ONLINE APPENDIX TO
Does a Detailed Model of the Electricity Grid Matter?
Estimating the Impacts of the Regional Greenhouse Gas Initiative
Representative hours
Figures A1 and A2 show the relative frequency and load scaling for representative hour types
across the entire Eastern Interconnect. Note that regional variations exist here, too. For example,
as one travels north, the different because the Summer peak and the Winter peak becomes less
extreme, as the needs for cooling in the summer diminish while the needs for heating in the
winter increase. In the far north, such as parts of Canada and the Midwest, the power system is
actually winter-peaking. (In Florida, the same also holds true because the population peaks in
the winter.)
18%
16%
14%
Peak
12%
10%
High
8%
6%
Medium
4%
2%
0%
Summer
Winter
Fall+Spring
Figure A1: Relative Frequency of Representative Hour Types
Low
100%
80%
60%
40%
20%
0%
Summer
Winter
Fall+Spring
Figure A2: Scaling of the Load in Each Representative Hour Type
Load growth
Figure A3 shows the relative load growth for the NERC regions in the Eastern Interconnect,
which are based on historical trends from 1990-2010. The already populous regions in the
NPCC and RFC, which include most of the population along the Northeast Atlantic growth will
experience relatively little growth in load, while most of the load growth is expected to occur in
the southeastern United States.
1.3
1.25
1.2
1.15
1.1
1.05
1
NPCC
RFC
SERC
FRCC
SPP
MRO
Figure A3: Load Growth by NERC Region
Data on Generators
We calculated each unit’s CO2 emissions per MWh from its heat rate, using CO2 emission rates
per million British thermal units of 0.102 short tons for coal, 0.86 for oil and petroleum coke,
and 0.59 for natural gas and other gasses from fossil fuels. We assumed net CO2 emission rates
of zero for all other energy sources.
The capacity factor of new wind and solar generation facilities varies by region. A generator’s
“capacity factor” is the ratio of its average generation to its maximum generation capacity. Table
3 shows the capacity factors we assume.
Table A1 shows the assumed capacity factor for new and pre-existing wind and solar generation
units. Because wind and solar resources vary across the Eastern Interconnect, it is important to
model the capacity factor for new generation. The capacity factor is the average amount of
generation a new unit will produce. For example, a new 1 MW wind turbine in the Southwest
Power Pool will produce approximately 4,380 MWh of electricity over the course of a year (1
MW x 8,760 hours x 0.5) while a new 1 MW solar generator in the SPP will produce 1,752
MWh of electricity in the same time (1 MW x 8,760 hours x 0.2.)
Table A1: Assumed Capacity Factors for Wind and Solar Generators, by Region
Fuel Type
Assumed Capacity Factor
for New Windfarms
Assumed Capacity Factor for
New Solar Generators
Northeast Power
Coordinating Council
.333
0.160
ReliabilityFirst Corporation
(Mid-Atlantic to Chicago)
.333
0.160
SERC Reliability Corporation
(southeast)
.333
0.192
Florida Reliability
Coordinating Council
.167
0.200
Southwest Power Pool
.500
0.200
Midwest Reliability
Organization
.500
0.167
We convert the overnight cost reported by the EIA into a total construction cost as of the day of
first operation by assuming that an equal portion of the overnight cost is spent at the beginning of
each year of the project’s construction, and that the debt accrues interest at an annual rate of 8%.
We assume construction times of 8 years for nuclear, 5 years for coal, 4 years for gas, and 1 year
for wind and solar.
We assume that a power plant will be built if and only if it will pay back its investment in the
first 10 years of operation. Toward this end, we calculate a capital recovery factor for each
technology, using equation (1).
(1)
CRF: Capital Recovery Factor
i:
Interest Rate (8%)
n:
Compounding Periods (10 Years)
The capital recovery factor for ten years and an annual interest rate of 8% is 14.9%. So a plant is
built if it can cover 14.9% of the total construction cost in the representative year in which it is
built.
Table A2 summarizes information about the characteristics of new electric generators. The first
column is the amount of capital that a new unit needs to recover in the first year of operation in
order to justify its construction. This is approximately 14.9% of its total construction cost,
including the cost of capital. The second column is the total annual cost to keep a unit, either
new or existing, in service for another year. The third column is the total variable cost of a unit,
including an estimate of fuel costs for fossil fuel units. The fourth column is the total MW of
capacity that are allowed to be built in a ten year period, which is based on historical trends for
fossil fuel and nuclear units and projections of building capacity for renewable units. Finally, the
last column details the construction time required for each unit, which affects the total capital
cost.
