78 125 00 51 3 4 66 9 64 6 82 84 35 14 3 0 98611 3 26720 120 00 24 00 0 00 0 00 1 112 40 24 18 2 4 66 9 70 5 12 26 9 51 4 1 00000 1 00557 120 00 24 00 150 00 43 49 1 35 10 0 31 5 65 79 19 18 6 0 69 0 00 5 0 99096 2 90781 120 00 24 00 0 00 0 00 3 12 26 9 64 4 65 79 16 83 6 66 47 16 81 6 1 00000 0 98583 0 00 0 00 150 00 58 55 2 82 84 39 35 4 0 69 0 00 5 66 47 19 20 Mw gen 606 908 Mvar gen 138 444 Mw load 605 000 Mvar load 120 000 Total I2R Mw losses 1 909 Total 12X Mvar losses 18 452 Generation Cost 585 146048 The AC power flow should also be used after DCOPF to check the operating point feasibility by checking the SSNetNRPF return values Jacobian rank number of iterations and P Q mismatches 52 5 4 DC Optimal Power Flow Flexible Load If there is demand control implemented there should be a way to account for the demand flexibility This Example shows such a hypothetical situation DCOPFFlexD example uses the same b6 raw as the above example Ipsys script is very similar to the previous example ipsys gt SSNet b6 b6 raw b6 sup b6 dmd ipsys gt SSNetDCOPFFlexD b6 ipsys gt SSNetPrintFlows b6 and the DCOPFFlexD result is Input data file 0 100 00 Tue Mar 27 16 21 28 2007 GIPSYS From Volt Volt Mw Mvar Mw Mvar Bus Mag Angle Load Load Ge
19. 0 0 1 10 0 0 0 0 0 0 0 sido O 3 l 0 0 0 0 0 0 0 10 1 31 0 0 0 0 0 0 Doo i 230 0 0 0 0 0 0 0 0 10 0 0 0 0 0 3 7 with the following solution z 5 45 4 07 3 70 5 82 1 75 3 70 5 63 11 91 16 79 20 04 18 42 13 02 DCOPFFlexGD function performs the DC optimization with respect to both generation and demand accounting for the flow constraint from bus number 1 to bus number 3 ipsys gt SSNet all allen raw allen sup allen dmd ipsys gt SSNetDCOPFFlexGD all ipsys gt SSNetNRPF all DCOPFFlexGD output is as the following Solution Output 59 Input data file 0 100 00 Tue Oct 25 16 18 59 2005 GIPSYS From Volt Volt Mw Mvar Mw Mvar Bus Mag Angle Load Load Gen Gen To Mw Mvar Bus Flow Flow 1 1 00000 0 00000 0 00 0 00 5 46 0 15 2 1 76 0 02 3 3 70 0 13 2 1 00000 1 00607 0 00 0 00 4 07 0 25 1 1 76 0 02 4 3 88 0 15 3 1 95 0 08 3 0 99939 2 12135 3 71 0 00 0 00 0 00 1 3 70 0 01 2 1 95 0 04 4 1 94 0 04 4 0 99922 3 23376 5 82 0 00 0 00 0 00 2 3 88 0 00 3 1 94 0 00 Mw gen 9 529 Mvar gen 0 394 Mw load 9 529 Mvar load 0 000 Total I2R Mw losses 0 000 Total I2X Mvar losses 0 394 Generation Cost 71 936386 IPSYS calculations agree closely with the hand calculations above and the results reported in 1 hand calculation paper reported IPSYS Gi MW 5 4552 5 4553 5 45517 Go MW 4 0734 4 0734 4 07342 L MW
20. 23 printf 3 3f a c General purpose functions include cos real or matrix sin real or matrix tan real or matrix abs real or matrix exp real or matrix srand real or none rand min real or matrix max real or matrix Rows matrix Cols matrix Range real real Rank matrix Cond matrix Idn real Trn matrix EDec matrix real Clust matrix real Matrix string Returns cos argument is in radians Returns sin argument is in radians Returns tan argument is in radians Returns abs Returns natural exponent Seeds random number generator Returns a random integer between 0 and RAND_MAX Returns minimum entry in a matrix Returns maximum entry in a matrix Returns number of rows of a matrix Returns number of columns of a matrix Returns a row vector of integer numbers in the range Returns rank of a matrix Returns condition number of a matrix Returns identity matrix of requested dimension Returns transpose of a given matrix Returns epsilon decomposition of a matrix Returns clustered matrix Returns a matrix read from file Table 2 2 General purpose functions Single argument math functions can take either a real number or a matrix In the case of a matrix input function is applied to each matrix entry The four arithmetic operations have the usual meaning for real numbers If one of the operands is a matrix the result depends on which of the operands is the matrix For addition subtr
21. 3 2f b a 0 50 0 00 0 50 1 Next is an example of the solution of a system Ax B where A is 3x3 and B 3x2 This is still a determined system of equations ipsys gt b 1 2 1 3 1 4 ipsys gt printf 3 2f b a 0 50 0 50 0 00 0 00 0 50 0 83 1 14 Note that one of the solutions is the same as in the previous example The following is an example of an over determined system of equations The so lution is an LSE approximation a 1 2 3 5 4 2 7 6 4 1 2 91 ipsys gt printf Rank a g n Rank a Rank a 3 b 1 1 1 1 ipsys gt printf 3 2f b a 0 38 0 65 0 01 The result of this operation is the LSE approximation of the solution Note that the rank of a must be equal to the number of columns for this operation to be possible These results are typical for QR decomposition used to solve systems of equations IPSYS also provides enough of the common programming constructs to be able to write general purpose scripts This includes condition tests IF THEN ELSE conditional execution WHILE and FOR loops With these basic constructs one should be able to perform other program flow controls Condition tests and logical operators implemented by IPSYS Equality test lt Less than test gt Greater than test lt Less than or equal test gt Greater than or equal test Not equal test amp amp Logical AND operator Logical OR o
22. 3 4 0 3 7059 5 8227 0 0 0 0 63 Chapter 7 Notes Bugs TODO list IPSYS and implementation of all the underlying algorithms are results of recent development not rigorously tested and work in progress Further development can be done in the area of fixing bugs and providing new func tionality User feedback is crucial for both of these areas This program is being actively developed Detecting reporting and dealing with errors is one of the most difficult aspects of interpreter and compiler design Detecting errors can cause the interpreter to just abort the entire program try to process as much of the input as possible and continue working and or report the error back to the user Helpful error reporting includes a fair amount of guessing of what is wrong with the input IPSYS is a simple interface to a number of complex algorithms and is more focused on algorithm implementation and less on er ror handling As the result error reporting and sometimes recovery is quite limited Following is the list of known bugs that are being worked on 1 Running loops from the command line will slow GUI display update if the network window is moved or overlapped during loop execution This seems to be operating system Windows related 2 If a name is not associated with a function or data the interpreter reports that there is no matrix with such a name This is a consequence of using the same parenthesis type for indexing matrices and calling
