Shift Factors: Methodology and Example
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1 Shift Factors: Methodology and Example CRR Educational Class #5 CAISO Market Operations
2 Why are Shift Factors Important to the Allocation Process? The shift factors are derived directly from the DC version of the FNM They are not an explicit input into the CRR allocation The shift factor is the actual component that determines the flows on branches and interfaces that results from the application of CRR Source(s) and Sink(s) The flow on a branch or interfaces as compared to the branch limit or interface limit impacts how many CRRs can be allocated The shift factors are calculated and used in the optimization process within the simultaneous feasibility test Market Ops - R. Treinen 12/6/2005 to 12/8/2005 2
3 Course Objectives Upon completion of this course, you will be able to: Understand the concept of a shift factor Given a power system FNM data set, determine a set of shift factors for this FNM Given a set of shift factors along with a balanced Source/Sink pair determine the flow on any line in the system Market Ops - R. Treinen 12/6/2005 to 12/8/2005 3
4 Agenda Definition of the shift factor Shift factors in the AC FNM Shift factors in the DC FNM Using shift factors Detailed calculation of the shift factors An example using shift factors Market Ops - R. Treinen 12/6/2005 to 12/8/2005 4
5 Definition of the Shift Factor Shift Factors are also known by other names Power Transfer Distribution Factors (PTDFs) Power Distribution Coefficients (PDCs) Effectiveness Factors Impedance Factors Market Ops - R. Treinen 12/6/2005 to 12/8/2005 5
6 Definition of the Shift Factor To provide the definition or concept of the shift factor, start with a sample FNM (buses and lines). Assume this network is a DC FNM. The arrow on line l represents the reference direction Injection at Bus i Bus Bus i Transmission Line Line l Withdrawal at the Reference Bus Market Ops - R. Treinen 12/6/2005 to 12/8/2005 6
7 Definition of the Shift Factor A shift factor has 4 attributes: A particular Line (with reference direction) A particular Bus Value of the shift factor A reference bus The value of the shift factor of line l with respect to bus i is defined to be: The change (or sensitivity) of active (MW) power flow in a reference direction on line l with respect to an change in injection at bus i and a corresponding change in withdrawal at the reference bus Δflow in line l / Δinjection at bus i Market Ops - R. Treinen 12/6/2005 to 12/8/2005 7
8 Definition of the Shift Factor Simple Example Suppose there is a generator at Bus i and a load at the reference bus If the load increased by 1 MW and this load was served by an additional MW of generation (assume lossless FNM) at bus i there would be changes in the transmission line flows throughout the FNM Assume that the change in the MW flow on line l (in the reference direction) increased by 0.4 MW The shift factor for this line with respect to the generator and load locations is Δflow in line l / Δinjection at bus i 0.4/1 = 0.4 = 40% Market Ops - R. Treinen 12/6/2005 to 12/8/2005 8
9 Agenda Definition of the shift factor Shift factors in the AC FNM Shift factors in the DC FNM Using shift factors Detailed calculation of the shift factors An example using shift factors Market Ops - R. Treinen 12/6/2005 to 12/8/2005 9
10 Shift Factors for an AC FNM Calculated for the system conditions around a solved power flow operating point A given generation and load pattern Calculated for a certain system topology Topology is the connectivity of the system Shift factors will change when Operating point changes Reactive power flow may change Losses may change Topology changes Line characteristics change, e.g., change in line impedance Market Ops - R. Treinen 12/6/2005 to 12/8/
11 Agenda Definition of the shift factor Shift factors in the AC FNM Shift factors in the DC FNM Using shift factors Detailed calculation of the shift factors An example using shift factors Market Ops - R. Treinen 12/6/2005 to 12/8/
12 Shift Factors for a DC FNM DC FNM is a linear system Shift factors calculated for a certain system topology No operating point necessary No reactive power flow modeled No transmission losses modeled Market Ops - R. Treinen 12/6/2005 to 12/8/
13 Shift Factors In the DC FNM Shift factors change when Topology changes Line impedance changes Shift factors do not change if the injection/withdrawal amount increases for any set of locations Market Ops - R. Treinen 12/6/2005 to 12/8/
