accurate energy transaction allocation using path
TRANSCRIPT
Accurate Energy Transaction Allocation
using Path Integration and Interpolation
A THESIS SUBMITTED
TO THE FACULTY OF GRADUATE SCHOOL OF
UNIVERSITY OF MINNESOTA
BY
Mandar Mohan Bhide
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
Bruce F. Wollenberg,Adviser
June 2013
Copyright © 2013 Mandar Mohan Bhide
All rights reserved
Acknowledgments
Foremost, I would like to express my sincere gratitude to my research adviser Prof.
Bruce Wollenberg for his continuous support during my Master’s study and Research. I am
thankful for his patience, motivation and immense knowledge in the subject matter which
helped me during my research, studies and also outside of the academia.
My sincere thanks to visiting Prof.Qian Chen from Hohai University,China. My re-
search wouldn’t have been possible without his dedicated effort to streamline the earlier
work.
Finally, I am also thankful to my parents and all the divine or unintelligible forces that
have brought me to where and what I am today.
i
Contents
Contents ii
List of Figures vi
List of Tables vii
I Introduction 1
1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Energy Markets before 1992 . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Energy Markets between 1992 and 1996 . . . . . . . . . . . . . . . 2
1.1.3 Energy Markets after 1996 . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Electricity as commodity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Importance of Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 The Transmission Infrastructure . . . . . . . . . . . . . . . . . . . 4
1.3.2 Ancillary Services . . . . . . . . . . . . . . . . . . . . . . . . . . 5
II Cost Allocation Methods Review 7
2 Cost Allocation Criteria and Complexity 7
2.1 Basic Criteria for Cost Allocation Criteria . . . . . . . . . . . . . . . . . . 7
2.2 Factors to consider in Cost allocation : . . . . . . . . . . . . . . . . . . . . 7
2.3 MW-Mile Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Power Flow Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Power Transfer Distribution Factor (PTDF) Method . . . . . . . . 9
2.4.2 Sensitivity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Tracing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ii
2.5.1 Bialek tracing method . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5.2 Kirschen Tracing Method . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Alternative Pricing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.1 Unused Transmission Capacity . . . . . . . . . . . . . . . . . . . . 15
2.6.2 MVA-Mile Methodology . . . . . . . . . . . . . . . . . . . . . . . 15
2.6.3 Pricing of Counter Flows . . . . . . . . . . . . . . . . . . . . . . . 15
III ALLOCATION METHODOLOGY 17
3 Derivation of ETA Factors 17
3.1 Formulation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.2 Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Derivation of ETA Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Transaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 ETA Factors Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
IV Correction Factors and State polynomials 28
4 Making Case for Correction Factors and their derivations 28
4.1 General Newton-Raphson(NR) Method: . . . . . . . . . . . . . . . . . . . 28
4.2 Newton-Raphson Method for power flow . . . . . . . . . . . . . . . . . . 29
4.3 Calculating accurate J−1(θ ,V ) . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.1 Introduction to Gamma factors (Γ) . . . . . . . . . . . . . . . . . . 31
4.4 Calculate accurate X(s) and dX(s)ds . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Calculating the X(s) polynomial . . . . . . . . . . . . . . . . . . . 32
4.4.2 Calculating dX(s)ds . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.3 Calculating Gamma (Γ) factors . . . . . . . . . . . . . . . . . . . 35
iii
4.4.4 Incorporating Gamma (Γ) into Energy Transaction Allocation Factors 36
V Conclusion and Future Work 38
5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Result Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References 41
Appendix A Methods to Solve Power Flow 43
A.1 Power Flow Model Formulation . . . . . . . . . . . . . . . . . . . . . . . 43
A.2 Full Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2.1 General Derivation: . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2.2 Applying Newton -Raphson to Power Flow . . . . . . . . . . . . . 47
A.3 Decoupled and DC Power Flow . . . . . . . . . . . . . . . . . . . . . . . 48
A.3.1 Decoupled Power Flow . . . . . . . . . . . . . . . . . . . . . . . . 48
A.3.2 DC Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Appendix B List of Formulas 51
B.1 Transmission line power flow . . . . . . . . . . . . . . . . . . . . . . . . 51
B.2 Real and Reactive Power Loss . . . . . . . . . . . . . . . . . . . . . . . . 53
B.2.1 Evaluating the δPLosssrδX and δQLosssr
δX . . . . . . . . . . . . . . . . . . 54
Appendix C Numerical Integration Methods 56
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
C.2 Trapezoidal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
C.3 Simpsons Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
C.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
iv
Appendix D Data Points Fitting with Polynomials 63
D.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
D.2 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
D.3 Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Appendix E Experimental Results 66
E.1 Model Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
E.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
v
List of Figures
2.1 Proportional Sharing sample node . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Kirschen Tracing Method example . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Acyclic Diagram for generator contribution . . . . . . . . . . . . . . . . . 14
4.1 Newton -Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . 29
C.1 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
C.2 Trapezoidal example for n=1 . . . . . . . . . . . . . . . . . . . . . . . . . 58
C.3 Trapezoidal example for n=2 . . . . . . . . . . . . . . . . . . . . . . . . . 58
C.4 Simpsons Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
E.1 IEEE Standard 14 Bus Case . . . . . . . . . . . . . . . . . . . . . . . . . 66
vi
List of Tables
1 Inflow contribution to outflow . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Inflow contribution to outflow . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Bus Types in Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Transactions Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Power Loss (MW) Allocation for Area 1 . . . . . . . . . . . . . . . . . . . 68
6 Power Loss (MW) Allocation for Area 2 . . . . . . . . . . . . . . . . . . . 68
7 Power Flow (MW) Allocation for Area 1 . . . . . . . . . . . . . . . . . . . 69
8 Power Flow (MW)Allocation for Area 2 . . . . . . . . . . . . . . . . . . . 69
9 Reactive Power Loss (MVAR) Allocation for Area 1 . . . . . . . . . . . . 70
10 Reactive Power Loss (MVAR) Allocation for Area 2 . . . . . . . . . . . . 70
11 Reactive Power Flow(MVAR) Allocation for Area 1 . . . . . . . . . . . . . 71
12 ReactivePower Flow (MVAR) Allocation for Area 2 . . . . . . . . . . . . . 71
vii
Part I
IntroductionIn last two decades, United States power sector has seen a major shift in the market and
the actual system operation. Traditionally electric utilities, electric power cooperatives
and industrial facilities retained control over limited geographical part of the electrical
power infrastructure which includes generation, transmission and distribution. As a result,
this monopolistic vertically integrated structure was very much against free market ideas.
In1996, the order No.888 and 889 by Federal Energy Regulatory Commission (FERC)
provided the set of rules and guidelines which shaped today’s deregulated power market
and its operation [Lamoureux, FERC].
Trading electric power as commodity has its own challenges. In this part we discuss the
issues with using the electric power as a commodity in a free market economy and then we
will discuss the infrastructure and support services required for the reliable power system
operation.
1.1 Historical Background
Two main events in 1992 and 1996 drastically changed the North American electricity
market.
1.1.1 Energy Markets before 1992
All the utilities were regulated monopolies that had their own generation, transmission and
distribution networks with exclusive right to serve to the customer in a given geographi-
cal area. With increasing customer demand and economical aspects, utilities entered into
1
power purchases or sale agreements with neighboring utilities. These agreements were
either on long-term or short-terms basis. The short-term agreements usually consisted of
day-ahead or real time agreements. Utilities were more likely to enter into day-ahead agree-
ments when one of the utilities might have forecasted insufficient generation for the next
day. 1 Real-time transactions between utilities were rare and happened only in cases of
forced outages or during unforecasted load increase.
1.1.2 Energy Markets between 1992 and 1996
In 1992, Energy Policy Act (EPAct) was introduced which laid down the deregulation foun-
dation. EPAct allowed external utilities and independent generators to access the electric
transmission system of other utilities by levying appropriate charges. Although the act did
not dismantle the vertical utilities, it provided utility companies to serve load (customer)
that wasn’t in their physical area. The charge levied on external generators was in purview
of the transmission service provider. As a result, due to lack of proper guidelines, utilities
had unfair advantage over other market players for accessing their networks.
1.1.3 Energy Markets after 1996
In 1996, Federal Energy Regulatory Commission (FERC) issued orders No.888 and 889.
These orders caused major overhaul in utility’s operation. Previously vertically integrated
utilities were dis-bundled into autonomous generation, transmission and distribution en-
tities. This facilitated the fair access to Independent Power Provider (IPP) and also the
implementation of the Open Access Same-Time Information System (OASIS) to prevent
any unfair treatment by external transmission system operators. Open access to transmis-
sion systems and open markets resulted in indiscriminate access to all market participants
irrespective of their parent company.
The dis-bundling of these services raised many important questions for Transmission
1Note that online generation capacity consists of actual load and reserve margin. Reserve Margin isdictated by FERC standards.
2
Service Providers (TSP). Most importantly, the question of collecting the revenue for the
provided transmission service. Since Load Serving Entities (LSE) and Resource Entities
(RE) are no longer directly associated with TSP’s profitability, TSP has to generate its own
revenue and profit. In next part, we will discuss the current and proposed methods for TSP
to recover its cost.
1.2 Electricity as commodity
Unlike other commodities like gold, currency and oil, electric power has its own opera-
tion constraints that limits its buy and sell. One of the major hurdles in electric power
transactions include
1. Electric power generation should match total load and system losses i.e
Total Generation = Total Load +System Losses
System losses mainly consist of transmission and distribution losses that also in-
cludes transformer losses. Transmission and distribution constitutes about 7% of the
total generation. On average, high voltage transmission lines (above 39kV) usually
make up 65-70% of the total system loss.
2. It is hard to trace the flow of electricity from a particular generator to a particular load.
Most of the electric grid is based on Alternative Current (AC) principles. Unlike
Direct Current (DC) linear circuits, superposition principles can’t be used AC circuit
especially on complex circuitry as that of grid.
3. Congestion plays a major role in recovering and displaying system cost and con-
straints respectively. In power system, in order to supply the given amount of power
we must have enough capacity of transmission between a generator and a load. In
other word, as we increase the capacity of generator and loads, there must be enough
3
transmission capacity to transfer that power. In many cases, especially during peak-
load condition, there might not enough capacity of transmission between desired
(low cost) generator and load hence called congestion. It is also important to note
that power transfer not only depend upon transmission capacity but also on the power
flow solution feasibility.
4. Unlike other commodities, electricity market operations have to be regulated for a
reliable operation. The Power system has the backup online generation, referred as
reserve margin, used in case of any generation failure and sudden load increase to
avoid cascading system failure.