23. Interactive Power Systems Simulator User s Manual Copyright Carnegie Mellon University December 16 2007 Contents 1 Introduction 2 Using IPSYS PAIN 2D 2 3 2 4 2 5 Data ANDO spin eh ee ee AAA Built in Operators and Functions Scripts and Functions ces Bee Power Systems Functions 0 Input Output and Miscellaneous POMS CoLa He oe rak MORE A os 3 Matlab Interface 3 1 3 2 3 3 TAB AUGHOM re re aora RE SEE ES EO Matlab Interface Data 02 22 2 2 0000 Matlab Interface Functions 4 GIPSYS Interface 4 1 4 2 Entering New Network ea 40 se a e hae Running IPSYS Algorithms 5 IPSYS Examples 5 1 5 2 5 3 5 4 Linear Algebra on lt x A wee 4 4 80808 Ke ana 874 9 1 1 Checking Rank and Condition Number 51 2 Sensitivity Matrix Extraction lt lt eosa ds a wy ws 5 1 3 Matrix Decomposition and Clustering Power Flow Examples au ar a rt ok AA Sal Example een bao erna tte DC Optimal Power Flow Flexible Generation DC Optimal Power Flow Flexible Load 12 18 22 27 u o a WW gt 5 5 DC Optimal Power Flow Flexible Generation and Load Matlab Examples 6 1 Matlab Input Output ar ccs 644 ee 64d ee eae ne 62 Matlab BURNS ss la ne ses a Ja ee at gn aa Notes Bugs TODO list Installation Description of the PTI Load Flow Data Form
24. KKKKRKKKKKKKKRKKKRKKKKKRKRKKRKEKRKKKKRKK KK KKK KKK KK 120 0 j24 0 ipsys gt ER zipsys File Edit Format Der H3 al X B O 100 00 Tue Mar 27 16 21 28 2007 GIPSYS 3 120 0000 24 0000 0 0000 0 0000 ll 0 0000 B1 125 0000 24 0000 0 0000 0 0000 iD 0 0000 B2 120 0000 24 0000 0 0000 0 0000 0 0 0000 B3 120 0000 24 0000 0 0000 0 0000 MN 0 0000 B4 120 0000 24 0000 0 0000 0 0000 1 0 9848 0 0000 B5 0 0000 0 0000 0 0000 0 0000 1 1 0000 0 0000 B6 END OF BUS DATA BEGIN GENERATOR DATA 1 3 2 1 3 1 4 2 5 1 6 2 Df Figure 4 1 GIPSYS windows 4 1 Entering a New Network At the start up GIPSYS presents user with the CLI command window from which text script and GUI editors can be opened Text editor can be used to open multiple text scripts at the same time while the GUI editor can be used to edit a single network at the time This network can also be accessed from the CLI window using net name gnet The network editor is started and closed from the main window with Tools Start GUI and Tools Close GUI To edit or start a new scrip use File gt Open Script or File New Script A new network can be entered with File gt New CTRL N or clicking on New icon on the network editor tool bar Once the user is presented with the blank GIPSYS screen network elements and connections can be entered by first clicking on the element type on the tool bar or E
25. action if both operands are matrices they must be of the same dimensions and the result is the regular matrix addi tion subtraction If one of the operands is a matrix and the other one is a 13 real number the addition subtraction is done element wise That is it is equivalent to addition subtraction of two matrices where one of the matrices has all the elements equal to the real number operand Two real number multiplication also has the usual meaning If one of the operands is a real number and the other one is a matrix the result is the matrix multiplied by a constant Division has the usual interpretation for real numbers If one of the operands is a matrix and the other one is a real number the operation is done element wise Since division is not commutative either the real num ber or matrix elements can be the divisor In the case of division there can also be a combination of two matrices of different dimensions Depending on the dimensions of the matrices the result can be a solution of a single system of linear equations a number of linear equations or a Least Square Estimation LSE Matrices still must satisfy dimension requirements to be able to perform these operations The following example finds a solution of a single system of linear equations AB X A is 3x3 and B is 3x1 This is a determined system of equations ipsys gt a Matrix 3x3 mat ipsys gt a 2 2 1 ipsys gt b 1 1 1 ipsys gt printf 2
26. alculates the LMPs assuming that the generation is fixed and the demand is flexible SSNetLMPSGD all calculates the LMPs assuming that both generation and demand are flexible The same transmission congestion considerations should be observed as in the previous two functions 25 2 row vector with two adjacent nodes and increment Function Required Optional Return Name Arguments Arguments Values SSNetNRPF 1 net name 2 Pm Qm Iter 1 rank Pm Qm Iter SSNetDCOPFFlexG 1 net name 1 cost SSNetDCOPFFlexD 1 net name 1 cost SSNetDCOPFFlexGD 1 net name 1 cost SSNetDF 1 net name 1 distribution factors SSNetDCFL 1 net name 1 DC power flows 2 source buses 3 destination buses SSNetMakeSens 1 net name 1 0 SSNetDPgDT 1 net name 1 ore matrix SSNetDPgDVg 1 net name 1 T matrix SSNetDPgDVI 1 net name 1 o matrix SSNetDPIDT 1 net name 1 a matrix SSNetDPIDVg 1 net name 1 de matrix SSNetDPIDVI 1 net name 1 a matrix SSNetDQIDT 1 net name 1 1 matrix SSNetDQIDVg 1 net name 1 do matrix SSNetDQIDVI 1 net name 1 F matrix MatrixLMPSG 1 net name 1 bus indexes and LMPS 2 row vector with optimizing generation two adjacent nodes and increment MatrixLMPSD 1 net name 1 bus indexes and LMPS 2 row vector with optimizing demand two adjacent nodes and increment MatrixLMPSGD 1 net name 1 bus indexes and LMPS optimizing generation and demand Table 2 6 IPSYS power syst
27. at Power Flow Review Optimal Power Flow Review DC Optimal Power Flow Review 55 58 58 62 64 67 68 73 77 79 Chapter 1 Introduction Academic research frequently suffers from programming limitations of gen eral purpose programming languages such as Fortran C C as well as interpreted languages such as Matlab General programming languages pro vide the ultimate flexibility at the price of long development cycles and are difficult to maintain once the access to the original developers is not avail able anymore On the other hand interpreted languages provide significant development and maintenance flexibility at the price of programming fea tures and frequently price After surveying both approaches to scientific programming it seemed that it could be possible to have the best of the both worlds that is an interpreted language expandable with natively coded modules and ability to call native functions from the interpreter and even the interpreter functions from the native functions and all that for a small price in flexibility and short learning curve The result of this conclusion is a generic Application Programming Interface API framework Appropriate power systems functionality is added to this framework producing Interactive Power Systems Simulator or IPSYS for short The framework allows adding new special purpose modules independently of existing modules and using the built in Command Line Interface CLI
28. atlab If a net structure is modified in some way or if it is just a result of a calcu lation it can be stored in a PTI23 string and or saved in a file for further processing gt gt raw pl_PrintRawStr nt and the same can be done for the cost functions gt gt pgcost pl_PrintPgCost nt gt gt plcost pl_PrintPLCostStr nt 6 2 Matlab Functions Table 3 2 lists all the Matlab interface functions available to a user In this section a few functions will be used to demonstrate how they can be used The most common algorithm that can be used is the Newton Raphson power flow and continuing with the system shown in Figure 6 1 we will change the generation of the non slack bus and rerun the power flow 62 gt gt b4 gen 2 PG 2 gt gt nt pl NRPF b4 gt gt nt bus I nt bus VM nt bus VA ans 1 0000 2 0000 3 0000 4 0000 1 0000 1 0000 0 9993 0 9993 O 1 7916 2 7955 3 7275 gt gt Note that due the change in generation distribution voltage magnitudes re mained the same but voltage angles changed as it should be expected The next example uses the same network to find DC optimal power con sidering both the generators and loads to be flexible gt gt b4 pl_pti23Net allen raw allen sup allen dmd gt gt nt pl DCOPFFlexGD b4 gt gt nt gen I nt gen PG nt gen QG ans 1 2 5 4552 4 0734 0 14527 0 24908 gt gt nt bus I nt bus PL nt bus QL ans 1 2
