Constrained Cost Optimization Under Uncertainty for
an Incompletely-Connected Electric Utility System
J. Andrew Howe & Chris Humphries
TransAtlantic Petroleum & UK National Grid
Presented at the
Palisade Risk Conference 2013
Frankfurt, Germany
Howe & Humphries
Utility Cost Optimization
June 2013
1 / 38
Presentation Outline
Introduction
Modeling Power Utilities
Our Problem
Constrained Optimization Model
The LP Formulation
The psuedo-LP Formulation
Adding in Uncertainty
Modeling with Palisade’s Tools
@Risk, Evolver, and VBA
Optimization Efficiency
Model Results
What We Now Know
Investment Decision Making
Concluding Remarks
Howe & Humphries
Utility Cost Optimization
June 2013
2 / 38
Introduction
I
I
I
I
Modeling Power Utilities
Like any other business, electric utilities have to operate under cost
constraints - even public utilities that don’t need to make a profit.
In fact, they can be more constrained. If your local supermarket runs
out of your favorite food, you’re unhappy. If your power company
runs out of power, it’s an entirely different thing.
When power prices rise dramatically (California, USA, 1990’s), all hell
breaks loose.
Furthermore, power utilities are constrained by more than just costs:
I
I
I
If a region’s winter is especially warm & dry, there will be little runoff
for dams the ensuing spring. If the utility relies on hydrological power
for cheap baseload, this will raise costs.
If a region’s populace turns against gas transport pipelines, gas-burning
CC / CT peaking units (basically a jet engine mounted on the ground)
may have to be left off. The utility may have to purchase more
off-network electricity at much higher prices - if even possible.
If enough people are swayed by the arguments against nuclear power,
clean-burning nuclear generators must be shut down. The utility will
then have to burn more coal or gas - at higher costs (and ultimately
more pollution).
Howe & Humphries
Utility Cost Optimization
June 2013
3 / 38
Introduction
next: our problem
I
In addition to usual operating costs, utilities often have further
constraints on investments.
I
If a utility’s region expands outward dramatically, it can be much
cheaper to extend high-voltage lines to the new outlying regions,
instead of building new capacity.
I
However, this requires extra wayleaves that landowners may fight.
I
Furthermore, expansion by adding new power plants can face
regulatory and environmental challenges, as well as the costs.
I
This bring us to our specific modeling problem.
Howe & Humphries
Utility Cost Optimization
June 2013
4 / 38
Introduction
Our Problem
I
In this case study, we demonstrate how Palisade’s @Risk and Evolver
products can work together to perform cost optimization and inform
investment decision modeling for a system of incompletely-connected
power grids.
I
The modeled utility system in the UK is composed of four notional
power-generating regions. Each region has its own power demand and
portfolio of types of power generating plants.
I
Because of the variation in generating assets, the generating cost
profile varies dramatically by region.
Power transmission lines connect the four regions subject to the
constraints that
I
a) some regions are not connected at all, and
b) the amount of electricity that is allowed to flow over each boundary is
limited.
I
Furthermore, power plant availability and total generating capacity
vary stochastically, as a function of many factors.
Howe & Humphries
Utility Cost Optimization
June 2013
5 / 38
Introduction
next: the LP formulation
I
We mathematically define an optimization problem that allows us to
meet the demand of all regions under the transmission constraints
while minimizing the total cost.
I
We develop this optimization problem in three stages.
I
It is first defined as a linear programming optimization, then extended
to be psuedo-linear, then finally nonlinear.
I
This is then implemented under the framework of uncertain
generation capacity so we can make probabilistic statements about
the costs and other relevant quantities.
I
The results from the final model can be used to guide and inform
investment decisions for demand reduction and capacity /
transmission expansion.
Howe & Humphries
Utility Cost Optimization
June 2013
6 / 38
Constrained Optimization Model
The LP Formulation
Power flow boundaries between notional regions North Ireland (N), Scotland (S),
England & Wales (E ), and Republic of Ireland (R)
Howe & Humphries
Utility Cost Optimization
June 2013
7 / 38
Constrained Optimization Model
I
In formulating our problem, first we define the region-specific
variables:
I
I
I
I
The LP Formulation
Unit Cost per region in £/MW: CN , CS , CE , CR ;
Generating capacity per region (MW): MN , MS , ME , MR ;
Peak power demand per region (MW): DN , DS , DE , DR .
Next we define the maximum flow constraints in Megawatts between
all pairs of regions (even those not physically connected get a
variable):
I
I
I
I
I
I
North Ireland ↔ Scotland: MNSN ,
North Ireland ↔ England & Wales: MNEN ,
North Ireland ↔ Republic of Ireland: MNRN ,
Scotland ↔ England & Wales: MSES ,
Scotland ↔ Republic of Ireland: MSRS ,
England & Wales ↔ Republic of Ireland: MERE .
Howe & Humphries
Utility Cost Optimization
June 2013
8 / 38
Constrained Optimization Model
I
The LP Formulation
Finally, we define variables holding the total amount of power
generated per region:
I
I
I
I
GN = GNN + GNS + GNE + GNR ,
GS = GSN + GSS + GSE + GSR ,
GE = GEN + GES + GEE + GER ,
GR = GRN + GRS + GRE + GRR ,
I
The first subscript indicates the generation region and the second
indicates the demand region.
I
For example, GSE is the power generated by the Scotland region that
flows into England & Wales; GES is the exact opposite.
I
Our LP optimization problem can then be formulated to minimize the
total cost, subject to the inter-region flow constraints, by varying the
G∗∗ variables.
I
This is shown on the next slide.
Howe & Humphries
Utility Cost Optimization
June 2013
9 / 38
Constrained Optimization Model
The LP Formulation
minimize CN GN + CS GS + CE GE + CR GR
subject to G∗∗ ≥ 0