1.3 Importance of Allocation
In any free market, it is of utmost importance that seller must be free to choose his or her
supplier of choice. In the deregulated electricity market, generator/loads sells/buys at a
particular bus or a hub (group of buses). Transmission and Ancillary services provides the
means and security to transport that block of power. Due to high volume of investments in
these services, there needs to be a revenue stream to support and build these services.
1.3.1 The Transmission Infrastructure
The transmission infrastructure mainly consists of
• Transmission Lines: Transmission lines are the major part of the transmission cost
(around 90%). Transmission lines forms the meshed network and are segregated
based on their voltage levels. Every transmission line is rated for a particular MVA
and voltage rating. Usually, highest voltage transmission line can be up to 765kV.
• Transformers: Power/Auto type Transformers are used for stepping up and down
the system voltage. These transformers are essential for interconnecting power sys-
tem network at different voltage levels. Sometimes, Phase shift transformers are used
4
for interconnecting two different power systems. They accounts for 5% of the total
transmission cost.
• Protection: Protection is essential in order to protect costly devices in power system
(like transformers and capacitor banks) and is also essential for safe operations of
power system. Protection devices mainly consist of logic devices called relays and
devices to interrupt the circuit called circuit breakers. Protection constitutes about
2% cost of total transmission system
• Supporting devices: For a stable and reliable operation of the power system, Reac-
tive power compensator and new FACTS (Flexible AC Transmission System) devices
were introduced. They usually constitute less than 3% cost of the total transmission
system.
– Reactive Power Compensator: They are essential to maintain voltage stability
of the system. They are usually provided near load resulting in decrease total
inductive and increased transmission capacity. The Reactive Power Compen-
sator usually is a shunt capacitor bank which is switched whenever they are
needed.
– FACTS (Flexible AC Transmission Services): These devices are used for var-
ious reasons such as reactive compensation, generator oscillation damping, or
increasing the transmission capacity. These devices make use of many of the
advances in high voltage power electronics.
1.3.2 Ancillary Services
Ancillary Services can be classified mainly into following categories
1. Regulation: is a service that addressees any short term changes in demand and some
minor outages.
5
2. Reserve Margins: Similar to regulation, reserve margin addresses the increase in
the demand. But in contrast to regulation, reserve margins are used to addresses the
increase in the demand in several seconds to few minutes. Usually they are standby
generators like online hydro and gas generators. Based on the response time (avail-
ability time) of these generators, their grade and cost is decided.
3. Black Start: This service is used for restoration in the unlikely event to bring back
the grid online after the complete black out. In case of black start,special generators
equipped with disel generators are used to bring the system online.
It is important to note that Transmission cost account about 10% and Ancillary services
about 6% of the total MW cost to the beneficiary2 of the service. [7]
For electricity markets to be fair it is vital to allocate costs associated with transmis-
sion and ancillary services in scientifically sound manner. As we will see in Part II, We will
review the popular and highly cited proposed allocation methods. For rest of the thesis, En-
ergy Transaction Allocation using Path integration method proposed by A.Fradi, S.Brigone
and B.Wollenberg is discussed [1, 8]and then modified version of correction factors origi-
nally proposed by Gildersleeve [Gildersleeve] is introduced.
2Beneficiary definition is discussed in more detail in part II
6
Part II
Cost Allocation Methods ReviewFrom part I, we have seen the various components of Transmission and Ancillary Services,
and need of cost allocation for these services. Cost allocation problem has two study com-
ponents i.e. power system engineering and economics of market operation. In this part, we
will discuss the criteria and factors that any transmission cost allocation methodology ide-
ally should satisfy. Then next we will discuss some of the allocation method and principles
briefly. Finally, some proposed adjustments to these methodologies are discussed.
2 Cost Allocation Criteria and Complexity
2.1 Basic Criteria for Cost Allocation Criteria
Any cost allocation method should (from [7])
1. Cover running costs like maintenance, operations etc.of the service
2. Descent Return on capital investments
3. Provides incentives for future expansions
4. Follow basic principles of the Power Systems engineering
2.2 Factors to consider in Cost allocation :
Factors affecting the cost allocation are as follows (from [7])
1. Service Beneficiary: In US, load are seen as the beneficiaries of the services and are
usually allocated all the transmission costs. On the other hand, in European Union
and many of the free power market countries considers both generation and loads as
7
beneficiaries. As a result, transmission cost is the allocated in certain proportion to
each beneficiary.
2. Usage: Allocating costs based on the annual megawatt-hours of consumption and/or
generation regardless of location and peak demand. This method is widely known as
the Postage-Stamp rate method.
3. Peak Usage: Cost allocation is based maximum amount the peak generation and
load in a certain time period. Location of generator/load is not considered as a fac-
tor in cost allocation. This method is practiced by Electric Reliability Council of
Texas(ERCOT).
4. Physical Power Flow: In this allocation method, cost is allocated based on the actual
transmission flow in the line due to particular load and generator.
It is also worth to note that in many cases allocation methodology varies based on trans-
mission system voltage level. There is still no majorly accepted cost allocation method and
many of the power flow methods are still in development stage.
2.3 MW-Mile Methodology
MW-Mile method was one of the first pricing strategy proposed to recover fixed transmis-
sion cost based on the actual use of transmission network [5, 6, 7]. Original methodology
uses the DC power flows to estimate the transmission usage. This method guarantees full
recovery and reasonably reflects the actual usage of transmission system. Following steps
gives the outline of MW-Mile method in multi-transaction environment.
1. For a given transaction t, calculate flows on network lines using DC power flow
model by considering power injections only involved in transaction t
8
2. Calculate cost for the each transaction as
MWMILEt = ∑kεK
ckLkMWt,k (2.1)
where ck=cost per MW per unit length of line kεK ( $/MW-Mile),
Lk= Length of line k (miles)
MWt,k= MW of flow on line k due to transaction t
3. Above process is repeated for all the transactions. Based on proportional cost sharing,
total transmission cost for each transaction tεT is given by
TCt = TotalCost ∗ MWMILEt
∑tεT MWMILEt(2.2)
2.4 Power Flow Based Methods
These methods primary make use of factors derived from different power flow methods that
represents change in electrical quantities. Based on the type of power flow they are broadly
classified as
1. Distribution Factors : these factors are based on the DC power flow calculation[7, 2]
2. Sensitivity Factors: these factors are derived for full AC power flow models[6].
2.4.1 Power Transfer Distribution Factor (PTDF) Method
This method is partially used by PJM and Midwest ISO3 . Distribution factors are calcu-
lated using Decoupled Power flow method. Decouple power flow is discussed in detail in
Appendix A. The PTDF for the line between sending bus s and receiving bus r is given by
3PJM uses distribution factors for below 500kV lines while Midwest ISO uses combination of distributionfactors and Peak Usage method to allocate
9
PT DFsr =4Psr4Pt
(2.3)
where
4Psr = change in active power (MW) flow in the line between sending bus sand receiv-
ing bus r.
4Pt= active power (MW) injected in the system by transaction t
PTDF, inherently has all the drawbacks associated with DC methodology i.e. they easy
to calculate but with diminished accuracy. Cost allocation becomes easier task, since each
transaction effect can be superimposed on each other.
2.4.2 Sensitivity Factors
Unlike PTDFs, Sensitivity factors uses the AC (Newton-Raphson) power flow to calculate
the changes.
Derivation:
From the Newton-Raphson Power flow (given in appendix A) , we know that,
4P
4Q
=
δPδV
δPδθ
δQδV
δQδθ
︸ ︷︷ ︸
Jacobian
4V
4θ
(2.4)
where
4P =
4P
4P2
...
4Pn
4Q =
4Q1
4Q2
...
4Qn
4θ =
4θ1
4θ2
...
4θn
4V =
4V1
4V2
...
4Vn
4V
4θ
=
δPδV
δPδθ
δQδV
δQδθ
−1 4P
4Q
(2.5)
Above Jacobian inverse matrix can be written as in following format,
10
4V
4θ
=
δVδP
δVδQ
δθ
δPδθ
δQ
4P
4Q
(2.6)
Each term in the above inverse Jacobian matrix i.e. δθ
δP ,δVδP ,
δVδQ and δθ
δQ are sensitivity
factors since they are change in states i.e Voltage and Phase Angle w.r.t P and Q. Based
on these factors many methods have been introduced [6, 5]. All of these method tries to
address the multi-transaction scenario.-
2.5 Tracing Methods
Bialek and Kirschen Tracing methods are widely cited in Tracing based methods. Both are
designed for the recovery of fixed transmission cost by allocating real and reactive power
in a pool based market in a non-incremental fashion.[6, 5, 4]
2.5.1 Bialek tracing method
This method is based proportional distribution of power.
Figure 2.1: Proportional Sharing sample node
This assumption states that network node act as a perfect “mixer” of incoming flows so
that nodal inflows are shared proportionally between the outflows. In the given example as
11
shown above,contribution of incoming flows(q j,qk) to the outward flow (qm,ql) is given
by
contribution of qm ql
q j (q j
q j+qk)∗qm (
q jq j+qk
)∗ql
qk ( qkq j+qk
)∗qm ( qkq j+qk
)∗ql
Table 1: Inflow contribution to outflow
This method lacks mathematical rigor as well as conclusive proof of the underlying
assumption which is explained below.
2.5.2 Kirschen Tracing Method
In this method, system is divided into mainly in three parts
1. Domain: These are set of buses that obtain power from particular buses
2. Commons: These are set of contiguous buses supplied by the same set of generators
3. Links:Thesear are set of branches that connects commons.
Method involves uses recursive procedure to allocate the contribution by each generator or
load based on the principle of proportional sharing at commons. For example, consider the
power system shown below
12
Figure 2.2: Kirschen Tracing Method example
In above example let Pi jbe the flow from bus i to bus j through branchi j
Using above definitions bus 1 and bus 3 will form the commons 1 and 3 respectively.
Since they are supplied by Gen1 and Gen 3 respectively.
On the other hand buses 2,4,5 will be in commons 2 .
Then the resulting acyclic Diagram for each generator contribution is given in the fol-
lowing
13
Figure 2.3: Acyclic Diagram for generator contribution
Then, the usage allocation to each generator will be as follows.
Branch Pi j PG1i j PG2
i j1-2 P12 P12 01-5 P15 P15 03-2 P32 0 P323-5 P35 0 P353-4 P34 0 P34
5-4 P54(P12+P15)∗P54
P12+P15+P32+P35+P34
(P32+P35+P34)∗P54P12+P15+P32+P35+P34
Table 2: Inflow contribution to outflow
As it can be seen from the above table, we applied the proportional sharing principle
only to branch 5-4 . The branch 5-4 connects the buses(5,4) that are in the same commons.