29. ch input output function can be used for both input and or output by allowing for optional inputs That is if the function is to be used just to get an output from the net the optional arguments are omitted If the same function is to be used as an input to the net the optional arguments contain the input values IPSYS network functions use a part of PTI23 data described in 4 Cur rently IPSYS uses bus generator switched shunts and transformer adjust ment data However both switched shunts and transformers are used as fixed elements that is their settings are not adjusted to control either bus voltage or branch real power flow Power system input output function description Function Name Function Description SSNet Defines a network from a PTI file and optional supply and demand files SSNetGen Sets real power generation level SSNetGenST Sets generator status SSNetLoad Sets real and reactive load level SSNet Volt Sets voltage magnitude and angle of a bus SSNetShunt Sets shunt value on a given bus in a given system SSNet Print Prints out PTI23 network data SSNetPrintFlows Prints out power flows SSNetJac Returns power flow jacobian SSNetG Returns G part of admittance matrix SSNetB Returns B part of admittance matrix SSNetCost Returns productions cost Table 2 4 Power system input output functions 22 Power system output functions calling options and return values Function Requ
30. ch takes the number of millisec onds during which the execution should be paused 28 Chapter 3 Matlab Interface 3 1 Introduction Matlab Interactive Power Systems Simulator MIPSYS consists of algo rithms described in 2 4 MIPSYS uses an older version of IPSYS algorithms and its further development has been abandoned in favor of GUI interface and its ability to export GIPSYS networks to Matlab environment All of the functionality provided by ipsys is also available through Matlab Al though there are other packages which have similar functionality all of them are rather difficult to use and integrate with the rest of Matlab Matlab interface to ipsys algorithms is designed with user friendliness in mind more than efficiency Both ipsys and Matlab interface are designed using object oriented approach While ipsys defines system as an object and and operates on it Matlab passes an object around from a function to a function Due to Matlab design all the data is passed by value This could result in lots of data copying between function calls Rather than using lots of static data and be able to analyze a single system at the time the performance penalty is accepted for the sake of flexibility and user friendliness The main interface design decisions were to use array of structures to define a network in Matlab and to use pairs of Matlab C mex functions to interface Matlab functions with regular C functions Since Matlab does not
31. chine or machines defaults to system MVA base ZR ZX Machine impedance pu on MBASE RT XT Step up transformer impedance p u on MBASE GTAP Step up transformer off nominal turns ratio STAT Machine status 1 in service O out of service 69 RMPCT Percent of total VARS required to hold voltage at bus IREG to come from bus I for remote buses controlled by several generators PT Max MW PB Min MW Branch Data Branch records ending with a record with from bus of zero I J CKT R X B RATEA RATEB RATEC RATIO ANGLE GI BI GJ BJ ST I From bus number J To bus number CKT Circuit identifier two character not clear if integer or alpha R Resistance per unit X Reactance per unit B Total line charging per unit RATEA MVA rating A RATEB RATEC Higher MVA ratings RATIO Transformer off nominal turns ratio ANGLE Transformer phase shift angle GI BI Line shunt complex admittance for shunt at from end I bus pu GJ BJ Line shunt complex admittance for shunt at to end J bus pu ST Initial branch status 1 in service O out of service Transformer Adjustment Data Ends with record with from bus of zero I J CKT ICONT RMA RMI VMA VMI STEP TABLE I From bus number J To bus number CKT Circuit number ICONT Number of bus to control If different from I or J sign of ICONT determines control Positive sign close to impedance untapped bus 70 of transformer Negative sign op
32. connections and two vectors of the original matrix coor dinates Figure 5 1 illustrates this approach to system reduction The second function EDec mat eps calculates the same decomposition but it returns a matrix with number of rows equal to the number of clusters where each row contains row indexes of the original matrix These two functions give equiv alent results except possibly different permutations within a cluster EDec combined with SSNetJac and SSNetNRPF was used in 8 to decompose and analyze the standard IEEE test network RTS 1996 42 30 02 04 30 0 0 01 50 02 05 20 20 0 20 03 20 0 0 50 Figure 5 1 Network clustering 43 5 2 Power Flow Examples 5 2 1 Example 1 The six bus network shown in figure 5 2 is used for the next 4 examples The examples start with a simple one and progressively become more complex The first example is just a power flow runs for a couple of different operating points First calculate the generation cost for the first operating point 120 0 j24 0 120 0 j24 0 1 0 2 9 0 2 4 150 0 j56 2 120 0 j24 0 Figure 5 2 Six bus system ipsys gt SSNet b6 b6 raw b6 sup b6 dmd ipsys gt SSNetNRPF b6 ipsys gt Pg4 SSNetGen b6 4 1 gt ipsys gt Pg6 SSNetGen b6 6 1 44 ipsys gt Pgl SSNetGen b6 1 1 21 ipsys gt GenCost SSNetCost b6 ipsys gt printf Pgi 7 2f tPg4 7 2f tPg6 7 2f tGenCost 7 2f n Pg1 Pg4 Pg6 GenCo
33. cos 0 0 Ba sin 0 0 Z EF Ex Gi sin 0 0 Bix cos 9 01 k Fori k m Qi BE P GED P Gu E Pi GuEj dQ 2 eA Qi Buk A 75 And in the matrix form AP AQ AP AQ A6 A E1 Ei A0 A Ez Ex C 10 Equation C 10 is in the format used by the Newton Raphson iterative solver The whole process starts by assigning an initial solution for complex voltages which is usually of a magnitude of 1 0 per unit and O phase The next step is to calculate AP and AQ as well as the Jacobian Using these values A9 A E E2 and 76 are calculated and a new solution for voltages is found This is repeated until change in P and Q is sufficiently small Appendix D Optimal Power Flow Review The Optimal Power Flow OPF problem has been a topic of intense research for the last forty years OPF is a static power flow problem which tries to determine the optimal control variables in a power network under constraint conditions The OPF can be mathematically formulated in general terms as in 2 Min f Z ZER f prin RI D 1 subject to G Z 0 G Rr RL D 2 H Z lt 0 H rr p D 3 where f Z is the objective function to be minimized G Z are the nonlin ear power flow equations and H Z are the nonlinear inequality constraints Vector Z U X is the vector of both state and control variables State variables vector X includes bus vol