14 Shift Factors In the DC FNM Linear superposition of flows holds This is because the DC FNM is a linear network For example, for a given line l Injection of 1 MW at bus A and withdrawal of 1 MW at reference bus Assume shift factor on line l in the reference direction = 30% The flow on line l is 1 MW * 30% = 0.3 MW Remove injection at Bus A and withdrawal at reference bus Market Ops - R. Treinen 12/6/2005 to 12/8/
15 Shift Factors In the DC FNM Injection of 1 MW at bus B and withdrawal of 1 MW at reference bus Assume shift factor on line l in the reference direction = 40% The flow on line l is 1 MW * 40% = 0.4 MW Apply both sets simultaneously Inject 1 MW at bus A and inject 1 MW at bus B Withdrawal 2 MW (1 + 1) from the reference bus Flow on line l in the reference direction = = 0.7 MW Market Ops - R. Treinen 12/6/2005 to 12/8/
16 Shift Factors In the DC FNM Injection of 1 MW at bus A and withdrawal of 1 MW at reference bus Bus A 1 MW 0.3 MW Line l Reference Bus 1 MW Market Ops - R. Treinen 12/6/2005 to 12/8/
17 Shift Factors In the DC FNM Injection of 1 MW at bus B and withdrawal of 1 MW at reference bus 1 MW Bus B 0.4 MW Line l 1 MW Reference Bus Market Ops - R. Treinen 12/6/2005 to 12/8/
18 Shift Factors In the DC FNM Injection of 1 MW at bus A and 1 MW at bus B and withdrawal of 2 MW at reference bus Bus A 1 MW 1 MW Bus B 0.7 MW Line l 1 MW Reference Bus 1 MW Market Ops - R. Treinen 12/6/2005 to 12/8/
19 Any Questions? Market Ops - R. Treinen 12/6/2005 to 12/8/
20 Agenda Definition of the shift factor Shift factors in the AC FNM Shift factors in the DC FNM Using shift factors Detailed calculation of the shift factors An example using shift factors Market Ops - R. Treinen 12/6/2005 to 12/8/
21 Using Shift Factors Task: Given a set of shift factors for a FNM Determine the flow on line l given a Source (injection) of x MW and a Sink (withdrawal) of x MW Steps to accomplish this task Understand the set of shift factors Understand how to use this set along with the given Source/Sink pair to determine the flow on line l Market Ops - R. Treinen 12/6/2005 to 12/8/
22 Using Shift Factors Understanding the set of shift factors Assume N number of buses in system Assume N th node is defined to be the reference bus Assume L number of transmission lines in the FNM each with a given reference direction Market Ops - R. Treinen 12/6/2005 to 12/8/
23 Using Shift Factors (continued) For each line l (1,, L) calculate the shift factor for each node n (1,, N- 1) Inject 1 MW at node n and withdraw 1 MW at reference node (node N) and determine the flow on line l The actual mathematical method of calculating the shift factors are presented later in this presentation Each shift factor has associated with it a line (l) and a node (n) Each shift factor is denoted by S ln Market Ops - R. Treinen 12/6/2005 to 12/8/
24 Using Shift Factors Understanding the set of shift factors Note that S ln = 0, l (1,, L) A injection of 1 MW at the reference bus and a withdrawal of 1 MW at the reference bus will result in a net of 0 MW injection at the reference bus No flow in the system The shift factors, S ln, l (1,, L) are all 0 Market Ops - R. Treinen 12/6/2005 to 12/8/
25 Using Shift Factors Understanding the set of shift factors The number of shift factors in the set for a FNM with N nodes and L branches Number of shift factors calculated are L * (N 1) All shift factors are defined with respect to a specified reference bus The number of shift factors for S ln is L A Total number of shift factors are L * (N 1) + L = L * N Market Ops - R. Treinen 12/6/2005 to 12/8/
26 Using Shift Factors Sample shift factor data set There would be L * N records in the set Each record in the data set would have Injection bus identifier (e.g., bus name) Line identifier From bus To bus Circuit ID The reference direction is in the direction of From bus to To bus Shift factor value Market Ops - R. Treinen 12/6/2005 to 12/8/
27 Using Shift Factors Given the set of shift factors what is the flow on line l in the reference direction for an injection of x MW at bus A (the Source location) and withdrawal of x MW at bus B (the Sink location)? Assume bus A is not the reference bus Assume bus B is not the reference bus Market Ops - R. Treinen 12/6/2005 to 12/8/