Table A2 also shows the limits we assume on capacity built after 2012. Limits for coal, natural
gas combined-cycle, and nuclear were calculated from historical data. For emerging renewable
technologies—wind and solar—the capacity limit is such that each region could generate 23% of
its electricity from wind power and 12% from solar in 2022 and 2032. The last column of Table
2 shows the resulting limits.
Table A2: Information About New Generators
Fuel Type
Capital Recovery
Required in the
Representative
Year of Entry
($/MW)
Annual Fixed
Operating and
Maintenance
Costs ($/MW)
Total
Variable
Cost
($/MWh)
Total Possible
Cumulative
Capacity
Additions in EI**
Construction
Time
Coal (Dual Unit
Advanced)
$495,245
$35,255
$29.05
34 GW
5 Years
Natural Gas
(Advanced
Combined
Cycle)
$167,859
$20,661
$35.26 (at
fuel cost
of $5 per
mmBtu)
110 GW
4 Years
Natural Gas
(Advanced
Combustion
Turbine)
$107,173
$12,741
$58.62 (at
fuel cost
of $5 per
mmBtu)
110 GW
4 Years
Wind
$392,322*
$30,710
$0
249 GW (Year 10)
285 GW (Year 20)
1 Year
Nuclear
$1,054,899
$95,571
$2.04
20 GW
8 Years
Solar
$765,175*
$17,548
$0
250 GW (Year 10)
285 GW (Year 20)
1 Year
* Excludes the production tax credit for wind and the investment tax credit for solar.
Sources: (USEIA, 2010b); (USEIA, 2011b) Solar costs are reduced by 32% in year 10 to account for
relative technology improvement.
**Capacity additions are based on historical rates for coal, natural gas and nuclear units. Wind units are
expected to be able to provide up to 23% of capacity in year 10 and solar up to 12%.
To calculate the total annual fixed cost of each generator, we include assumed tax and insurance
payments in the annual fixed operating and maintenance costs specified in Table 2. The assumed
tax payments are $2585 per MW per year for coal-powered generators, $3041 per MW per year
for natural gas-fired generators, $3421 per MW per year for nuclear-powered generators, $1140
per MW per year for wind-powered generators, and $832 per MW per year for solar-powered
generators. The assumed insurance payments are $3000 per MW per year for coal-powered
generators, $3000 per MW per year for natural gas-fired generators, $3400 per MW per year for
nuclear-powered generators, $1500 per MW per year for wind-powered generators, and $5000
per MW per year for solar-powered generators. These insurance payments buy business
interruption and replacement-in-kind insurance.
In our scenarios, we vary the price of natural gas, as we will describe in section Error!
Reference source not found.. We left coal prices constant across all three representative years
because the projections in the 2011 Annual Energy Outlook (USEIA, 2011b) had them only 2%
lower in 2022 and only 2% higher in 2032 than in 2012. All other fuel prices remain constant at
the levels specified in the Energy Visuals data.
Regional system operators in the US require that each region have more generation capacity than
the expected peak quantity demanded, typically by approximately 10%. Our model imposes
such a requirement, so that in each modeled year the total generating capacity is at least 10%
more than the maximum quantity demanded.
Additional Description of Network Reduction or “Equivalencing” Method
The electric power system, is made up of generators, electrical nodes (or buses) at which power
is either generated or consumed, electrical loads, and branches (transmission lines, transformers,
etc.) which interconnect the electrical buses. The electric power network, which is comprised of
the buses and branches, is used to transport electric power; however it is qualitatively different
from other transportation networks in many respects. For example, branches have flow limits
based on the magnitudes of complex-valued numbers and generators have complex-valued
power-capability limits that must be enforced if accurate simulation of the electric power system
is to be achieved. Further, branch flows are governed nonlinearly by bus voltages rather than
power injections.
Fig. 3 One-line diagram of the EI network, Courtesy of Power World
The electric power grid in the United States is divided into three weakly connected grids, the
largest of which is the Eastern Interconnection (EI), shown in Fig. 3. The EI consists of
approximately 62,000 buses, 8,000 generators, 80,000 branches, and 37 High-Voltage DirectCurrent (HVDC) lines. The total generation and load in the system are approximately 668,000
MW and 651,000 MW, respectively, with power losses of 2.6 % (of the total generation). As
with all electric power system models of this extent, network features below (about) 69 kV are
suppressed and their effect is modeled, with only a negligible loss of accuracy, by appropriate
placement of the loads in the suppressed network.
Despite these simplifications, the 62,000-bus EI model, when embedded within an
optimization problem, is still too complex, leading to computation times that are unacceptable.
To reduce execution time, network reduction/simplification techniques are used.