34. d IPSYS capabilities One is by using include statement and the other one is by writing new scripting and or hard coded functions New scripting functions can be either typed from the command prompt or more commonly using the include statement Include statement simply includes a file that is automatically interpreted as an IPSYS script An include function can contain any statements that might be used during an interactive IPSYS session Such script files can be used for function definition repetitive data entry and or execution As an example the following is the execution of an include statement ipsys gt include Solve ipsi c 0 93 1 32 1 81 0 43 0 75 0 44 where Solve ipsi file contents is a rand 3 3 b rand 3 2 c b a printf 5 2f c In this case include is executed at the top level and the created variables are global ipsys gt lsVars stack level O a 00A095F8 a 3x3 matrix b 00A096C8 b 3x2 matrix c 00409888 es 3x2 matrix IPSYS can use scripting and built in hard coded functions From users perspective they are the same Users can program both scripting and built in C functions However built in functions must be defined in IPSYS and added to a lookup table in addition to compiling and linking the function 18 These process is straight forward and explained in more details in the API Manual Scripting functions must be defined before used Once defined
35. dit menu and then clicking at the desired location on the screen Elements can be arranged 36 on the screen by dragging them around with the mouse Two buses can be connected with a branch by clicking on the branch edit menu entry and then clicking on from and to bus in the network A branch can be broken into segments by first clicking on the branch to select it and then SHIFT RIGHT CLICK at the location of the desired joint After the branch has one or more joints they can be dragged around with the mouse to arrange branch path in a visually pleasing way Elements can be attached to buses by choosing an element and then clicking on the bus with which the element is associated Typical start of a new network is shown in Figure 4 2 All system elements E EE File Edit Run wiew Help S ca Gd A Fe ERA gt lt gt 0O 0 j20 0 61 Zoom 10044 Figure 4 2 New GIPSYS defined network can have its typical measurements as a label and buses can have custom la bels as well This is done by using right mouse click on an element Element parameters dialog can be accessed by pressing right mouse button on an ele ment and choosing Properties from the pop up menu or by using left double click Edit gt Preferences defines the initial label option for each element GIPSYS GUI follows standard GUI design and users can expect similar fea tures as in other applications with GUI interface Network can be saved by using File gt Save As
36. e controlled bus P E known and generator bus P E known or P Q if Q exceeds generator limits A power system is connected with lines of known complex impedance between the buses and the ground The system can be described with a set of equations similar to any electrical circuit However since the known variables are complex power quantities the power system equations are nonlinear The following equations are the derivation of Newton Raphson solution 7 used in this work The Newton Raphson solution is based on the first order multivariable Taylor series expansion which requires one time calcu lation of the system s impedance matrix and an iterative calculation of the Jacobian If the power system s nonlinear equations are represented by F V const C 1 we choose a starting value for the independent variable V The starting value will cause an error e in function F F V const C 2 The error can be decreased using Taylor series expansion dF V F V9 dF V ay const C 3 73 solving for independent variable correction AV by setting error e to zero and reiterating with the new value of the independent variable until the error is acceptable A RR AV const F V C 4 In power systems case the independent variable is the set of complex value bus voltages the derivative in the Taylor series expansion is the Jacobian of the power injection equations at each bus and the err
37. e is transfered it must be the left most one and in that case it does not need brackets as shown above Also if the function does not return any variable the default return value is zero The following function is perfectly acceptable and useless but it does produce a zero as the return value ipsys gt function zero F ipsys gt printf g n zero 0 Each IPSYS function uses its own memory space and recursive calls can be used as expected The following example defines a recursive Fibonacci function and shows its use ipsys gt include fib ipsi ipsys gt f fib 14 ipsys gt printf g n f 233 ipsys gt where fib ipsi file contains function f fib n if n lt 1 return 0 else if n 2 return 1 else 20 return fib n 2 fib n 1 Application Programming Interface API and development of natively com piled functions is described in the API User s Manual As far as function user is concerned there is no difference between calling a scripting function or compiled function 21 2 4 Power Systems Functions IPSYS is based on the framework potentially providing interactive interface for many different problems To differentiate between different algorithm types power systems functions are prefixed with SSNet meaning steady state network Power systems functions can be grouped into two groups network input output functions and network computational functions Ea
38. e power balance equation Equation E 6 is the real power transmission constraint AP is the vector of net generations of all the buses that is AP P Pp DCOPF functions take into account the power generation limits for all of the buses including the slack bus The cost function is minimized with with respect to all of the generators and or loads including the slack bus This is different from the power flow calculations where the slack bus is used to cover for the losses regardless of its limits and cost 80
39. em algorithms calling options and return values 26 2 5 Input Output and Miscellaneous Functions The API framework used for IPSYS can easily provide any functionality the native C C programming language provides IPSYS implements screen and file input output internal data structures listing and some additional miscellaneous functions As demonstrated throughout the previous examples results can be displayed using built in printf function Similarly fprintf function is used to write to a file rather than to the screen Both of these functions are similar to C functions with the same names Additionally printf can print out a single matrix using given format applied to each matrix entry Format string is just the regular C printf format string Printing to a file is done using fprintf which can also print a single matrix like printf fprintf takes a file name as an argument rather then a file descriptor To be able to write to a file file must first be opened with fopen function which does not return file descriptor but makes it globally visible using the file name After writing to a file the file should be closed with fclose Writing a matrix of random numbers between 0 and 1 in Matlab format to a file could be performed as ipsys gt srand ipsys gt fopen rand5x5 m w ipsys gt fprintf rand5x5 m 7 3f rand 5 5 RAND_MAX ipsys gt fclose rand5x5 m resulting in rand5x5 m file more rand5x5 m 0 847