GNS , GSN ≤ MNSN GNE , GEN ≤ MNEN
flow limits GNR , GRN ≤ MNRN GSE , GES ≤ MSES


GSR , GRS ≤ MSRS GER , GRE ≤ MERE
(
0 ≤ GN ≤ MN 0 ≤ GS ≤ MS
generation up to capacity
0 ≤ GE ≤ ME 0 ≤ GR ≤ MR


GNN + GSN + GEN + GRN = DN



G + G + G + G = D
NS
SS
ES
RS
S
demand is met

GNE + GSE + GEE + GRE = DE



G + G + G + G = D
NR
SR
ER
RR
R
Howe & Humphries
Utility Cost Optimization
June 2013
10 / 38
Constrained Optimization Model
next: the psuedo-LP formulation
I
In the first constraint, G∗∗ indicates all power generation quantities
previously mentioned. This constraint forces power generation to be
positive
I
The six constraints starting in the first block limit the amount of
power that can flow over all possible boundaries.
I
To prevent power flowing between two disjoint regions, the RHS here
just needs to be set to 0 (such as North Ireland and England &
Wales, MNEN = 0).
I
The next block of four constraints force the total power generated in
each region to not exceed the capacity.
I
The last four constraints guarantee all region’s power demands are
met
I
This works, and can be solved extremely easily with the Solver add-in
to Microsoft Excel, but has one major shortcoming.
Howe & Humphries
Utility Cost Optimization
June 2013
11 / 38
Constrained Optimization Model
The psuedo-LP Formulation
I
The major drawback of this optimization problem is that it will allow
power to flow bidirectionally between two adjacent regions.
I
For example, maybe Scotland will generate some power to meet the
demand in England & Wales, while England & Wales sends power to
Scotland.
I
The resulting minimum cost is the same as if the two flows simply
netted out, but the solution requires some manual adjustment to be
realistic.
I
To solve this, we instead formulate a psuedo-LP problem by adding
six binary constraints
I
Define
GNSN =
(
1 if GNS > 0 and GSN > 0
.
0 otherwise
The quantities GNEN , GNRN , GSES , GSRS , and GERE are similarly
defined.
Howe & Humphries
Utility Cost Optimization
June 2013
12 / 38
Constrained Optimization Model
I
next: adding in uncertainty
We can then optimize the psuedo-LP problem shown here
minimize CN GN + CS GS + CE GE + CR GR
subject to G∗∗ ≥ 0