Network Equivalencing Procedure
There are many methods for performing network reduction, each appropriate for different
objectives. For the objectives in this work, and for most network reduction needs, Ward-type
equivalents (and its variants) are used [1]-[5]. The major steps in this process are: 1) determine
the electrical buses and/or transmission lines of the (internal) network that need to be modeled
with fidelity; 2) use mathematical procedures to replace the external network with a system of
“equivalent” branches that approximates the effect of this less important external network. Fig. 4
is a Venn diagram showing the connectivity-relationship between the internal, external and
boundary systems.
E
E – External System
B
I – Internal System
B – Boundary Buses
I
Fig. 4. Venn Diagram of internal and external systems
For this work, the objective was to develop equivalent models of reduced size that accurately
reflected the flow of bulk power on the high voltage transmission lines as well as any lines
whose flow limitations were likely to be exceeded under the heavy-load conditions experienced
either today or in the future. In this work, two reduced network equivalent models were produced
to determine the effect of using more and less refined models in the optimization process.
In the less refined model, containing 300 buses and 1300 branches, the criteria for selecting
branches to retain was based on the likelihood of line congestion as reported in the National
Electric Transmission Congestion Study commissioned by DOE [6]-[7]. In the more refined
model, containing 5200 buses and 14,000 branches, all lines rated at 230 kV and above were
retained. Each of these reduced equivalent models was generated using the same approach which
follows.
Once the bus/line retention criteria were established, the next step was to determine a reduced
model of the remaining network that would approximately model the effect of this external
network. In theory, this process is simple and yields an exact equivalent of the external networks
in terms of branch and generator models; however an equivalent produce this way requires each
generator to be fragmented and a small portion of it modeled at many buses throughout the
system. In this process it is not uncommon to have a physical generator split into 100 smaller
nonphysical generators. This yields a model that is too computationally complex because the
number of optimization variables is correspondingly increased and the imposition of a limit on
the combined generation of these 100 small nonphysical generators (which matches the limit of
the corresponding physical generator) is similarly unwieldy. Correcting the problem requires a
four-step procedure.
In the first step, the Ward equivalencing method was applied to remove all the buses but those to
be retained and to calculate the equivalent branch models that accurately represented this
external network. This process resulted in the desired equivalent network model. To determine
an optimal way to locate generators without breaking them into small nonphysical generators, the
next two steps were needed.
In the first of these two steps, the Ward equivalencing process was conducted again but with a
new set of retained buses: all internal buses and all external buses at which generators were
located. When the Ward reduction process is used in this way, all generators remain whole since
the buses at which they are located are retained. This model is useful only as an intermediate step
in optimal generator placement.
In the next step, each external generator is moved integrally to an internal bus using a “shortest
electrical distance” metric. Electrical distance is a measure of strength of the connection between
any two points in a power system. If the connection is strong, the electrical distance is short; if
the connection is weak, the electrical distance is long. Using this metric, generators at external
buses were moved integrally to the internal bus to which they had the strongest electrical
connection.
Determining the shortest electrical distance between each external generator buses and its closest
internal bus became an important subproblem. Based on graph theory, the generator-moving
problem can be formulated as the shortest path problem [8]-[9], the objective of which is to find
a path between each generator bus and an internal bus such that the sum of the weights of
branches in the path is minimized. The weight on each branch i-j is defined as:
wi  j  ri2 j  xi2 j
(1)
where ri-j and xi-j are the resistance and reactance of branch i-j, respectively. The solution to the
shortest path problem was obtained using Dijkstra’s algorithm [8]-[9].
After generators were optimally moved, external loads (which one could think of as negative
generation,) also needed to be moved; however, unlike generators, loads do not have identities in
our model, so they may be moved, split and coalesced as needed to match a desired operating
point. In generating the reduced equivalents, loads were moved so that the line flows calculated
in the reduced models exactly matched the corresponding line flows in the 62,000 bus model.
The 300- and 5200-bus models produced by this process are shown in Fig. 5 and Fig. 6,
respectively.
Fig. 5 The 300-bus equivalent of EI
Fig. 6 The 5200-bus equivalent of EI
CORRESPONDENCE BETWEEN THE COLOR OF THE LINES AND THEIR VOLTAGE LEVELS
Line color
Voltage level (kV)
Red
345
Orange
500
Yellow
500 (retained TLs)
Green
735, 765
Purple
765 (retained TLs)
References
[1] J. B. Ward, “Equivalent Circuits for Power Flow Studies,” AIEE Trans. Power Appl. Syst., vol. 68, pp.
373–382, 1949.
[2] W. Snyder, “Load-Flow Equivalent Circuits–An Overview,” IEEE PES Winter Meeting, New York, Jan.
1972.
[3] F.F. Wu and A. Monticelli, “Critical Review of External Network Modeling for Online Security
Analysis,” Electrical power & energy systems, vol.5, no. 4, 1983.