40. end function is called through pl_ cpp_ calling mechanism the entire network is passed to the back end functions and treated as a brand new study case In Matlab it is very difficult if not impossible to pass data to dynamically linked functions by reference This causes performance penalty but in this case it also ensures that a network is always treated as completely new one There is a potential problem with functions returning the Jacobian and ad mittance matrix The problems is the internal bus and branch indexing If a computation depends on these two matrices and their ordering order the buses in an increasing order starting with the slack bus in the PTI23 input file In that case back end internal ordering will be the same as in the network structure formed by Matlab input routine The sensitivity and clustered matrices are not a problem contain indexes in the first row and column as they do in ipsys Table 3 2 contains the list of all the functions Use help function in Matlab for the detailed description of input and output parameters and the examples in chapter 6 for a quick start 33 Function Name Input Output Data I O Functions pl_pti23Net pl printPGCostStr pl printPLCostStr pl_printRawStr pl_NetFlows PTI 23 file name Net structure Net structure Net structure Net structure Network structure P cost functions string P cost functions string PTI 23 string Net flows string Power Flow Algorithm
41. functions 3 A function must be defined before called This will be changed by 64 introducing function declaration since two functions cannot call each other unless they are the same function that is unless if it is a recursive call A defined function cannot be redefined within the same session Multi line commands in GUI must be ended with ENTER or SHIFT RETURN except the last one which is ended with the usual RETURN key The error reporting needs to be improved This is just a time consuming task There are surely many more bugs Any bug reports will be greatly appreciated and addressed There are also a few unfinished things 1 2 SSNetPrint function not completed Transformer editing and use is incomplete in GUI Introduce function declaration the lack of it affects both GUI and CLI interface History saving menu is not functional yet 65 Bibliography 1 Eric Allen Marija Ilic and Ziad Younes Providing for transmission in times of scarcity an iso cannot do it all Electrical Power and Energy Systems 21 147 163 1999 55 57 2 IEEE Tutorial Course Optimal Power Flow Solution Techniques Re quirements and Challenges 1996 77 3 Visual Numerics IMSL C Math Library User s Manual Visual Numer ics Inc 2000 40 4 University of Washington Department of Electrical Engineering http www ee washington edu research pstca formats pti txt 3 22 5 Roldan Pozo Template numerical to
42. i brary IMSL but they could use any other commercial and non commercial C C F77 library GIPSYS adds to IPSYS a GUI interface from which a network can be entered modified and a few common algorithms executed GIPSYS is a semi inde pendent front end library developed for a different purpose and illustrates the flexibility with which new modules with very little interdependence can be added to the framework The only difference between IPSYS run without GUI and with GUI is that the GUI version does not handle more than one network at the time MIPSYS is the Matlab interface to IPSYS back end routines Since Matlab provides much richer general computational language capabilities the only IPSYS routines that are exported to Matlab are the power systems specific which are not part of Matlab This development branch uses an older IPSYS version and is abandoned in favor of IPSYS and GIPSYS user interfaces and ability to export power system networks to Matlab environment Chapter 2 Using IPSYS In this chapter IPSYS data structures operators program control and pro gramming are discussed in detail These are the core features that are avail able for power systems functions and GIPSYS use both of which are discussed in subsequent chapters IPSYS CLI is started by simply typing ipsys exe command at the command prompt provided that the the IPSYS executable is in the systems PATH Exiting from IPSYS is done by typing quit command prompt
43. ine entries can be real variables or expressions for example ipsys gt a M_PI cos M_PI_2 0 1 2 ipsys gt printf 7 3f a a 3 142 0 000 0 000 0 500 When a matrix is read from an ASCI data file the first two integers are the number of rows and the number of columns of the matrix and the rest are the matrix entries There can be more than enough matrix entries but it is an error if there are fewer then rows x columns entries 3 x 3 9 in the example Numbers in the data file can be arranged in any way and can be either in floating point or integer number format Matrix entry indexing is one based meaning that row and columns counting starts from 1 The following script swaps matrix entries a 1 1 and a 2 2 ipsys gt tmp a 1 1 ipsys gt a 1 1 a 2 2 ipsys gt a 2 2 tmp In the case of the matrix data type IPSYS also allows for a matrix block extraction or assignment For example if test mat contains a 8 x 7 matrix ipsys gt t Matrix test mat ipsys gt printf 3g t t 1 2 3 4 B B 7 3 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 1 the following commands show how matrix block extraction can be used ipsys gt a t 1 2 3 2 31 ipsys gt printf 4 3g a a ipsys gt a 1 2 1 2 t 7 8 6 71 ipsys gt printf 3g a a E
44. ired Optional Return Name Arguments Arguments Values SSNet 1 net name 02 2 input file 3 supply file 4 demand file SSNetGen 1 net name 4 new P 1 old P 2 bus number 5 new Q 2 old Q 3 generator ID SSNetGenST 1 net name 4 new ST 1 old ST 2 bus number 3 generator ID SSNetLoad 1 net name 4 new P 1 old P 2 bus number 5 new Q 2 old Q 3 load ID SSNet Volt 1 net name 3 new mag 1 old Vmag 2 bus number 4 new ang 2 old Vang SSNetShunt 1 net name 3 new Gl 1 old Gl 2 bus number 4 new Bl 2 old Bl SSNetPrint 1 net name 0 SSNetPrintFlows 1 net name 1 output file 0 SSNetJac 1 net name 1 Jacobian SSNetG 1 net name 1 matrix G SSNetB 1 net name 1 matrix B SSNetCost 1 net name 1 cost Table 2 5 Power system input output functions arguments The implemented power system algorithms are described in Table 2 4 Power system algorithm calling options and return values are shown in Table 2 6 SSNet function requires four arguments where some or all of the last two can be empty strings Functions SSNetGen SSNetGenST SSNetLoad SSNet Volt and SSNetShunt are used to get set or get old and set new values For example the following command at the same time reads the old generator status value and sets the 23 Function Name Function Description SSNetNRPF Newton Raphson power flow SSNetDCOPFFlexG DC OPF with dispatchable generation SSNetDCOPFFlexD DC OPF with dispa