28 Using Shift Factors Injection of x MW at bus A and withdrawal of x MW at bus B What is the flow on line l? Bus A x MW x MW Bus B Line l Flow =? Market Ops - R. Treinen 12/6/2005 to 12/8/
29 Using Shift Factors Note that from the set of shift factors given An injection of x MW at bus A and withdrawal at the reference bus of x MW will give the flow on line l in the reference direction as Flow on line l = (x MW) * S la An injection of x MW at bus B and withdrawal at the reference bus of x MW will give the flow on line l in the reference direction as Flow on line l = (x MW) * S lb Market Ops - R. Treinen 12/6/2005 to 12/8/
30 Using Shift Factors Injection of x MW at bus A and withdrawal of x MW at reference bus Injection of x MW at bus B and withdrawal of x MW at reference bus Bus A x MW x MW Bus B x * S la x * S lb x MW Reference Bus x MW Market Ops - R. Treinen 12/6/2005 to 12/8/
31 Using Shift Factors Instead of having an injection of x MW at bus B and x MW of withdrawal at the reference bus, what if the injection is at the reference node and withdrawal at bus B? Shift factor in the reference direction is (- S lb ) The flow on line l in the reference direction is (x MW) * (- S lb ) Note that the shift factor has a negative sign indicating a flow opposite to the reference direction Market Ops - R. Treinen 12/6/2005 to 12/8/
32 Using Shift Factors Injection of x MW at bus A and withdrawal of x MW at reference bus Injection of x MW at reference bus and withdrawal of x MW at bus B Bus A x MW x MW Bus B x * S la x * (-S lb ) x MW Reference Bus x MW Market Ops - R. Treinen 12/6/2005 to 12/8/
33 Using Shift Factors The injection and withdrawal at the reference bus cancels out! What remains is an injection of x MW at bus A and a withdrawal of x MW at bus B The flow on line l in the reference direction is x * S la + x * (-S lb ) = x * (S la S lb ) Market Ops - R. Treinen 12/6/2005 to 12/8/
34 Using Shift Factors Thus given the set of shift factors with The calculated shift factors, L * (N 1) with respect to a reference node With S ln = 0, l = (1,, L) To determine the flow on line l in the reference direction with an injection of x MW at bus A and a withdrawal of x MW at bus B perform a subtraction of the associated shift factors S la S lb Market Ops - R. Treinen 12/6/2005 to 12/8/
35 Using Shift Factors (continued) Apply this value to the injection (or balanced withdrawal amount) to determine the flow on the line in the given reference direction Flow on line in the reference direction = x * (S la S lb ) Does not matter where the reference bus is located! Do not even need to know the location of the reference bus! Market Ops - R. Treinen 12/6/2005 to 12/8/
36 Using Shift Factors Injection of x MW at bus A and withdrawal of x MW at bus B What is the flow on line l? Bus A x MW x MW Bus B Line l Flow = x * (S la - S lb ) Market Ops - R. Treinen 12/6/2005 to 12/8/
37 Any Questions? Market Ops - R. Treinen 12/6/2005 to 12/8/
38 Agenda Definition of the shift factor Shift factors in the AC FNM Shift factors in the DC FNM Using shift factors Detailed calculation of the shift factors An example using shift factors Market Ops - R. Treinen 12/6/2005 to 12/8/
39 Detailed Calculation of the Shift Factors The shift factor is conceptually obtained in three steps based for a DC FNM: Step 1: Sensitivity of phase angles with respect to bus injections: θ = [ B ] 1 P Step 2: Sensitivity of line flows with respect to phase angles: P l = [ H ] θ Step 3: Combine Step 1 and Step 2: P l = [ H ] [ B ] 1 P The shift factors are the entries of the matrix: [ S ] = [ H ] [ B ] 1 Bold elements are vector or matrices Elements in brackets [ ] are matrices Market Ops - R. Treinen 12/6/2005 to 12/8/
40 Detailed Calculation of the Shift Factors - Details of Step 1 DC load flow model in matrix form P = [ B ] θ P i, the i th element of vector P, is the net injection into the system from bus I θ i, the i th element of θ, is the phase angle at bus i [ B ] is the bus admittance matrix and is determined directly from the FNM line data B(i,i) = j (1/x ij ), for j is over all lines connected to i B(i,j) = -1/x ij [ B ] θ is the vector whose i th row is the sum of the power flows over all the lines originated from bus i Market Ops - R. Treinen 12/6/2005 to 12/8/