2.6 Alternative Pricing Strategies
In this section we will discuss the main issues with unused transmission capacity, MVA-
Mile methodology, and pricing of counter flows for equal access. These issues are rather
less technical and deals more with transmission operator’s cost recovery capabilities. [6, 5]
14
2.6.1 Unused Transmission Capacity
Unused transmission capacity is defined as the difference of transmission capacity and the
actual flow on that line. Recall that MW-Mile methodology allocates the cost only based
on the proportion of power flow due to corresponding transaction. As a result, transmission
owners can recover cost only for MW flow in thel line/. 4
In case of allocation based on Unused Transmission Capacity, MW-Mile method can
be modified as equation 2.7
Et,c = Ec ∗MWMILEt
∑tεT MWMILEt(2.7)
where Ec= total embedded cost (dollars)
Et,c=Embedded cost allocated to transaction t (dollors)
2.6.2 MVA-Mile Methodology
One of the contentions with the MW-Mile method is that it dosen’t truly represents the
actual loading of the line . Since the MVA flow trully trepresnts the actual loading of the
line, it’s been proposed to use the MVA loading as the cost allocation parameter instead of
MW loading.
MW-Miles method can be modified to MVA-Miles methodology easily as given brlow
TCt = TotalCost ∗ MVA−MILEt
∑tεT MVA−MILEt(2.8)
2.6.3 Pricing of Counter Flows
In all of the power flow methods, any transaction can result in counter flow in any of the
transmission elements. Counter flow is the flow component contributed by the transaction
that goes in the opposite direction of the net power flow. Transmission service providers
4In a recent court ruling, MISO is ordered not to use cost allocation based on usage for transmission linesabove 500kV
15
have resisted paying for such counter flows. As a result, the zero counter flow pricing was
introduced which states that only those transaction that use the transmission facility in the
same direction of the net flow should be charged in proportion to their contributions to the
total positive flow.
16
Part III
ALLOCATION METHODOLOGYIn this part we discuss the methodology originally proposed in [1] to allocate the non-linear
transmission system energy quantities to multiple transactions. Formulation and derivation
of the Energy Transaction Allocation (ETA) factors are introduced next.
3 Derivation of ETA Factors
3.1 Formulation and Assumptions
3.1.1 Formulation
At any given instant any electric quantity in a power system is a function of system states.
These electrical quantities f can be any of the following
• Branch MW loss and MW flow
• Branch MVAR gain/loss and MVAR flow
• MVA loading
• Current magnitude
• Bus voltage magnitude
• Bus shunt MVAR gain/loss
Formula for some of the above quantities is given in Appendix B.
System States are nothing but Voltage and Phase Angle at each bus.
Mathematically,
17
f = F(V1,θ1, . . . ,Vn,θn) (3.1)
Two identical systems with identical states will have identical f value
If f baseand f f inal are the base (initial) and final values after all the transactions tεT then
any of the above electric quantity can be represented by their corresponding states,
f base = F(V base1 ,θ base
1 , . . . ,V basen ,θ base
n ) (3.2)
f f inal = F(V f inal1 ,θ
f inal1 , . . . ,V f inal
n ,θ f inaln ) (3.3)
If4 f is the resultant change in the electrical quantity
f f inal = f base +4 f (3.4)
Our objective is to allocate this4 f to each transactions.
Let 4 f t is change due to each transaction t (dependent variable) and letSt be vector
which represents the injected power (input independent variable) at the bus in transaction.
We will define a parameter η called the Energy Transaction Allocation (ETA) factor Such
that
4 f t = η .St (3.5)
Now that we have allocated4 f t to each transaction t, it is imperative that,
4 f = ∑tεT4 f t (3.6)
In short, our goal is to calculate 4 f t due to each transaction t such that it satisfies equa-
tion 3.4, 3.5 and 3.6 given above.We will define transaction later as we go along with the
18
derivation. But for the time being let’s assume that a transaction term involves input and
output of the power from certain of buses.
3.1.2 Assumption
We will assume injected Real and Reactive power at any bus are linear function of a pa-
rameter s such that
PN(s) = αPNP f inal
N s+βPN (3.7)
QN(s) = αQN Q f inal
N s+βQN (3.8)
Parameter constraints are
0≤ s≤ 1 (3.9)
Power Injection vector given below
S(s) =
P1(s)
Q1(s)...
Pn(s)
Qn(s)
(3.10)
Substituting Equation3.7, 3.8 in 3.10 , we get,
19
S(s) =
P1(s)
Q1(s)...
Pn(s)
Qn(s)
=
αP1 P f inal
1 s+β P1
αQ1 Q f inal
1 s+βQ1
...
αPn P f inal
n s+β Pn
αQn Q f inal
n s+βQn
S(s) =[
αP1 α
Q1 · · · αP
n αQn
]
P f inal1
Q f inal1...
P f inaln
Q f inaln
s+
β P1
βQ1...
β Pn
β Pn
(3.11)
S(s) = αS f inals+β (3.12)
For s=0, using equation 3.7 and 3.8
PN(0) = PbaseN = β
PN (3.13)
and
QN(0) = QbaseN = β
QN (3.14)
Or from equation3.11 ,
P1(0)
Q1(0)...
Pn(0)
Qn(0)
=
Pbase1
Qbase1...
Pbasen
Qbasen
=
β P1
βQ1...
β Pn
β Pn
(3.15)
Similarly,
20
For s=1, from equation 3.7 and 3.15 ,
PN(1) = P f inalN = α
PNP f inal
N +βPN
∴ P f inalN = α
PNP f inal
N +PbaseN
∴ αPN =
P f inalN −Pbase
N
P f inalN
(3.16)
and similarly,
αQN =
Q f inalN −Qbase
N
Q f inalN
(3.17)
IWe can use the MATLAB element wise operator (.) to convert above equation in
equivalent compact matrix form.
α =
αP1
αQ1...
αPn
αPn
=
P f inal1
Q f inal1...
P f inaln
Q f inaln
−
Pbase1
Qbase1...
Pbasen
Qbasen
./
P f inal1
Q f inal1...
P f inaln
Q f inaln
α = (S f inal−Sbase)./S f inal (3.18)
3.2 Derivation of ETA Factors
Recall that our objective is to calculate 4 f t i.e change in the desired electrical quantity
from states at s = 0 to s = 1 due to transaction tεT
Using the definite integral,4 f (= ∑4 f t ) is given by
21
4 f =� 1
0
d fds
ds (3.19)
The state variable X is given by
X =
θ1
V1
...
θn
Vn
(3.20)
Now, calculating the d fds using the Chain Rule,
d fds
=δ fδX
.dXds
(3.21)
ButdXds
=δXδS
dSds
(3.22)
∴d fds
=δ fδX
.δXδS
dSds
(3.23)
Note all the partial derivatives involved in above equation 3.23. Derivation of δ fδX for all the
electrical quantities is discussed in Appendix B.
From Newton -Raphson Power Flow method expression (Discussed in Appendix A),
δXδS
= J−1(θ ,V ) (3.24)
5
Substituting the above Equation 3.24in equation 3.22, we get
5Although given NR expression has some drawbacks, we will study the corrections and reasons behind itin the next part.
22
dXds
= J−1(θ ,V ).dSds
(3.25)
Substituting Equation 3.24in equation 3.23,
d fds
=δ fδX
.J−1(θ ,V ).dSds
(3.26)
From 3.12,
δSδ s
= α.S f inal (3.27)
But, from 3.18, the above equation can be rewritten as,
δSδ s
= α.S f inal = S f inal−Sbase =4S (3.28)
where
4S = S f inal−Sbase =
P f inal1
Q f inal1...
P f inaln
Q f inaln
−
Pbase1
Qbase1...
Pbasen
Qbasen
=
4P1
4Q1
...
4Pn
4Qn
(3.29)
d fds
=δ fδX
.J−1(θ ,V ).4S (3.30)
From equation 3.30 and 3.19,
4 f =� 1
0
δ fδX
.J−1(θ ,V ).4Sds (3.31)
Above equation forms the important part of the derivation since we were able to relate
the objective function 4 f with the parameter s. As we will see in next subsection, once
we define transaction then from equation 3.31, we can find out4 f t i.e change in electrical
23
quantity due to a transaction.
3.3 Transaction
In real-time operations, the power system is going through simultaneous changes in load
and generation. Based on the real-time buy and sell, one can assign power input into or
out of the given bus. Note that whenever market participants agrees to buy and sell partic-
ular block of energy it is always on specific bus/hub(group of buses). Any of these actual
power transfer into or out of the system constitutes as a transaction. Thus any transac-
tion can consist of increase/decrease by multiple power generation at multiple buses or
decrease/increase by loads at multiple buses.
Let us callStas the transaction tεT in input matrix form:
St=
Pt1
Qt1
...
Ptn
Qtn
(3.32)
For example, let us consider a single transaction in 14 bus case where we increase
generation at bus 1 and 3 by 2 MW and 3 MW respectively and increase the load at bus 13
by 5 MW respectively. Then transaction can be represented as
24
St=
P11
Q11
P12
Q12
...
P113
Q113
P114
Q114
=
2
0
3
0...
−5
0
0
0
Observe that Load increase represented as negative value with the same magnitude as
that of the total generation. Also note that all the generation and load increase sum to zero
which brings us to an important assumption when defining transactions.
Transactions are only defined by increase in load and generator which sum up to zero
(for any balanced and lossless system this is an implicit condition). As a result, generators
defined in transaction don’t compensate for any of system losses. The slack bus makes up
for all the system losses. This idea is very much similar to that of the Local Marginal Pric-
ing (LMP). Since, in LMPs calculation we assume that the system losses are compensated
by the slack bus. [2]
3.4 ETA Factors Calculations
Transaction defined in previous section will now help us to calculate4 f t
All the transactions when summed together gives us total changes in the system i.e.
4S = S1 +S2 + · · ·+St (3.33)
Where S1,S2, · · · ,St are all the transactions that occurred concurrently.
i.e.
25
4S =
4P1
4Q1
...
4Pn
4Qn
=
P11
Q11
...
P1n
Q1n
+ · · ·+
Pt1
Qt1
...
Ptn
Qtn
(3.34)
Thus, from equation 3.34 and 3.31,
4 f =� 1
0
δ fδX
.J−1(θ ,V ).
P11
Q11
...
P1n
Q1n
+ · · ·+
Pt1
Qt1
...