45. n Gen To Mw Mvar Bus Flow Flow 1 1 00000 0 00000 64 39 24 00 0 00 0 00 2 10 25 10 25 3 0 00 33 40 4 0 00 0 00 2 0 97950 0 00000 10 00 24 00 0 00 0 00 1 10 04 10 04 3 0 00 7 44 6 0 00 40 16 53 3 0 98330 0 00000 50 00 24 00 0 00 0 00 1 0 00 32 84 2 0 00 7 47 5 0 00 2 95 4 1 00000 0 00000 200 00 24 00 186 36 24 69 1 0 00 0 00 5 0 00 30 40 6 0 00 0 00 5 0 98480 0 00000 10 00 24 00 0 00 0 00 3 0 00 2 95 4 0 00 29 94 6 0 00 29 94 6 1 00000 0 00000 0 00 0 00 148 03 76 96 2 0 00 41 00 4 0 00 0 00 5 0 00 30 40 Mw gen 334 394 Mvar gen 101 646 Mw load 334 394 Mvar load 120 000 Total I2R Mw losses 0 210 Total 12X Mvar losses 2 566 Generation Cost 370 716924 54 5 5 DC Optimal Power Flow Flexible Gen eration and Load As an example of optimal generation and demand calculations calculations of 1 on pages 157 158 will be done by hand using IPSYS and compared with the results in the above paper The four bus network under consideration is shown in figure 6 1 Power flow shows that flow Pari exceeds the branch maximum flow Pi given as 3 7pu Using this fact as a binding con straint and setting up the Lagrange equation for constrained optimization the following set of equations results 2 0 0 0 0 0 0 10 0 0 0 j 0 4 0 0 0 0 0 0 10 0 0 0 5 0 0 2 0 0 0 0 0 0 10 0 94 1667 0 0 0 24 0 0 0 0 0 0 10 158 1667 0 0 0 0 0 0 0 13 11 0 0 d 0 00 231 at 11 20 Tod 10
46. olkit an interface for scientific com puting in c http math nist gov tnt index html 40 6 D D Shiljak Decentralized Control of Complex Systems Academic Press 1991 42 7 Allen J Wood and Bruce F Wollenberg Power Generation Operation and Control John Wiley amp Sons Inc 1996 73 79 8 Le Xie Jovan Ili and Marija Ili Towards grid modernization through enhanced communications and computing Novel performance index and information structure for monitoring voltage problems Carbondale Il 2006 North American Power Symposium 42 66 Appendix A Installation GIPSYS and MIPSYS probably require Windows 2000 or Windows XP while IPSYS could run on a less powerful computer and operating system Installa tion of IPSYS and accompanying interfaces is almost automatic in Windows environment An auto installation executable is provided which copies all necessary files to a user chosen directory All the necessary environment variables needed to run IPSYS and GIPSYS are entered into Windows reg istry However if MIPSYS is to be used user must add in Matlab path to package s pl m and cpp dll files This is done using addpath Matlab command To remember the new path between different Matlab sessions run savepath command after the last addpath At this point all three interfaces should be functional 67 Appendix B Description of the PTI Load Flow Data Format This is the description of an earlier
47. or clicking on the standard save icon on the tool bar It is users responsibility to provide one and only one slack bus for each network 37 A network entered in the GUI editor window does not have to be saved to be able to use it Of course the changes will not be available for the next session unless they are saved on the hard drive GUI editor can export the edited network to PTI23 format files and two additional files containing demand and generator cost functions These files can be used with non GUI IPSYS interface described in chapter 2 or to use it with another package that understands the PTI23 format GUI editor can also export the network as a picture for inclusion in publications and to Matpower format The export functions can be accessed with File gt Export menu entry 4 2 Running IPSYS Algorithms Once a network is opened it can be used either to run common algorithms from the GUI editor interface or from the GUI CLI interface The text in terface provides much finer control over the execution of the algorithms then the GUI interface can Whatever is done through the GUI or command win dow is reflected in both environments For example if the Newton Raphson power flow is run from the command window by issuing NRPF command the results are reflected in the GUI window after the power flow is executed Similarly if the Newton Raphson power flow is executed from the GUI menu the IPSYS network object is modified accordingly This
48. or is the set of P and Qi mismatches A power system s currents depend on the bus voltages in a linear manner T Yu Yr Ns Yu Ys E Io Ya Yoo Y Ya Yo Ez A gt 4 C 5 In Ym Yn2 Yn3 Yna Yns En where Yij Yij and Ya D Yij Yig j over all j connected buses and i g ground connections At each bus the injected complex power is given by P jQi El C 6 where N L gt Yin Ei k 1 then N P jQ Ei Ya k 1 and N P jQi EYE X YE E k 1 kri 74 The last equation can be expanded into P iQ ENEE Ga jB 00 00 2 E Ex Gik cos 0 Ox Bix sin 0 O C 7 j 2 Eil Er Gis sin 0 0k Bix cos 0 8x where Oi Ok the phase angles at buses and k il Ex the bus voltage magnitudes Gi HjBir Yi is the ik term in the Y matrix The system of equations C 7 is a nonlinear system which is solved using stan dard first order Taylor series approximation The Jacobian of the Equations in C 7 is given by the following matrix equation OPL OP OP OP AP 39 A pA Bal AB ub 1 1 1 An _ 8 EE Aa 2 2 AP 90 DE 265 Js Ad C 8 2 AQ OQ2 Q2 IQ2 A 001 E 002 O E2 To further simplify the derivatives instead of A E we use a which results in oo Ei Ex Gis sin 0 6 Bix 0 0s N E Ex Gis cos 0 0 Bix sin 0 Qi C 9 Z E Ex Gis
49. perator Table 2 3 General purpose logical operators Typical script with conditional tests 15 ipsys gt a 0 ipsys gt if a 0 b 1 0 else b 1 a ipsys gt printf Zg n a 0 IPSYS understands usual program flow control constructs Loops are imple mented using while and for constructs and the tests with if then else logical test All the logical operators are discussed in section 2 2 The fol lowing is an example of typical use of test and looping in IPSYS ipsys gt a 0 ipsys gt b 10 ipsys gt while a lt 5 amp amp b gt 0 printf a hg tb g n a b a a 1 b b 1 a 0 b 10 a 1 b 9 a 2 b 8 a 3 b 7 a 4 b 6 Another way of looping through a series of calculations is by using for loops ipsys gt for i 0 i lt 5 i i 1 printf 3 2g n i 5 0 oooo on N 16 Flow control using continue break and goto statement are not implemented since they are not necessary Switch case statement can be emulated with with a more flexible if else if construct ipsys gt a M_PI ipsys gt if a M_PI_2 printf a is g n M_PI_2 else if a M_PI printf a is g n M_PI else printf a is not known n ais 3 14159 Previous examples use printf function which is part of the I O and is de scribed in more detail in section 2 5 17 2 3 Scripts and Functions There are two mechanisms to automate and expan
50. posite RMA Upper limit of turns ratio or phase shift RMI Lower limit of turns ratio or phase shift VMA Upper limit of controlled volts MW or MVAR VMI Lower limit of controlled volts MW or MVAR STEP Turns ratio step increment TABLE Zero or number of a transformer impedance correction table 1 5 Area Interchange Data Ends with I of zero I ISW PDES PTOL ARNAM I Area number 1 100 ISW Area interchange slack bus number PDES Desired net interchange MW out PTOL Area interchange tolerance MW ARNAM Area name 8 characters enclosed in single quotes DC Line Data Ends with I of zero Each DC line has three consecutive records I MDC RDC SETVL VSCHD VCMOD RCOMP DELTI METER IPR NBR ALFMAX ALFMN RCR XCR EBASR TRR TAPR TPMXR TPMNR TSTPR IPI NBI GAMMX GAMMN RCI XCI EBASI TRI TAPI TPMXI TPMNI TSTPI I DC Line number MDC Control mode O blocked 1 power 2 current RDC Resistance ohms SETVL Current or power demand VSCHD Scheduled compunded DC voltage KV VCMOD Mode switch DC voltage KV switch to current control mode below this 71 RCOMP Compounding resistance ohms DELTI Current margin per unit of desired current METER Metered end code R rectifier I Inverter IPR Rectifier converter bus number NBR Number of birdges is series rectifier ALFMAX Maximum rectifier firing angle degrees ALFMN Minimum rectifier firing angle degrees RCR