GNS , GSN ≤ MNSN GNE , GEN ≤ MNEN
flow limits GNR , GRN ≤ MNRN GSE , GES ≤ MSES


GSR , GRS ≤ MSRS GER , GRE ≤ MERE
(
0 ≤ GN ≤ MN 0 ≤ GS ≤ MS
generation up to capacity
0 ≤ GE ≤ ME 0 ≤ GR ≤ MR


GNN + GSN + GEN + GRN = DN



G + G + G + G = D
NS
SS
ES
RS
S
demand is met

GNE + GSE + GEE + GRE = DE



G + G + G + G = D
NR
SR
ER
RR
R
GNSN , GNEN , GNRN , GSES , GSRS , GERE = 0
Howe & Humphries
Utility Cost Optimization
June 2013
13 / 38
Constrained Optimization Model
I
I
I
I
Adding in Uncertainty
But we aren’t quite done yet - we must consider the uncertainty in
generation capacity and cost.
For a power plant, the Nameplate Capacity is the maximum amount
of power it can generate safely under optimal conditions. A utility
system’s Total Nameplate Capacity would be the sum of the
Nameplate Capacities for all the plants.
The maximum power a plant can generate at any given time will
probably be less than its Nameplate Capacity; the same can be said
for the total utility system.
A utility system’s total generating capacity will generally vary from
this theoretical maximum for reasons such as:
I
I
I
I
planned outages (for maintenance) and / or unplanned outages
(something breaks)
if gas prices rise too high, gas-burning combustion turbine units may
not be run
if lower quality coal is used, a coal-burning plant may not be able to
run at full power
dam spillways maybe be partially closed during spawning season in
spring to increase viability of fish roe
Howe & Humphries
Utility Cost Optimization
June 2013
14 / 38
Constrained Optimization Model
I
I
I
I
Adding in Uncertainty
But we aren’t quite done yet - we must consider the uncertainty in
generation capacity and cost.
For a power plant, the Nameplate Capacity is the maximum amount
of power it can generate safely under optimal conditions. A utility
system’s Total Nameplate Capacity would be the sum of the
Nameplate Capacities for all the plants.
The maximum power a plant can generate at any given time will
probably be less than its Nameplate Capacity; the same can be said
for the total utility system.
A utility system’s total generating capacity will generally vary from
this theoretical maximum for reasons such as:
I
I
I
I
planned outages (for maintenance) and / or unplanned outages
(something breaks)
if gas prices rise too high, gas-burning combustion turbine units may
not be run
if lower quality coal is used, a coal-burning plant may not be able to
run at full power
dam spillways maybe be partially closed during spawning season in
spring to increase viability of fish roe (damn fish and dam spillways!)
Howe & Humphries
Utility Cost Optimization
June 2013
14 / 38
Constrained Optimization Model
Adding in Uncertainty
I
The cost to generate one Megawatt of power from a power plant is
largely, but not solely, a function fuel prices.
I
A very generic generating cost hierarchy for some major plant types
in the UK would be
Hydrological ≤ Nuclear < Gas < Coal < Oil < Diesel
I
In other places, such as the US, coal and gas may trade places.
I
If generating capacity from power plants that are lower in the
hierarchy is curtailed or completely unavailable, the total cost will be
increased.
I
Hence, since generating capacity (M∗ ) will vary from the Nameplate
Capacity at any given period, so will unit cost (C∗ ).
I
In this case study, we have several types of power plants, with
capacity modeled as shown on the next slide.1
1
Plant running costs and availability data is for illustrative purposes only,
and does not represent information for any particular generating plant.
Howe & Humphries
Utility Cost Optimization
June 2013
15 / 38
Constrained Optimization Model
Adding in Uncertainty
Stochastic modeling of plant capacity; N = Nameplate (MW), U = # plants.
Type
Offshore Wind
Onshore Wind
Wave & Tidal
Hydrological
Nuclear
France*
Imera/BritNed*
Base Gas
Base Coal
Biomass
Marginal Gas
Marginal Coal