Ptn
Qtn
ds (3.35)
Each transaction column vectors are not functions of s and can be split as follows
4 f t =
� 1
0
δ fδX
.J−1(θ ,V ).
Pt1
Qt1
...
Ptn
Qtn
ds (3.36)
It is important to note that above equation 3.36 ideally should satisfy equation 3.6
i.e.4 f = ∑4 f t
Comparing the equation 3.5(4 f t = η .St ) and 3.36 , we get
η =
� 1
0
δ fδX
.J−1(θ ,V ).ds (3.37)
Numerical integrations techniques are introduced in Appendi C for calculation of η .
Now that we have seen the original derivation of ETA factors, in the next part we will
26
discuss why the ETAs given by equation 3.37 are not very accurate estimate and introduc-
tion of correction factors based on interpolation techniques.
27
Part IV
Correction Factors and State
polynomialsIn derivation of the ETA factors (refer to equations 3.24 and 3.37, we assumed that deriva-
tive of the state X with respective to S is a equal to J−1(θ ,V ). This assumption introduces
significant errors during the ETA calculations. In this part we will introduce the reason
behind these errors and provides a possible solution to the problem.
4 Making Case for Correction Factors and their deriva-
tions
4.1 General Newton-Raphson(NR) Method:
In NR method, similar to other iterative methods, we start with an initial guess which is
reasonably close to the true root Next, we get the tangent ( f ′(x))at the current guess . X
co-ordinate of the intercept between the tangent ( with slope f ′(x) ) and y = f (x)desired will
be our new guess xnew.
f (x)desired− f (x0) = f ′(x)(xnew− x0)
or
4 f (x) = f ′(x)4x (4.1)
4x = f ′(x)−14 f (x) (4.2)
28
We will then repeat this procedure until4x is less than or equal to the desired tolerance.
As shown in the following figure, in NR method f ′(x) has to be evaluated at every xnew
Figure 4.1: Newton -Raphson Method
In the above figure, x0 and xfinal are initial and final solution. It can be seen from
above example, slope calculated by the NR method is not exactly equal to desired slope.
Otherwise , we would have gotten the Power flow results in a single iterations. But if we
know our x0 and xfinal we can calculate the desired slope and that is the basis of proposed
modification.
4.2 Newton-Raphson Method for power flow
In appendix we have seen NR application in solving the power flow.
From equation 3.24,dXdS
= J−1(θ ,V ) (4.3)
which can be rewritten as
4X = J−1(θ ,V )4S (4.4)
29
Comparing equations 4.4 and 4.2, J(θ ,V ) is equivalent to the f ′(x). As mentioned
earlier, in order to get the final state (X f inal) we have to iterate equation 4.2 several times
until change in solution is less than or equal to the desired tolerance.
X f inal−X0 =4X
where4X is the total change from intial to final state.
X f inal = X0+4X = X0+(J−1(θ ,V )4S)X=X0 +(J−1(θ ,V )4S)X=X0+(J−1(θ ,V )4S)X=X0+ . . .
(4.5)
Hence, from 4.2, and 4.4, it is a gross approximation to use J−1(θ ,V )4S as4X since
we are neglecting the subsequent Jacobian terms Newton-Raphson(NR).
From the previous discussion, it is possible to calculate our desired slope if we know
the initial and final values. The same idea can be used to calculated desired J−1(θ ,V ) if
we know X0 andX f inal .
4.3 Calculating accurate J−1(θ ,V )
Now by integration method used to calculate in appendix it is clear that we know value of
X0,X1,...,XM−1,XM which are states vectors at each M steps of the integration.
4s =n
Number.O f .Steps=
nM
(4.6)
Refere Appendix C to know more about numerical integration.
For the reason mentioned in section 3.2, we will now treat dXds in equation 3.25 as an
estimate
[dXds
]est
= J−1(θ ,V )dSds
(4.7)
30
which is constant
4.3.1 Introduction to Gamma factors (Γ)
Gamma Factors are based on the simple principle that if we know the correct value of
variable (let’s call it xcorrect)and the incorrect value (xincorrect) then we can introduce the
correction factors \Gamma such that
xcorrect = Γ.xincorrect
or
Γ =xcorrect
xincorrect(4.8)
From above simple equation its clear that we can modify xincorrect to correct value using
Γ factors. Let us apply the strategy to column matrix dXds
From4.7,
[dXds
]est
=
dθ1ds
dV1ds...
dθnds
dV nds
est
= J−1(θ ,V )dSds
(4.9)
Now suppose we know more accurate(correct) ([dX
ds
]corrrect) which is discussed in next
section.
Lets us now introduce Γ vector such that
[dXds
]corrrect
= Γ
[dXds
]est
(4.10)
Note that both[dX
ds
]matrices are the column matrices and Γ value for given state de-
31
pends on the Correct and Estimate values of that resepctive state and is independent of
other state variables.
Hence, we will define Gamma (Γ) is a diagonal vector as
Γ =
Γ1 · · · · · · 0... Γ2
...... . . . ...
0 · · · · · · Γn
(4.11)
dθ1ds
dV1ds...
dθnds
dV nds
correct
=
Γ1 · · · · · · 0... Γ2
...... . . . ...
0 · · · · · · Γn
dθ1ds
dV1ds...
dθnds
dV nds
est
(4.12)
4.4 Calculate accurate X(s) and dX(s)ds
4.4.1 Calculating the X(s) polynomial
Appendix D gives a brief overview of calculating m order polynomial in terms of s.
To evaluate 3.35, we can either use Bisection integration method or Simpsons Rule. In
both cases, we evaluate the value of all the state X at each step i.e. s(0), s(1),..., s(m-1) are
value of s at each step given by 4.6. i.e.
Note that ifm = numbers o f steps
s(k) = s(0)+ k4s
but s(0) = 0
∴ s(k) = k4s (4.13)
32
Now, Let X1 be a single state whose value we know for every step
X1(s(0))
X1(s(1))...
X1(s(m−1))
=
a1ms(0)m +a1(m−1)s(0)m−1 + · · ·+a10
a1ms(1)m +a1(m−1)s(0)+ · · ·+a10
...
a1ms(m−1))m +a1(m−1)s(m)m−1 + · · ·+a10
(4.14)
The above equation can be rewritten as,
X1(s(0))
X1(s(1))...
X1(s(m−1))
=
s(0)m s(0)m−1 · · · 1
s(1)m s(1)m−1 · · · 1...
... . . . ...
s(m−1)m s(m−1)m−1 · · · 1
︸ ︷︷ ︸
known s
a1m
a1(m−1)...
a10
︸ ︷︷ ︸
Co f f icient
(4.15)
Now that we have separated the know s and unknown part (cofficient column matrix),
taking the inverse of known s matrix
a1m
a1(m−1)...
a10
=
s(0)m s(0)m−1 · · · 1
s(1)m s(1)m−1 · · · 1...
... . . . ...
s(m−1)m s(m−1)m−1 · · · 1
−1
X1(s(0))
X1(s(1))...
X1(s(m−1))
(4.16)
If we have multiple n variables X the above equation4.16 can be modified in the ex-
panded matrix form as below,
If nbus is the total number of buses in a system, we will have n number of state variables
where n = 2∗nbus. Cofficient of all the state varible then can be given by
33
a1m a2m · · · anm
a1(m−1) a2(m−1) · · · an(m−1)...
... . . . ...
a10 a20 · · · an0
=
s(0)m s(0)m−1 · · · 1
s(1)m s(1)m−1 · · · 1...
... . . . ...
s(m−1)m s(m−1)m−1 · · · 1
−1
X1(s(0)) X2(s(0)) · · · Xn(s(0))
X1(s(1)) X2(s(1)) · · · Xn(s(1))...
... . . . ...
X1(s(m−1)) X2(s(m−1)) · · · Xn(s(m−1))
(4.17)
4.4.2 Calculating dX(s)ds
Since Equation4.17 gives the cofficients of the polynomial X(s),e now know the polyno-
mial X(s) expression . Resulting state vector polynomial can be written as
X(s) =
X1(s)
X2(s)...
Xn(s)
=
a1m a1(m−1) · · · a10
a2m a2(m−1) a20
... . . .
anm an(m−1) · · · an0
︸ ︷︷ ︸
Co f f icients
sm
sm−1
s0
︸ ︷︷ ︸
variable
(4.18)
Taking the derivative of each state X w.r.t to s,
dX1(s)ds
= m∗a1msm−1 +(m−1)∗a1(m−1)sm−2 + · · ·+a11
...
34
dXn(s)ds
= m∗anmsm +(m−1)∗an(m−1)sm−1 + · · ·+an1 (4.19)
Thus , in vector form,
dX(s)ds
=
m∗a1m (m−1)∗a1(m−1) · · · a11
m∗a2m (m−1)∗a2(m−1) · · · a21
...... . . . ...
m∗anm (m−1)∗an(m−1) · · · an1
sm−1
sm−2
...
s0
(4.20)
4.4.3 Calculating Gamma (Γ) factors
Recall the equation 4.12,
dθ1ds
dV1ds...
dθnds
dV nds
correct
=
Γ1 · · · · · · 0... Γ2
...... . . . ...
0 · · · · · · Γn
dθ1ds
dV1ds...
dθnds
dV nds
est
(4.21)
Now, From 4.20 we will get the[dX
ds
]correct and from4.9 we will get the value of
[dXds
]est
at certain step lets call it XN
[dXds
]correct
= Γ.J−1(θ ,V )dSds
(4.22)
Expanding the above equation,
35
∴
dθ1ds
dV1ds...
dθnds
dVnds
XN
=
Γ1 · · · · · · 0... Γ2
...... . . . ...
0 · · · · · · Γn
δP1δθ1
δQ1δθ1
· · · δQnδθ1
δP1δV1
δQ1δV1
· · · δQnδV1
δP1δVn
δQ1δVn
· · · δQnδVn
−1
XN
αP1 P1
αQ1 Q1
...
αPn Pn
αQn Qn
(4.23)
From equation 3.28,
∴
dθ1ds
dV1ds...
dθnds
dVnds
XN
=
Γ1 · · · · · · 0... Γ2
...... . . . ...
0 · · · · · · Γn
δP1δθ1
δQ1δθ1
· · · δQnδθ1
δP1δV1
δQ1δV1
· · · δQnδV1
δP1δVn
δQ1δVn
· · · δQnδVn
−1
XN
4P1
4Q1
...
4Pn
4Qn
(4.24)
Note that Gamma factors have to be calculated for every step of the integration i.e we
will evaluate dXds and J−1(θ ,V )dS
ds at every step of the integration. Also ote that since we
have to calculate X(s), it is necessary that we know the states at each step of the numerical
integration beforehand .