51. s pl_Admitt pl_DFFlows pl_DFactors pl DPgDT pLDPgDVg pLDPgDVI pLDPIDT pLDPIDVg pLDPIDVI pLDQIDT pl DQIDVg pl DQIDVI pl GSPF pl_Jacobian pLNRPF Net structure Net structure Net structure Net Structure Net structure Net structure Net structure Net structure Net structure Net structure Net structure Net structure Net structure Net structure Net structure Net admittance matrix distribution factors flow distribution factors 2 matrix matrix matrix matrix de matrix matrix Dy matrix matrix L matrix Gauss Seidel power flow Net Jacobian Newton Raphson power flow OPF Algorithms pl DCOPFFlexD pl DCOPFFlexG pl DCOPFFlexGD Net structure Net structure Net structure DCOPF with flexible Py DCOPF with flexible P DCOPF with flexible P and P4 pLLMPSD Net structure LMPS with flexible P4 node connected to ref node ref node increment pLLMPSG same as pl LMPSD LMPS with flexible P pLLMPSGD same as pl LMPSD LMPS with flexible P and P4 Miscellaneous Functins pl MClust Real matrix Clustered matrix pl_netUpdate Net structure Updates network variables Table 3 2 Matlab Interface Functions 34 Chapter 4 GIPSYS Interface GIPSYS provides a GUI interface to all of the IPSYS functionality How ever GIPSYS GUI editor can be used to edit only a single network at the time GIPSYS consists of IPSYS CLI interface script editor and GUI po
52. son power flow on the above network and prints out the voltages gt gt nt pl NRPF b4 gt gt nt bus I nt bus VM nt bus VA ans 1 0000 2 0000 3 0000 4 0000 1 0000 1 0000 0 9993 0 9993 O 1 4332 2 5803 3 4406 Network structure data can be at any point used to extract and or change data using usual Matlab methods Matlab provides a large number of func tions to extract whatever information is needed and present it in whatever 59 len 3 0 j0 0 9 0 0 0 5 04 00 Figure 6 1 Four Bus System 60 format is desired with great flexibility If it is desired to get the same line flows output as with ipsys interface the following function would return a Matlab string with the flows gt gt flows pl_NetFlows nt gt gt flows Input data file unknown 0 100 00 Tue Oct 25 16 18 59 2005 GIPSYS From Volt Volt Mw Mvar Mw Mvar Bus Mag Angle Load Load Gen Gen To Mw Mvar Bus Flow Flow 1 1 00000 0 00000 0 00 0 00 7 00 0 20 2 2 50 0 03 3 4 50 0 17 2 1 00000 1 43318 0 00 0 00 3 00 0 25 1 2 50 0 03 4 3 50 0 13 3 2 00 0 09 3 0 99932 2 58029 5 00 0 00 0 00 0 00 1 4 50 0 03 2 2 00 0 05 61 4 1 50 0 01 4 0 99930 3 44065 5 00 0 00 0 00 0 00 2 3 50 0 01 3 1 50 0 01 Mw gen 10 000 Mvar gen 0 451 Mw load 10 000 Mvar load 0 000 Total I2R Mw losses 0 000 Total I2X Mvar losses 0 451 Generation Cost 0 000 and at this point the string can be stored in a file or just analyzed in M
53. st Pgl 272 20 Pg4 186 36 Pg6 148 03 GenCost 580 91 Now change the generator output of the generator on bus four and calculate the generation cost again ipsys gt SSNetGen b6 4 1 150 ipsys gt SSNetNRPF b6 ipsys gt Pgi SSNetGen b6 1 1 gt ipsys gt Pg4 SSNetGen b6 4 1 gt ipsys gt Pg6 SSNetGen b6 6 1 gt ipsys gt GenCost SSNetCost b6 ipsys gt print Pgl 7 2f tPg4 7 2f tPg6 7 2f tGenCost 7 2f n Pg1 Pg4 Pg6 GenCost Pel 308 91 Pg4 150 00 Pg6 148 03 GenCost 585 23 Newton Raphson Power Flow solution for the second operating point is Input data file 0 100 00 Tue Mar 27 16 21 28 2007 GIPSYS From Volt Volt Mw Mvar Mw Mvar Bus Mag Angle Load Load Gen Gen To Mw Mvar Bus Flow Flow 1 1 00000 0 00000 120 00 24 00 308 91 0 00 45 2 39 73 19 09 3 113 13 31 09 4 36 05 0 33 2 0 98112 3 43660 125 00 24 00 0 00 0 00 1 37 78 21 03 3 5 00 9 71 6 82 22 35 32 3 0 98608 3 28854 120 00 24 00 0 00 0 00 1 113 13 24 21 2 5 00 9 78 5 11 87 9 57 4 1 00000 1 03295 120 00 24 00 150 00 0 00 1 36 05 0 33 5 65 97 19 20 6 0 08 0 00 5 0 99095 2 94059 120 00 24 00 0 00 0 00 3 11 87 9 69 4 65 97 16 84 6 65 89 16 84 6 1 00000 1 03526 0 00 0 00 148 03 0 00 2 82 22 39 47 4 0 08 0 00 5 65 89 19 20 Mw gen 606 942 Mvar gen 0 000 Mw load 605 000 Mvar load 120 000 Total I2R Mw losses 1 942 Total I2X Mvar losses 18 532
54. tages reference bus angle non controlled generator MVA and MVAR outputs and loads fixed bus voltages etc Sim ilarly U is the vector of control variables such as real and reactive power generation phase shifter angles net interchange load shedding DC trans mission line flows control voltage settings LTC settings etc Equality and inequality constraints include limits on all control variables power flow equa tions generation load balance power flow limits bus voltage limits reserve limits MVAR limits etc The optimization objective could include power cost optimization loss minimization minimum control shift minimum num ber of controls rescheduled etc With the introduction of the free market TT a different set of constraints and different objectives are being introduced depending on the deregulation model OPF solutions are based on both linear and nonlinear numerical methods Linear programming methods are based on different variants of Linear Pro gramming LP while the nonlinear methods include Sequential Quadratic Programming Augmented Lagrangian Methods Generalized Reduced Gra dient Projected Augmented Lagrangian Successive Linear Programming Interior Point Methods among others The number of the different solutions and variants is a clear indication of the complexity of the problem All of the OPF algorithms in use in some way simplify the original problem 78 Appendix E DC Optimal Power Flow Revie
55. tchable demand SSNetDCOPFFlexGD DC OPF with dispatchable generation and demand SSNetDF returns distribution factors matrix SSNetDCFL returns calculated DC flows using distribution matrix SSNetMakeSens calculates sensitivity matrices SSNetDPgDT Returns Zi and index vectors SSNetDPgDVg Returns ae and index vectors SSNetDPgDVI Returns S and index vectors SSNetDPIDT Returns oF and index vectors SSNetDPIDVg Returns h and index vectors SSNetDPIDVI Returns 7 and index vectors SSNetDQIDT Returns a and index vectors SSNetDQIDVg Returns E and index vectors SSNetDQIDVI Returns e and index vectors MatrixLMPSG Returns LMPS while optimizing flexible generation MatrixLMPSD Returns LMPS while optimizing flexible demand MatrixLMPSGD Returns LMPS while optimizing flexible generation and demand IPSYS Power system algorithms new value ipsys gt st4 SSNetGenST b6 4 1 0 ipsys gt printf previous generator on bus 4 id 1 status was g n st4 previous generator on bus 4 id 1 status was 1 while the next example just reads the status of the same generator ipsys gt st4 SSNetGenST b6 4 1 gt ipsys gt printf current generator on bus 4 id 1 status is g n stA4 current generator on bus 4 id 1 status is 0 The following group of functions calculate various sensitivity matrices For efficiency reasons SSNetMakeSens must be called to actually calculate all of the sensitivities for the current operating conditions Afterward a