[4] H. E. Brown, Solution of Large Networks by Matrix Methods. New York: John Wiley & Sons Inc, ISBN:
0-4718-0074-0, 1985.
[5] H. Duran and N. Arvanitidis, “Simplification for Area Security Analysis: A New Look at Equivalencing,”
IEEE Trans. Power App. Syst., vol. PAS-91, pp. 670-679, 1972.
[6] Department of Energy, National Electric Transmission Congestion Study, Aug. 2006, [Online].
Available:
http://nietc.anl.gov/documents/docs/Congestion_Study_2006-9MB.pdf.
[7] Department of Energy, National Electric Transmission Congestion Study, Dec. 2009, [Online].
Available:
http://energy.gov/sites/prod/files/Congestion_Study_2009.pdf.
[8] E. W. Dijkstra, "A Note on Two Problems in Connection with Graphs," Numerische Math, vol. 1, no.
1, pp. 269-271, 1959.
[9] B. V. Cherkassky, A. V. Andrew and T. Radzik, "Shortest Paths Algorithms: Theory and Experimental
Evaluation", Mathematical Programming, Ser. A 73(2), pp. 129-174, 1996.
Additional Results
Carbon Dioxide Emissions (in annual short tons)
1 Node - No Policy
Non RGGI
RGGI
Total
Year 0
1,602,629,337
109,274,886
1,711,904,223
Year 10
2,367,990,231
94,961,366
2,462,951,597
Year 20
2,642,679,113
114,869,381
2,757,548,494
1 Node - Policy*
Non RGGI
RGGI
Total
1,681,655,604
54,254,639
1,735,910,243
2,422,277,584
46,426,309
2,468,703,893
2,698,414,536
60,461,374
2,758,875,910
79,026,267
-55,020,247
54,287,354
-48,535,057
55,735,423
-54,408,007
144%
112%
102%
300 Node - No Policy
Non RGGI
RGGI
Total
1,643,835,229
80,873,100
1,724,708,329
2,349,552,389
93,475,259
2,443,027,648
2,655,852,913
104,760,955
2,760,613,868
300 Node - Policy
Non RGGI
RGGI
Total
1,680,114,219
55,439,574
1,735,553,794
2,387,708,289
57,964,776
2,445,673,065
2,699,182,364
55,006,769
2,754,189,133
36,278,990
-25,433,526
143%
38,155,900
-35,510,483
107%
43,329,451
-49,754,186
87%
5K Node - No Policy
Non RGGI
RGGI
Total
1,645,975,837
86,637,372
1,732,613,209
2,361,113,762
99,770,654
2,460,884,416
2,670,220,183
123,609,657
2,793,829,840
5K Node - Policy
Non RGGI
RGGI
Total
1,672,184,758
68,354,796
1,740,539,555
2,394,619,108
68,901,179
2,463,520,286
2,713,532,232
62,346,980
2,775,879,211
26,208,921
5k Node – Policy Effect
-18,282,576
Leakage Percent
143%
*$10 permit required for each short ton of CO2 emissions
33,505,346
-30,869,475
109%
43,312,049
-61,262,677
71%
1 Node – Policy Effect
300 Node – Policy Effect
Outside RGGI
In RGGI
Leakage
Percent**
Outside RGGI
In RGGI
Leakage Percent
Outside RGGI
In RGGI
**Leakage Percent =
𝐶𝑂2 𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 𝑂𝑢𝑡𝑠𝑖𝑑𝑒 𝑅𝐺𝐺𝐼
𝐶𝑂2 𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝐷𝑒𝑐𝑟𝑒𝑚𝑒𝑛𝑡 𝐼𝑛𝑠𝑖𝑑𝑒 𝑅𝐺𝐺𝐼
Average Wholesale Price of Electricity in RGGI Region (in $)
Model
1 Node
300 Node
Scenario
Policy*
No Policy
Policy Effect
Year 0
$29.39
$28.98
$0.41
Year 10
$40.37
$39.73
$0.64
Year 20
$63.89
$63.81
$0.08
Policy
No Policy
Policy Effect
$30.73
$28.28
$2.45
$43.35
$40.79
$2.56
$62.68
$61.83
$0.85
$46.16
$42.22
$3.94
$61.30
$59.83
$1.47
Policy
$32.62
No Policy
$28.44
Policy Effect
$4.18
*$10 permit required for each short ton of CO2 emissions
5,000 Node
Summary Statistics of the Four Models
Number of Buses
1 Node Model
300 Node Model
5K Node Model
60K Node Model
1
293
5,222
62,000
Number of Transmission
Lines
0
1,328
14,228
79,766
Number of Transmission
Constraints
0
216
8,596
49,730