4.4.4 Incorporating Gamma (Γ) into Energy Transaction Allocation Factors
Substituting value of dXds from the equation4.24 into the equation 3.35,
36
∇ f =� 1
0
δ fδX
.Γ.J−1(θ ,V ).
P11
Q11
...
P1n
Q1n
+ · · ·+
Pt1
Qt1
...
Ptn
Qtn
ds (4.25)
Similarly, we can divide the
∇ f t =
� 1
0
δ fδX
.Γ.J−1(θ ,V ).
Pt1
Qt1
...
Ptn
Qtn
ds (4.26)
As a result, ETA factors (η) can be given by,
η =
� 1
0
δ fδX
.Γ.J−1(θ ,V )ds (4.27)
As it can be seen from the equations 3.37 and 4.27, ETA calculations in original and
proposed method are very much similar other than the introduction of the Gamma Γ factors.
Hence, the ETAs can be calculate similar to original ETAs using same numerical integration
methods given in Appendix C.
Results for improved ETA calculations are given in Appendix E.
37
Part V
Conclusion and Future WorkIn this part, we will first discuss the current trends, need for ETAs and the observations
made based on proposed ETA calculation method. Later, we will also discuss the possible
future work.
5.2 Summary
From our discussion in Part I and II, it is quite clear that allocation problem is two folds i.e.
power systems constraints and economics. On the power system part, we have seen that
MW-Mile, usage or tracing algorithms based methodologies hardly comply with the power
system engineering principles. Although these methods are popular due to their simplicity,
these methods affects negatively to the competition and cost recovery in the deregulated
market. For example, usage based method completely neglects location of generator and
loads served which creates disadvatage to local load serving generatirs. Although PTDFs
and Sensitivity methods were introduced to comply with Power Systems principles, they
partially addressed the multi-transaction scenario. Energy Transaction Allocation (ETA)
introduced here are not only are in accordance with all of the power system principles but
also they fully comply with the multi-transaction scenario. The proposed methodology
here addresses two main issues with the original ETA calculations i.e. high number of
calculation and lower accuracy.
5.3 Result Discussion
Experiment involves using a standard IEEE 14 bus case and simulating the three transac-
tions as given in the table 4 of Appendix E. Table from 5-10 shows the real and reactive
power flow and losses allocation using the proposed methodology.
38
Two of the main advantages of using Gamma factors proposed methodology are
1. Significantly lower number of calculations: The proposed method calculates the
ETA factors with very few steps.For example, in Original ETA calculation number
steps required were eight but by using proposed alogorithm E the more accurate
results were achieved by just three number of steps i.e. less than half of the original
computations
2. Proposed method drastically improved the calculation accuracy. Original ETA
calculation method, in general, accuracy was limited as explained in Part IV.
5.4 Future Work
Transmission Service Providers (TSPs), in general, are important but traditionally been
passive participants of deregulated power market. As a result, they have always resisted to
use the cost allocation method that involves TSP active participation or the bookkeeping of
the positive and negative prices. Recent advances in Energy Management Systems (EMS)
and Market Management Systems (MMS) have helped tremendously to Generator owners
and Loads to worry only about the real time market prices and leave the system operations
in the hands of Regional Transmission Operators(RTOs) and Independent System Opera-
tors(ISOs). It would be certainly advantageous if we could integrate the application ETAs
with these applications.
1. Integrating the Transmission cost with Local Marginal Prices: As we have seen
in part III, LMPs and proposed cost allocation methodology share many of the basic
assumptions like role of slack bus and transaction definition. Successful integration
LMPs with ETA would not only be superior original LMPs calculation but also would
be less complicated for market players.
2. Integrating the ETAs with EMS: As we know the ETAs doesn’t require any special
formulation and only require the power flow and information about transactions in
39
the system. If the ETA calculations could be integrated with EMS’s power flow and
real time information, cost allocation would be a very easy task to handle without
any significant changes in the system.
3. Possible ETAs applications: Application of using ETAs has been originally dis-
cussed in [8, 3]. The ETAs can be used in many of the diverse application like
ancillary services and economic dispatch. More avenues of using ETAs in power
system should be looked at.
40
References
[1] A.Fradi. Calculation and Application of Energy Transaction Allocation Factors in
Electric Power Transamission Systems. PhD thesis, University of Minnesota, 2000.
[2] Bruce F. Wollenberg Allen J. Wood. Power Generation,Operation and Control. Wiley-
Interscience, 1996.
[3] D.Gildersleeve. The application of the path integration methodologyto the accurate
allocation of transmission system support services. Master’s thesis, University of Min-
nesota, 2006.
[4] J.Bialek. Tracing the flow of electricity. In IEEE ProccGener. Transm. Distrib. Vol.
143 No. 4, July 1996.
[5] Saifur Rahman Fellow IEEE Jiuping Pan, Yonael Teklu and Koda Jun. Review of
usage-based transmission cost allocation methods under open access. In IEEE TRANS-
ACTIONS ON POWER SYSTEMS VOL. 15 NO. 4, NOVEMBER 2000.
[6] Zuyi Li Mohammad Shahidehpour, Hatim Yamin. Market Operations in Electric
Power Systems: Forecasting, Scheduling, and Risk Management. Wiley-IEEE Press,
2002.
[7] PJM. A survey of transmission cost allocation issues, methods and practices. page 58,
2010.
[8] S.Brignone. Accurate Calculation of Power Systems Ancillary Services. PhD thesis,
University of Mineesota, 2010.
41
APPENDIX
42
Appendix A Methods to Solve Power Flow
As we have seen from the introduction of the thesis, transmission network plays impor-
tant role in differentiating electricity from other commodities. Unlike DC power flows,
solving AC circuits such as transmission systems involves solving of series of non-linear
equations.6 Each bus has at least one non-linear equations associated with it. The biggest
transmission network under Midwest ISO/ PJM operations models has buses more than
40,000. As a result, these advanced power flow solving packages involves solving on av-
erage 70,000 equations simultaneously. A power flow algorithm computes the voltage
magnitude and phase angle at each bus in a power system under balanced three-phase
which in turn enables us to calculate the real and reactive power flows for all transmission
lines and transformers, as well as losses in the different components in the system. System
components are transmission lines, generators, loads, transformers, phase shifters, shunt re-
active support components and FACTS (Flexible AC Transmission Systems) devices such
as Static Var Compensator(SVCs),STATatic COMpensators (STATCOM),Thyristor Con-
trolled Series Compensator etc. These devices such as tap changers, FACTS devices which
changes need to be modeled based on various constraints in order to incorporate in power
flow model. We will restrict our discussion to simple power flow models involving trans-
mission lines and transformers. Reference [2] covers the power flow in more detail.
A.1 Power Flow Model Formulation
Voltage (magnitude and phase) at each bus of the network and line impedance interconnec-
tion these buses determines power flow along the lines. (Complex) Power flow at each bus
(called nodes) should satisfy the conservation of energy law i.e input power node should
equal to output power at the node. This can be written as
6Note that although elements are linear, resulting power flow equations are non-linear.
43
Sk =n
∑i
Ski (A.1)
where Sk = Pk + jQk is the power that might be power injected (generator bus) or taken
out of the bus (load bus). Ski is the complex power flowing from bus k to bus i through
branch connecting bus k and bus i. If Ik is the current injected at node k, by Kirchhoff
current law,
Ik =n
∑i
Iki (A.2)
If we know the Yki, i.e admittance of braces connecting bus k and i ,
Iki = yki ∗ (Vi−Vj) (A.3)
substituting A.3 into A.2 we get
Ik = (yk1 + · · ·+ ykk + · · ·+ ykn)Vi− yk1V1− yk2V2−·· ·− yknVn (A.4)
Note that ykk is the self admittance of the bus
Lets call
Ykk = yk1 + · · ·+ ykk + · · ·+ ykn =n
∑i6=k
yki + ykk (A.5)
for k 6= i
Yki =−yki (A.6)
Equation A.1 can be written as
Sk = Pk + jQk = Vk
n
∑i
I∗ki = Vk
n
∑i(YkiVi)
∗ (A.7)
44
Also note that Equation A.4 can be written as
I1
I2
...
In
=
Y11 Y12 · · · Y1n
Y13 Y22 · · · Y2n
...... . . . ...
Y1(n−1) Y2(n−1) · · · Ynn
︸ ︷︷ ︸
Admittance(Y )−Matrix
V1
V2
...
Vn
(A.8)
which can be written as
Now that we have formulated the current in terms of system impedance and voltage,
we will look at the known and unknown for given particular bus. At each bus we have 4
inter-dependent variables i.e Voltage magnitude (V ), phase angle (θ ), real power (P) and
imaginary power(Q)
Type of Bus Alternative Name Known Variable Unknown VariablesSlack – V,θ P,Q
Generator PV P,V Q,θLoad PQ P,Q V,θ
Table 3: Bus Types in Power Flow
Slack Bus is a bus with a generator that plays major role in providing insufficient gen-
eration and compensating system losses. Usually slack bus is one of the biggest generator
in the system. System has only one slack bus. 7
Let there be nPQ,nPV buses in the system. Thus we know real power values at nPQ+nPV
buses and reactive power at nPQ buses. Similarly we don’t know value of θ at nPV + nPQ
buses and Voltage at nPQ buses. Thus we have same number of known and unknowns(2nPQ+
nPV ) which is crucial for solving any equation.
7Sometimes slack buses are defined for each area in case of islanding.
45
A.2 Full Newton-Raphson Method
A.2.1 General Derivation:
Let f (x) be any linear or non-linear function at any point x provided that it is infinitely
differentiable can be written as power series as follows
f (x) = f (x0)+f ′(a)1!
(x− x0)+f ′′(x0)
2!(x− x0)
2 + · · · (A.9)
If value of x0is very much near to the solution such that we can ignore higher orders
terms of (x− x0) i.e we will assume
(x− x0)n ≈ 0
for all n>1
As a result equation A.9 reduces to
f (x) = f (x0)+f ′(x0)
1!(x− x0)
Note that our objective is to calculatex that satisfies given f (x)
x = x0 + f ′(x0)−1( f (x)− f (x0)) (A.10)
or
∴4x = f−1(x0)4 f (x) (A.11)
x0 |new= x0 +4x (A.12)
we evaluate f−1(x0) and f (x) for new x0 until we reach the desired solution margin.