41 Detailed Calculation of the Shift Factor - Details of Step 1 [ B ] is the submatrix of [ B ] obtained by eliminating the row and column associated with the reference bus [ B ] cannot be inverted because it is singular [ B ] is not singular θ and P are obtained from θ and P by eliminating the elements associated with the reference bus θ reference = 0 radians Reduced DC load flow model P = [ B ] θ θ = [ B ] 1 P Market Ops - R. Treinen 12/6/2005 to 12/8/
42 Detailed Calculation of the Shift Factors - Details of Step 2 The relationship between the line flow and phase angles across the line is as follows P ij = (θ i θ j ) /x ij P ij is the power flow on the line connecting bus i and j in the direction from bus i to bus j θ i is the phase angle at bus i θ j is the phase angle at bus j x ij is the reactance of the line connecting buses i and j In vector form, this can be written as P lines = [ H ] θ where P lines, k = P ij is the k th element (i.e., the k th line) of P lines H(k,i) =1/x ij, H(k, j) = - 1/x ij and H(k, m) = 0 for m i, m j The values of [ H ] are determined directly from the FNM line data Market Ops - R. Treinen 12/6/2005 to 12/8/
43 Detailed Calculation of the Shift Factor - Details of Step 3 Denote [ Z ] = [ B ] 1 Then P lines = [ H ] θ = [ H ] [ B ] 1 P = [ H ] [ Z ] P Define [ S ] = [ H ][ Z ] P lines = [ S ] P Gives the relationship between the flows on lines and the injection/withdrawals at buses Market Ops - R. Treinen 12/6/2005 to 12/8/
44 Detailed Calculation of the Shift Factor - Details of Step 3 The shift factors are the entries of matrix [ S ] where the (l, i) th element of [S] is the sensitivity of the flow on the l th line with respect to a resource at the i th node. The slack node balances out the system S li = (z mi -z ni ) / x l where x l is the reactance for line l which connects bus m and bus n. z mi is the (m,i) th element of [Z] Market Ops - R. Treinen 12/6/2005 to 12/8/
45 Any Questions? Market Ops - R. Treinen 12/6/2005 to 12/8/
46 Agenda Definition of the shift factor Shift factors in the AC FNM Shift factors in the DC FNM Using shift factors Detailed calculation of the shift factors An example using shift factors Market Ops - R. Treinen 12/6/2005 to 12/8/
47 An Example of Using Shift Factors Example 7 bus system with Main commercial transfer corridor External loop Inject 10 MW at bus 1 Withdraw 10 MW at bus 5 Find the resulting flows on line 4 (bus 3 to bus 4 is the reference direction) line 9 (bus 2 to bus 7 is the reference direction) Use Bus 7 as the reference bus Market Ops - R. Treinen 12/6/2005 to 12/8/
48 An Example of Using Shift Factors Inject 10 MW Line Line reactances are in italics Line Withdraw 10 MW Market Ops - R. Treinen 12/6/2005 to 12/8/
49 An Example of Using Shift Factors To determine the flows on these lines First Determine the shift factors with Bus 7 as the reference bus Second From this set of shift factors and the amount of injection and withdrawal determine the flow on these lines Market Ops - R. Treinen 12/6/2005 to 12/8/
50 B = An Example of Using Shift Factors Determine the set of shift factors - Form the B matrix B As can be seen, the sum of all the columns and the sum of all the rows are zeros. Select bus 7 as the ref bus; delete row 7 and column 7 to obtain B Market Ops - R. Treinen 12/6/2005 to 12/8/
51 An Example of Using Shift Factors Z = [B ] -1 = Invert the B matrix to get the Z matrix Market Ops - R. Treinen 12/6/2005 to 12/8/
52 An Example of Using Shift Factors Remember the form of the shift factor is S line,bus The relevant shift factors are: S 4,1 = 0.45 S 4,1 = (z 3,1 z 4,1 ) / x 4 = ( )/(1/800) S 4,5 = S 4,5 = (z 3,5 z 4,5 ) / x 4 = ( )/(1/800) S 9,1 = 0.55 S 9,1 = (z 2,1 z 7,1 ) / x 9 = /(1/100) S 9,5 = 0.45 S 9,5 = (z 2,5 z 7,5 ) / x 9 = /(1/100) Market Ops - R. Treinen 12/6/2005 to 12/8/
53 An Example of Using Shift Factors Shift Factor data with only relevant records Bus From bus To Bus Circuit ID Shift Factor Value Line 4 is from bus 3 to bus 4 with circuit ID = 1 Line 9 is from bus 2 to bus 7 with circuit ID = 1 Market Ops - R. Treinen 12/6/2005 to 12/8/
54 An Example of Using Shift Factors The flow on Line 4 = 10 MW * (S 4,1 + ( S 4,5 )) = 10 MW * ( (-(-0.45))) = 10 MW * 0.9 = 9 MW The flow on Line 9 = 10 MW * (S 9,1 + ( S 9,5 ) ) = 10 MW * ( ( 0.45) ) = 10 MW * 0.1 = 1 MW Market Ops - R. Treinen 12/6/2005 to 12/8/
55 An Example of Using Shift Factors Inject 10 MW MW Line 4 Line 9 1 MW 7 4 Withdraw 10 MW 5 6 Market Ops - R. Treinen 12/6/2005 to 12/8/
56 Any Questions? Market Ops - R. Treinen 12/6/2005 to 12/8/
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