56. uch as matrix input matrix manipulation gen eral purpose functions and linear algebra algorithms available The second section demonstrates input output facilities The rest of the examples deal with power systems manipulation and analysis Implementation details and the theory behind this algorithms can be found in the appendices All of the examples will use the following data files 1 b3 raw is three bus PTI raw data file b3 sup generator cost curves b3 dmd demand cost curves data b6 raw six bus PTI raw data file GV AeA WwW N b6 sup generator cost curves 6 b6 dmd demand cost curves and should be included with this manual The examples emhasize power systems features more than how to use IPSYS interface which should be intuitive enough All of the algorithms are implemented using dense matrix methods from IMSL library 3 version 6 0 academic edition and Template Numerical Toolkit TNT 5 40 5 1 Linear Algebra This section demonstrates general numerical and linear algebra ipsys features 5 1 1 Checking Rank and Condition Number This example shows how a matrix can be constructed element by element and checked for the condition number and rank ipsys gt h Matrix 5 5 0 ipsys gt for i 1 i lt Rows h i i 1 for j 1 j lt Cols h j j 1 h i j 1 1 3 1 gt ipsys gt printf n g n Cond h 476607 ipsys gt printf n g n Rank h 5 5 1 2 Sensitivity Matri
57. version of PTI input format and it can be found at the Power Systems Test Case Archive Note that PTI reserves the right to change the format at any time For use with the IEEE 300 bus test case in PTI format Case Identification Data First record IC SBASE Ic 0 for base case 1 for change data to be added SBASE System MVA base Records 2 and 3 two lines of heading up to 60 characters per line Bus data records terminated by a record with a bus number of zero I IDE PL QL GL BL IA VM VA NAME BASKL ZONE I Bus number 1 to 29997 68 IDE Bus type 1 Load bus no generation 2 Generator or plant bus 3 Swing bus 4 Islolated bus PL Load MW QL Load MVAR GL Shunt conductance MW at 1 0 per unit voltage BL Shunt susceptance MVAR at 1 0 per unit voltage reactor IA Area number 1 100 VM Voltage magnitude per unit VA Voltage angle degrees NAME Bus name 8 characters must be enclosed in quotes BASKV Base voltage KV ZONE Loss zone 1 999 Generator Data Generator data records terminated by a generator with an index of zero I ID PG QG QT QB VS IREG MBASE ZR ZX RT XT GTAP STAT RMPCT PT PB I Bus number ID Machine identifier 0 9 A Z PG MW output QG MVAR output QT Max MVAR QB Min MVAR VS Voltage setpoint IREG Remote controlled bus index must be type 1 zero to control own voltage and must be zero for gen at swing bus MBASE Total MVA base of this ma
58. w DC Optimal Power Flow is based on the decoupled power flow without the Q V equations 7 The decoupled power flow neglects the P and any Ek interactions as well as Q and any O interactions Also neglecting Q V equations results in a DC power flow that is a linear system of power flow equations with all voltages set to 1 0 This system of equations can be solved for voltage angles The following discussion of DC OPF takes advantage of this linearity to formulate the OPF with respect to generation and load lev els Optimization with respect to either only the generation or only the load can be easily derived from the following equations The objective function is give in equation E 1 Ng N Min A Ys Ci Pa gt U Pi E 1 i j such that simple constraints are Pru lt P Gestal E 2 LE xt eai E 3 2 2 do const E 4 and linear combination constraints are Bx0 AP E 5 DF x AP lt Foz E 6 Where P is the generation of the it bus 6 is the voltage angle of the it bus do is the voltage angle of the slack bus Matrix B is the imaginary part of the admittance matrix and DF is the distribution matrix Equations E 2 E 3 and E 4 are simple variable constraints Real power generation is limited by minimum and maximum values all voltage angles except the slack bus angle are limited by 5 Slack bus angle is fixed to the original value read from the input data Linear constraint equation E 5 is th
59. wer systems network editor IPSYS CLI interface is the same CLI interpreter de scribed in chapter 2 with one difference When entering multi line data or commands user must use SHIFT RETURN or ENTER key to end each line except the last one which is ended with the usual RETURN key This requirement is due to unexpected parser design restrictions but it will also allow for much better error handling and reporting The script plain text and GUI editors are almost independent modules or libraries developed inde pendently The framework s modular design makes addition of new modules and interfaces quite easy GIPSYS functionality is achieved through a num ber of menu entries and the CLI interface with the rest of IPSYS GIPSYS GUI design follows standard GUI interface as much as possible including file opening saving closing and exporting to other formats and or graphics Edit menu includes adding new system elements such as buses branches etc but copying cutting and pasting is not implemented yet Run menu allows running the most common IPSYS algorithms View menu provides zooming in out snap on and choosing element label Figure 4 1 shows all three parts of the graphical user interface 35 ERE A GIPSYS b6 gps File Edit Tools File Edit Run view Help Pee bSa AM OMOM CHK OSH FRB 1 gt tt Interactive Power Systems Simulator Carnegie Mellon University Nov 3 2007 12 47 20 opa 10 29 RERERXERERERKRERKKK
60. x Extraction In the power systems operations it is important to be able to extract parts of a matrix For example some of the sensitivity matrices that ipsys calculates can be extracted from the Jacobian matrix Here is an example of how 29 can be extracted from the Jacobian matrix ipsys gt SSNet b6 b6 raw ipsys gt SSNetNRPF b6 0 000001 0 000001 501 ipsys gt jac SSNetJac b6 ipsys gt dqldt jac Range 6 Rows jac Range 1 5 ipsys gt SSNetMakeSens b6 ipsys gt print g SSNetDQ1DT b6 6 0712 0 0268085 O O 0 877069 0 0268085 1 2 O 0 190515 O O 0 190515 0 725736 1 2 0 664779 41 ipsys gt printf g daldt dqldt 6 0712 0 0268057 O O 0 877071 0 0268057 1 20001 O 0 190516 O O 0 190516 0 725739 1 20001 0 664782 According to the above example SSNetDQIDT function is not really neces sary but is provided for user s convenience 5 1 3 Matrix Decomposition and Clustering Epsilon decomposition and clustering can be used replacing one large network with a few smaller loosely coupled networks 8 IPSYS provides two almost equivalent function for this task The first one Clust mat eps implements a general network clustering algorithm described in 6 which takes a weighted undirectional graph and a weight threshold below which connections are ig nored This function returns a clustered matrix resulting from ignoring small below eps threshold

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