46
A.2.2 Applying Newton -Raphson to Power Flow
Recall the equation A.7 ,
Pk + jQk = Vk
n
∑i(YkiVi)
∗
This equation can be expanded for bus i as follow
Pi + jQi =n
∑k=1
ViVk(Gik− jBik)e j(θi−θk) (A.13)
where
Vi,Vk - are the the bus voltage magnitude at bus i and k
θi,θk- are phase angles at bus i and k
and Yik = Gik + jBik which is i-k term of the formed Y matrix formed earlier A.8
Now we will take derivatives of Pi and Qi (function) with respect to the variables θk and
Vk which is given below,8
δPi
δθk=ViVk[Giksin(θi−θk)−Bikcos(θi−θk)]
δPiδVkVk
=ViVk[Gikcos(θi−θk)−Biksin(θi−θk)]
δQi
δθk=−ViVk[Gikcos(θi−θk)+Biksin(θi−θk)]
δQiδVkVk
=ViVk[Giksin(θi−θk)−Bikcos(θi−θk)] (A.14)
Derivatives Pi and Qi (function) with respect to the variables θi and Vi,
8Note we calculate4P and4Q w.r.t to 4VV since it simplifies the equation
47
δPi
δθi=−Qi−BiiV 2
i
δPiδVkVk
= Pi +GiiV 2i
δQi
δθi= Pi−GiiV 2
i
δQiδVkVk
= Qi−BiiV 2i (A.15)
Now we can equation equivalent to A.11 for power flow
4θ
4V
=
δPδθ
δPδV
δQδθ
δQδV
−1 4P
4Q
(A.16)
In expanded form,
4θ1
4V1
...
4θn
4Vn
=
δP1δθ1
δP1δV1
· · · δP1δθn
δP1δVn
δQ1δθ1
δQ1δV1
· · · δQ1δθ
δQ1δVn
......
... . . . ...
δQnδθ1
δQnδV1
· · · δQnδθn
δQnδVn
−1
4P1
4Q1
...
4Pn
4Qn
(A.17)
A.3 Decoupled and DC Power Flow
A.3.1 Decoupled Power Flow
In decoupled power flow we will use following real-world approximations
1. sin(θi−θk)≈ 0 since all the adjacent buses have very small phase difference
2. Gik is negligible compared to Bik
48
As a result equation A.14 reduces to
δPi
δθk=−ViVkBik
δQiδVkVk
=−ViVkBik (A.18)
4P =δPi
δθk4θ
4Q =
δQi(δVkVk
)4Vk
Vk(A.19)
From equation A.19 it’s clear that we have decoupled the change in active and reactive
power from each other and are now dependent on the changes in bus angle and voltage
respectively.
A.3.2 DC Power Flow
In DC power flow , we assume that
1. All the bus voltage is near to 1p.u which is usually the case for normal operation.
2. Additionally, we will assume that rik ≈ 0 or rik is negligible to line reactance xik
As a result, from equation A.18,A.5and A.6
4P1
4P2
4Pn
=[B′]
4θ1
4θ2
4θn
(A.20)
and
49
4Qi = 0 (A.21)
where
B′ii = ∑1
xik
B′ik =−1
xik
It becomes easy to calculate the change in power flow in case of DC power flow, if
θi and θk are the phase angle oh bus i and k.
Then
Pik =θi−θk
xik
Notice that DC power flow gives no information actual reactive line flows, but gives
very fast approximate on change in real power flows.
Detailed AC derivation power flow across the line is discussed in next Appendix.
50
Appendix B List of Formulas
It is clear from the equation C.9 that we need to evaluate δ fδX . This appendix will the give
the equation for electrical quantity ( f ) and its respective derivative w.r.t V,θ .
B.1 Transmission line power flow
If Vs and Vr is the bus voltages at bus s and r, and ysr = gsr + jbsr is the admittance of the
branch 9,
Current from sending end to receiving end is given by
¯Isr = ysr(Vs−Vr) (B.1)
Power flow from sending end(s) to receiving end (r) are calculated as follows
Ssr = VsI∗sr = Vs [ysr(Vs−Vr)]
∗ (B.2)
which can then be expanded as Vs =Vse jθs and Vr =Vre jθr
Psr + jQsr =Vse[(gsr + jbsr)(Vse jθs−Vre jθr)
]∗(B.3)
Psr = gsr[V 2s −VsVrcos(θs−θr)]−bsr[VsVrsin(θs−θr)] (B.4)
Qsr =−bsr[V 2s −VsVrcos(θs−θr)]−gsr[VsVrsin(θs−θr)] (B.5)
Let us also consider branch charging suceptance bch and we will ignore shunt conduc-
tances (glr and gls) . Since shunt suceptance only inject/produce reactive power , Equation
B.59Since we know YBUS there’s no need to consider taps phase shift and taps which we already considered
when calculating YBUS for power flow
51
Qsr =−bsr[V 2s −VsVrcos(θs−θr)]−gsr[VsVrsin(θs−θr)]−V 2
sbch
2(B.6)
For simplicity, we will consider notation V as |V | as we don’t have any vectors in the
above expressions
Our objective is to calculate
δPsr
δX=
[· · · δPsr
δθs
δPsr
(δVs/Vs)· · · δPsr
δθr
δPsr
(δVs/Vs)· · ·]
(B.7)
and
δQsr
δX=
[· · · δQsr
δθs
δQsr
(δVs/Vs)· · · δQsr
δθr
δQsr
(δVs/Vs)· · ·]
(B.8)
Note that Psr,Qsr are only function θs,θr,Vs,Vr hence
for i 6= s,r;
δPsr
δθi=
δPsr
(δVi/Vi)=
δQsr
δθi=
δQsr
(δVi/Vi)= 0 (B.9)
evaluating δPδX
δPsr
δθs=VSVr (gsr[sin(θs−θr)]−bsr[cos(θs−θr)]) (B.10)
δPsr
δθr=VSVr (−gsr[sin(θs−θr)]+bsr[cos(θs−θr)]) (B.11)
δPsr
(δVs/Vs)= 2gsrV 2
s −VsVr[gsrcos(θs−θr)−bsrsin(θs−θr)] (B.12)
δPsr
(δVr/Vr)=−VsVr(gsrcos(θs−θr)+bsrsin(θs−θr)) (B.13)
evaluating δQsrδX
52
δQsr
δθs=−VSVr (gsr[cos(θs−θr)]+bsr[sin(θs−θr)]) (B.14)
δQsr
δθr=VSVr (gsr[cos(θs−θr)]−bsr[sin(θs−θr)]) (B.15)
δQsr
(δVs/Vs)= 2gsrV 2
s +VsVr[gsrsin(θs−θr)−bsrcos(θs−θr)] (B.16)
δQsr
(δVr/Vr)=VsVr(gsrsin(θs−θr)+bsrcos(θs−θr)) (B.17)
B.2 Real and Reactive Power Loss
In any branch transmission losses are given by
PLosssr = I2srRsr (B.18)
where Rsr =resistance of transmission line
similarly,
QLosssr = I2srXsr−
bch
2(V 2
s +V 2r ) (B.19)
10
where Xsr =inductive reactance of transmission line
bch= charging reactance transmission line
Recall the expression for ¯Isr give in equation B.1
¯Isr = ysr(Vs−Vr)
10We will use the π model for QLossCalculation
53
Isr = |Isr|= |ysr(Vs−Vr)| (B.20)
Now,
Vs =Vse jθs =Vs(cos(θs)+ jsin(θs))
and
Vr =Vre jθr =Vr(cos(θr)+ jsin(θr))
I2sr = |ysr|2 [|Vs−Vr|2] = |ysr|2 [V 2
s +V 2s −2VsVrcos(θs−θr)] (B.21)
Substituting above equation in B.18 and B.19
PLosssr = |ysr|2 [V 2s +V 2
r −2VsVrcos(θs−θr)]Rsr (B.22)
QLosssr = |ysr|2 [V 2s +V 2
r −2VsVrccos(θs−θr)]Xsr−bch
2(V 2
s +V 2r ) (B.23)
B.2.1 Evaluating the δPLosssrδX and δQLosssr
δX
δPLosssr
δX=
[· · · δPLosssr
δθs
δPLosssr
(δVs/Vs)· · · δPLosssr
δθr
δPLosssr
(δVs/Vs)· · ·]
(B.24)
δQLosssr
δX=
[· · · δQLosssr
δθs
δQLosssr
(δVs/Vs)· · · δQLosssr
δθr
δQLosssr
(δVs/Vs)· · ·]
(B.25)
for i 6= s,r;
δPLosssr
δθi=
δPLosssr
(δVi/Vi)=
δQLosssr
δθi=
δQLosssr
(δVi/Vi)= 0 (B.26)
NowδPLosssrδX
54
δPLosssr
δθs= |ysr|2 [2VsVrsin(θs−θr)]Xsr (B.27)
δPLosssr
δθr= |ysr|2 [−2VsVrcos(θs−θr)]Xsr (B.28)
δPLosssr
(δVs/Vs)= |ysr|2 [2V 2
s −2VsVrcos(θs−θr)]Rsr (B.29)
δPLosssr
(δVr/Vr)= |ysr|2 [2V 2
r −2VsVrcos(θs−θr)]Rsr (B.30)
for δQLosssrδX
δQLosssr
δθs= |ysr|2 [2VsVrsin(θs−θr)]Xsr (B.31)
δQLosssr
δθr= |ysr|2 [−2VsVrcos(θs−θr)]Xsr (B.32)
δQLosssr
(δVs/Vs)= |ysr|2 [2V 2
s −2VsVrcos(θs−θr)]Xsr−bch(V 2s ) (B.33)
δQLosssr
(δVr/Vr)= |ysr|2 [V 2
r −2VsVrcos(θs−θr)]Xsr−bch(V 2r ) (B.34)
55
Appendix C Numerical Integration Methods
Recall from equation for transaction
S(s) = αS f inals+β (C.1)
we defined all real and reactive power change to a single variable s. Also recall the ETA
factor (η) calculation
η =
� 1
0
δ fδX
.Γ.J−1(θ ,V )ds (C.2)
It can be seen that in order to calculate the ETA factors η we using definite integral of s
over 0 to 1. Thus in this appendix, we will only focus on the ETA factor calculations. Also
note that all the method being described based on Newton Cotes formula but have different
popular names. 11
C.1 Introduction
Our objective in numerical integration is to calculate the solution for
A =
b�
a
f (x)dx (C.3)
which is equivalent of finding area under the curve of f (x) from a to b as shown below,
11At high degrees sometime in Newton-Cotes error grows exponentially large but observation in the simu-lation showed that this is not the case for proposed ETA factor calculation
56
Figure C.1: Definite Integral
Also note that our function f (x) is continuous between interval [a,b]
C.2 Trapezoidal Method
In trapezoid methods, we will divide the interval [a,b] into n equal section, such that width
would be
4x =b−a
n
Then on each sub interval we will approximate the function with a straight line that is
equal to the function values at either endpoint of the interval.
As per trapezoidal rule, we will approximate A by summing up the area of each trape-
zoid as given below
b�
a
f (x)dx = A≈ 4x2
( f (x0)+2 f (x1)+2 f (x2) · · ·+ f (xn)) (C.4)
For example if n=1,
4x = b−a
57
Figure C.2: Trapezoidal example for n=1
Figure C.3: Trapezoidal example for n=2
∴
b�
a
f (x)dx = A≈ (b−a)2
( f (a)+ f (b))
since,
Now if n=2
4x =b−a
2
By trapezoidal rule,
58
∴
b�
a
f (x)dx=A≈ (b−a)4
( f (a)+2 f (a+4x)+ f (b))=4x2
( f (a)+ f (a+4x))+4x2
( f (a+4x)+ f (b))
(C.5)
As it can be seen from the above figure as we increase the number of steps, our error
goes on decreasing.
Sample MATLAB Code
h=(b-a)/n
x=a : h : b % where a and b are boundaries
y=0; % This is the Area A to be estimated under fx from a to b
for i=0:size(x,1)
if((i==0)||(i==nstep))
y= (1/3)*feval(fx,x(i+1)); % when i is either 0 or n
else
y= (2/3)*feval(fx,x(i+1)); % when i is intermediate points
end
end
Error in case of Trapezoid Method is given by,
Error =−(b−a)3
12n2 f ′′(ε) (C.6)
where ε is the number that exists between a and b.
As it can can be seen from the above derivation error is asymptotically proportional to
(b−a)3.
59
Figure C.4: Simpsons Method
C.3 Simpsons Method
In Simpsons Method, we will again divide up the interval into n sub-intervals. However,
unlike the previous two methods we need to require that n be even
4x =b−a
n
That because Simpson Rule uses the quadratic interpolation to approximate the func-
tion, Area under interval of [x0,x1]and [x1,x2] is given by
Shaded area in above figure represents the area A1
A1 =4x3
( f (x0)+4 f (x1)+ f (x2))
As a result, Area A (or� b
a f (x)dx) will be summation of all the areas , given by
∴
b�
a
f (x)dx = A =∑i
Ai =4x3
( f (x0)+4 f (x1)+2 f (x2)+ · · ·+4 f (xn−1)+ f (xn)) (C.7)
60
Sample MATLAB Code
h=(b-a)/n
x=a : h : b % where a and b are boundaries
y=0; % This is the Area A to be estimated under fx from a to b
for i=0:size(x,1)
if((i==0)||(i==nstep))
y= (1/3)*feval(fx,x(i+1)); % when i is odd
elseif(rem(i,2)==0) y= (4/3)*feval(fx,x(i+1)); % when i is even
else y= (2/3)*feval(fx,x(i+1));
end
end
Numerical Error in case of Simpson Method is given by,
Error =(b−a
2 )5
90
∣∣∣ f (4)(ε)∣∣∣ (C.8)
where ε is the number that exists between a and b.
As it can can be seen from the above derivation error is asymptotically proportional to
(b−a)5.
C.4 Discussion
Recall ETA factors calculation,
η =
� 1
0
δ fδX
.Γ.J−1(θ ,V )ds (C.9)
We can use both Trapezoid and Simpsons method as discussed in previous sections
methods to calculate the above integral.
Error observations in actual results between Trapezoidal and Simpsons showed clear
61
benefits of using Simpson methods for integrating state variables due to variables high
order nature.
62
Appendix D Data Points Fitting with Polynomials
In this section we will discuss theory behind curve fitting (polynomial calculation) through
the given sample data points.
D.1 Formulation
If we know n sample points of the given function f (x) given by (x1,y1),(x2,y2), . . . ,(xn,yn)
then there exists a unique polynomial of degree <= n−1
y = cn−1xn−1 + cn−2xn−2 + · · ·+ c1xn + c0 (D.1)
suhc that it will represent the a curve that will pass through all given sample points.
D.2 General Approach
Easiest approach ( but that require more computation amount) is to substitute value of x in
the polynomial and covert the resultant system into linear set of equation
y1 = cn−1xn−11 + cn−2xn−2
1 + · · ·+ c1x1 + c0
y2 = cn−1xn−12 + cn−2xn−2
2 + · · ·+ c1x2 + c0
...
y1 = cn−1xn−11 + cn−2xn−2
1 + · · ·+ c1x1 + c0 (D.2)
In matrix Form,
63
y1
y2
...
yn
=
xn1 xn−2
1 · · · 1
xn2 xn−2
2 · · · 1...
... . . . ...
xnn xn−2
n · · · 1
︸ ︷︷ ︸
Vandermonde
cn−1
cn−2
...
c0
(D.3)
∴
cn−1
cn−2
...
c0
=
xn1 xn−2
1 · · · 1
xn2 xn−2
2 · · · 1...
... . . . ...
xnn xn−2
n · · · 1
−1
y1
y2
...
yn
(D.4)
It can be observed Vandermonde matrix is dense in nature. In any program, to calcu-
late the inverse of VandermondeD.4 it has tocoverts the matrix into Gaussian upper/ lower
triangular form. Thus given matrix inverse requires series of manipulation to calculate the
inverse.
D.3 Newtons Method
Newton methods circumvents above problem of matrix manipulation to convert upper/
lower triangular form. Expression y is rewritten as follows,
y = c0 + c1(x− x1)+ c2(x− x1)(x− x2)+ · · ·+ cn−1(x− x1) · · ·(x− xn−1) (D.5)
where x1,x2, . . . ,xn are the data points
which coverts observed data points into following equations
f or x = x1; y1 = c0
64
f or x = x2; y2 = c0 + c1(x− x1)
f or x = xn; y2 = c0 + c1(x− x1)+ · · ·+ cn−1(x− x1) · · ·(x− xn−1)
Clearly above equation forms a lower triangle matrix,
y1
y2
...
yn
=
0 · · · 0 1
0 · · · (x− x1) 1... . . . ...
...
(x− x1) · · ·(x− xn−1) · · · (x− x1) 1
cn−1
...
c1
c0
(D.6)
cn−1
...
c1
c0
=
0 · · · 0 1
0 · · · (x− x1) 1... . . . ...
...
(x− x1) · · ·(x− xn−1) · · · (x− x1) 1
−1
y1
y2
...
yn
(D.7)
Note that resultant polynomial from equation D.7 and D.4 will result in the same poly-
nomial equation. It is clear from that inverse to calculate in Newton method will require
less computations compared to general approach.12
12Although Newton Approach has its advantages, general approach is used to calculate the polynomicalbecause of its easier formulation in MATLAB.
65
Appendix E Experimental Results
Due to space constraints, we will simulate results on IEEE standard 14 bus case and allocate
active and reactivepower transfer and loss shown below.
Figure E.1: IEEE Standard 14 Bus Case
E.1 Model Data
IEEE 14 Bus Case is shown in figure 9. In figure 9, branch arrangements are represented by
the lines that inter-connect each buses. Generators and loads are represented by the circles
and arrows at a given bus. As it can be seen generators are allocated at buses 1,2,3,6 and 8.
Instead of giving each bus and generator detail it would be apt the initial and final condition
after the transaction of the system. Transactions data is as shown below, Gen_Area and
Load_Areas are areas in which generator and load buses are situated.
ETA Calculation are carried out for nstep=3 which is significantly lower than original
ETA calculation.Simpsons integration is used to calculate the integral since they are more
accurate for quadratic functions.
66
From Gen To Load Gen_Area Load_Area Power Transfer(MW)6 14 1 2 126 9 1 2 88 2 2 2 10
Table 4: Transactions Data
Observe that all Errors are in the range of 10−9% (i.e. less than a watt) which is negli-
gible.
67
E.2
Res
ults
Pow
erL
oss
Allo
catio
nR
esul
tsFr
om_B
usTo
_Bus
Are
a_C
ode
Full_
Cas
e_PL
oss
Bas
e_C
ase_
PLos
sC
hang
e_in
_PL
oss
Tran
sact
ion_
1Tr
ansa
ctio
n_2
Tran
sact
ion_
31
21
0.13
2001
0.12
2932
0.00
9069
0.00
2074
0.00
1247
0.00
5747
15
10.
1344
780.
1625
50-0
.028
072
0.00
0535
-0.0
0089
3-0
.027
714
56
10.
0000
000.
0000
000.
0000
000.
0000
000.
0000
000.
0000
006
111
0.31
4352
0.17
1465
0.14
2887
0.09
4395
0.08
0887
-0.0
3239
56
121
0.28
8183
0.27
5279
0.01
2904
0.01
1143
0.00
2386
-0.0
0062
56
131
0.49
0526
0.33
8596
0.15
1930
0.12
2835
0.04
7538
-0.0
1844
212
131
0.18
0827
0.16
3920
0.01
6907
0.01
2650
0.00
7660
-0.0
0340
313
141
0.39
4709
0.14
3269
0.25
1440
0.19
9569
0.08
7325
-0.0
3545
4
Tabl
e5:
Pow
erL
oss
(MW
)Allo
catio
nfo
rAre
a1
From
_Bus
To_B
usA
rea_
Cod
eFu
ll_C
ase_
PLos
sB
ase_
Cas
e_PL
oss
Cha
nge_
in_P
Los
sTr
ansa
ctio
n_1
Tran
sact
ion_
2Tr
ansa
ctio
n_3
23
20.
1414
330.
1395
140.
0019
19-0
.000
411
-0.0
0030
30.
0026
332
42
0.08
6973
0.12
2781
-0.0
3580
80.
0086
310.
0064
08-0
.050
847
25
20.
0797
380.
1264
32-0
.046
694
-0.0
0242
9-0
.002
982
-0.0
4128
43
42
0.20
6599
0.23
1481
-0.0
2488
1-0
.003
687
-0.0
0284
5-0
.018
349
45
20.
0010
440.
0001
940.
0008
500.
0006
340.
0005
27-0
.000
310
47
20.
0000
000.
0000
000.
0000
000.
0000
000.
0000
000.
0000
004
92
0.00
0000
0.00
0000
0.00
0000
0.00
0000
0.00
0000
0.00
0000
78
20.
0000
000.
0000
000.
0000
000.
0000
000.
0000
000.
0000
007
92
0.00
0000
0.00
0000
0.00
0000
0.00
0000
0.00
0000
0.00
0000
910
20.
0427
280.
0247
110.
0180
180.
0119
810.
0105
96-0
.004
560
914
20.
0839
460.
0429
800.
0409
670.
0455
48-0
.007
112
0.00
2530
914
20.
0839
460.
0429
800.
0409
670.
0455
48-0
.007
112
0.00
2530
1011
20.
1908
440.
0904
830.
1003
610.
0663
510.
0570
65-0
.023
055
Tabl
e6:
Pow
erL
oss
(MW
)Allo
catio
nfo
rAre
a2
68
Pow
erFl
owA
lloca
tion
Res
ults
From
_Bus
To_B
usA
rea_
Cod
eFu
ll_C
ase_
Pflow
Bas
e_C
ase_
Pflow
Cha
nge_
in_P
flow
Tran
sact
ion_
1Tr
ansa
ctio
n_2
Tran
sact
ion_
31
21
17.9
9246
715
.297
386
2.69
5081
0.61
3344
0.36
9736
1.71
2002
15
113
.859
861
15.9
0218
0-2
.042
319
0.00
2024
-0.0
8934
4-1
.954
998
56
115
.971
249
24.1
7166
2-8
.200
413
-3.2
5610
0-2
.924
250
-2.0
2006
36
111
19.3
9031
414
.339
226
5.05
1088
3.33
5521
2.85
3226
-1.1
3765
96
121
8.21
8823
6.89
5593
1.32
3230
1.06
4374
0.42
6690
-0.1
6783
46
131
27.1
6211
221
.736
843
5.42
5269
4.34
4005
1.79
5834
-0.7
1457
012
131
1.83
0640
0.52
0315
1.31
0326
1.05
3231
0.42
4304
-0.1
6720
913
141
14.8
2139
98.
2546
416.
5667
585.
2617
512.
1649
40-0
.859
934
Tabl
e7:
Pow
erFl
ow(M
W)A
lloca
tion
forA
rea
1
From
_Bus
To_B
usA
rea_
Cod
eFu
ll_C
ase_
Pflow
Bas
e_C
ase_
Pflow
Cha
nge_
in_P
flow
Tran
sact
ion_
1Tr
ansa
ctio
n_2
Tran
sact
ion_
32
32
1.89
0420
3.13
3181
-1.2
4276
10.
2633
710.
1949
69-1
.701
101
24
212
.297
326
14.9
6785
1-2
.670
526
0.54
3812
0.40
2286
-3.6
1662
42
52
11.9
7272
115
.373
422
-3.4
0070
1-0
.195
913
-0.2
2876
7-2
.976
021
34
27.
5489
888.
7936
67-1
.244
680
0.26
3782
0.19
5273
-1.7
0373
44
52
-2.0
4607
20.
7852
37-2
.831
309
-3.0
6347
0-2
.609
487
2.84
1648
47
2-4
7.80
3400
-45.
9837
80-1
.819
620
2.51
5908
2.08
0459
-6.4
1598
74
92
21.6
0221
320
.805
800
0.79
6413
1.35
0213
1.12
3024
-1.6
7682
37
82
-70.
0000
00-6
0.00
0000
-10.
0000
000.
0000
000.
0000
00-1
0.00
0000
79
222
.196
600
14.0
1622
08.
1803
802.
5159
082.
0804
593.
5840
139
102
-6.3
4239
0-1
.552
568
-4.7
8982
2-3
.162
794
-2.7
0467
91.
0776
509
142
6.32
0601
3.43
7294
2.88
3308
3.51
4457
-1.0
4591
90.
4147
709
142
6.32
0601
3.43
7294
2.88
3308
3.51
4457
-1.0
4591
90.
4147
7010
112
-15.
3851
18-1
0.57
7278
-4.8
0784
0-3
.174
775
-2.7
1527
51.
0822
10
Tabl
e8:
Pow
erFl
ow(M
W)A
lloca
tion
forA
rea
2
69
Rea
ctiv
ePo
wer
Los
sA
lloca
tion
Res
ults
From
_Bus
To_B
usA
rea_
Cod
eFu
ll_C
ase_
QL
oss
Bas
e_C
ase_
QL
oss
Cha
nge_
in_Q
Los
sTr
ansa
ctio
n_1
Tran
sact
ion_
2Tr
ansa
ctio
n_3
12
1-5
.446
4435
-5.4
7411
740.
0276
739
0.00
6329
80.
0038
064
0.01
7537
71
51
-4.8
3641
28-4
.717
1335
-0.1
1927
930.
0028
453
-0.0
0342
41-0
.118
7006
56
11.
1433
289
1.82
2244
8-0
.678
9159
-0.2
8213
52-0
.251
2649
-0.1
4551
586
111
0.65
8153
40.
3589
930
0.29
9160
40.
1976
335
0.16
9350
9-0
.067
8239
612
10.
5998
141
0.57
2956
20.
0268
579
0.02
3192
60.
0049
656
-0.0
0130
046
131
0.96
5491
60.
6664
502
0.29
9041
40.
2417
729
0.09
3567
2-0
.036
2987
1213
10.
1636
369
0.14
8337
20.
0152
997
0.01
1447
40.
0069
321
-0.0
0307
9913
141
0.80
3737
50.
2917
362
0.51
2001
30.
4063
782
0.17
7817
9-0
.072
1948
Tabl
e9:
Rea
ctiv
ePo
wer
Los
s(M
VAR
)Allo
catio
nfo
rAre
a1
From
_Bus
To_B
usA
rea_
Cod
eFu
ll_C
ase_
QL
oss
Bas
e_C
ase_
QL
oss
Cha
nge_
in_Q
Los
sTr
ansa
ctio
n_1
Tran
sact
ion_
2Tr
ansa
ctio
n_3
23
2-4
.029
731
-4.0
3781
40.
0080
83-0
.001
732
-0.0
0127
70.
0110
922
42
-3.7
7142
8-3
.663
324
-0.1
0810
50.
0288
270.
0214
34-0
.158
366
25
2-3
.429
063
-3.2
8428
7-0
.144
776
-0.0
0697
1-0
.008
916
-0.1
2888
93
42
-3.0
8151
0-3
.018
518
-0.0
6299
2-0
.006
970
-0.0
0542
1-0
.050
601
45
2-1
.362
532
-1.3
6452
00.
0019
890.
0030
580.
0024
04-0
.003
474
47
24.
2900
413.
9669
420.
3230
99-0
.437
064
-0.3
6183
21.
1219
954
92
2.31
4539
2.12
9287
0.18
5252
0.29
5291
0.24
4789
-0.3
5482
87
82
8.10
2218
5.98
3005
2.11
9213
0.06
6978
0.05
1113
2.00
1122
79
20.
4853
720.
2016
770.
2836
950.
0836
840.
0694
110.
1306
009
102
0.11
3539
0.06
5662
0.04
7877
0.03
1837
0.02
8156
-0.0
1211
79
142
0.17
8593
0.09
1437
0.08
7155
0.09
6902
-0.0
1513
00.
0053
839
142
0.17
8593
0.09
1437
0.08
7155
0.09
6902
-0.0
1513
00.
0053
8310
112
0.44
7087
0.21
1974
0.23
5113
0.15
5439
0.13
3684
-0.0
5401
0
Tabl
e10
:Rea
ctiv
ePo
wer
Los
s(M
VAR
)Allo
catio
nfo
rAre
a2
70
Rea
ctiv
ePo
wer
Flow
Allo
catio
nR
esul
tsFr
om_B
usTo
_Bus
Are
a_C
ode
Full_
Cas
e_Pfl
owB
ase_
Cas
e_Pfl
owC
hang
e_in
_Pflo
wTr
ansa
ctio
n_1
Tran
sact
ion_
2Tr
ansa
ctio
n_3
12
118
.028
622
18.8
9648
4-0
.867
863
-0.1
9749
0-0
.119
056
-0.5
5131
71
51
6.60
1726
6.47
4151
0.12
7574
0.05
8459
0.04
3674
0.02
5441
56
117
.402
559
17.4
3066
6-0
.028
108
-0.1
9993
1-0
.150
073
0.32
1896
611
1-1
.690
937
-1.0
1422
0-0
.676
717
-0.4
6188
1-0
.459
117
0.24
4281
612
114
.174
395
14.4
5311
0-0
.278
715
-0.1
9934
6-0
.147
655
0.06
8287
613
110
.514
895
10.6
3465
4-0
.119
759
0.00
1029
-0.2
6504
20.
1442
5412
131
-9.1
0682
7-8
.836
095
-0.2
7073
2-0
.192
072
-0.1
4706
50.
0684
0513
141
-5.5
2106
1-4
.816
229
-0.7
0483
2-0
.444
263
-0.5
1260
70.
2520
38
Tabl
e11
:Rea
ctiv
ePo
wer
Flow
(MVA
R)A
lloca
tion
forA
rea
1
From
_Bus
To_B
usA
rea_
Cod
eFu
ll_C
ase_
Pflow
Bas
e_C
ase_
Pflow
Cha
nge_
in_P
flow
Tran
sact
ion_
1Tr
ansa
ctio
n_2
Tran
sact
ion_
32
32
15.6
3730
515
.338
037
0.29
9268
-0.0
6343
2-0
.046
955
0.40
9655
24
21.
4575
020.
5538
150.
9036
870.
2354
680.
1798
550.
4883
652
52
1.21
2446
0.56
8571
0.64
3875
0.13
4205
0.10
0711
0.40
8959
34
2-1
7.81
3698
-18.
3511
910.
5374
940.
3037
870.
2303
020.
0034
044
52
-2.7
1230
5-1
.645
229
-1.0
6707
6-0
.392
632
-0.3
0396
6-0
.370
478
47
21.
1083
87-0
.524
510
1.63
2897
0.25
4601
0.18
3817
1.19
4479
49
22.
3961
311.
3514
471.
0446
830.
6469
630.
5079
12-0
.110
192
78
2-1
5.63
0653
-14.
8564
78-0
.774
174
-0.9
4659
6-0
.722
833
0.89
5255
79
2-0
.975
899
-3.1
3741
82.
1615
191.
6797
061.
3001
31-0
.818
317
910
210
.509
716
9.25
0849
1.25
8867
0.84
6791
0.79
0308
-0.3
7823
29
142
5.84
0992
5.14
5420
0.69
5572
0.52
2223
0.33
0083
-0.1
5673
39
142
5.84
0992
5.14
5420
0.69
5572
0.52
2223
0.33
0083
-0.1
5673
310
112
4.59
6178
3.38
5187
1.21
0990
0.81
4954
0.76
2152
-0.3
6611
5
Tabl
e12
:Rea
ctiv
ePow
erFl
ow(M
VAR
)Allo
catio
nfo
rAre
a2
71