Pumped Storage
Oil
OCGT
Howe & Humphries
Cost (£/MW)
−0.5
−0.3
−0.7
−0.1
0
0
0
40
50
55
70
70
120
200
300
Capacity
Discrete(Uniform)×N
Discrete(Uniform)×N
Binom(n = U, p = 0.20) × 30
Gauss(µ = 0.6, σ = 0.04) × N
Binom(n = U, p = 0.70) × 600
Binom(n = 4, p = 0.50) × 500
Binom(n = 9, p = 0.50) × 500
Binom(n = U, p = 0.80) × 500
Binom(n = U, p = 0.80) × 500
Binom(n = U, p = 0.85) × 100
Binom(n = U, p = 0.80) × 500
Binom(n = U, p = 0.80) × 500
Binom(n = U, p = 0.90) × 300
Binom(n = U, p = 0.80) × 500
TGauss(µ = 0.95, σ = 0.03) × 677
Utility Cost Optimization
June 2013
16 / 38
Constrained Optimization Model
I
I
I
I
I
I
Adding in Uncertainty
Some plant types have slightly negative costs. These are designed to
run either intermittently (Wind) or predictably (Marine / Hydro), and
contractual obligations and fixed costs are structured accordingly.
Though fuel costs are zero, there are commercial advantages for these
plant to always run (hence the negative cost).
The two starred sources indicate submerged power cables coming
from the European continent into the England & Wales region.
Note the cost difference between Base Gas / Coal and Marginal Gas
/ Coal. Perhaps the marginal units are older or thermally less
efficient; hence they should be used less.
OCGT is a group of auxillary peaking diesel generators used only
when absolutely necessary. Note that the OCGT capacity is modeled
using a Truncated Normal distribution.
The availability rates vary seasonally; what we used was estimated by
SMEs as availability rates for the winter.
This table, with the simulated generating capacities, forms a Merit
Order.
Howe & Humphries
Utility Cost Optimization
June 2013
17 / 38
Constrained Optimization Model
I
I
I
Adding in Uncertainty
The Merit Order for each of the four generating regions is specific,
with the number of each plants varying by region.
For example, North Ireland has only 3 gas plants and no coal-burning
generators; England & Wales has 66 and 28, respectively.
Each time we simulate the complete Merit Order for a region, we can
sum the capacities and costs, then compute the average unit cost.
For example:
Example England & Wales partial Merit Order
Type
Nuclear
Base Gas
Base Coal
Biomass
Pumped Storage
Total
Average
Howe & Humphries
Unit Cost
0
40
50
55
120
Capacity
6, 000
17, 500
5, 500
1, 100
2, 100
ME = 32, 200
Total Cost
0
700, 000
275, 000
60, 500
252, 000
1, 287, 500
CE = 39.89
Utility Cost Optimization
June 2013
18 / 38
Constrained Optimization Model
I
I
I
I
I
I
Adding in Uncertainty
Hence we come back to our psuedo-linear optimization problem,
which we must now call nonlinear.
We allow each region’s Merit Order to be dynamically generated, in
that the optimization routine can turn on / turn off individual plant
types in each region independently.
Thus, the generating capacity M∗ and unit cost C∗ in each region can
be varied.
To extend our example from the table on the previous slide, if the
2, 100 MW of expensive Pumped Storage are turned off, the
generating capacity drops to 30, 100 MW and the unit cost to 34.40
£/MW.
While this allows expensive power generators to be turned off to
minimize the costs, this must be balanced against meeting all
demand.
Our final nonlinear optimization problem is shown on the next slide,
where M∗d and C∗d indicate that generating capacities and unit costs
are dynamic.
Howe & Humphries
Utility Cost Optimization
June 2013
19 / 38
Constrained Optimization Model
next: modeling with Palisade’s tools
minimize CNd GN + CSd GS + CEd GE + CRd GR
subject to G∗∗ ≥ 0


GNS , GSN ≤ MNSN GNE , GEN ≤ MNEN
flow limits GNR , GRN ≤ MNRN GSE , GES ≤ MSES


GSR , GRS ≤ MSRS GER , GRE ≤ MERE
(
0 ≤ GN ≤ MNd 0 ≤ GS ≤ MSd
generation up to capacity
0 ≤ GE ≤ MEd 0 ≤ GR ≤ MRd


GNN + GSN + GEN + GRN = DN



G + G + G + G = D
NS
SS
ES
RS
S
demand is met
GNE + GSE + GEE + GRE = DE



G + G + G + G = D
NR
SR
ER
RR
R
GNSN , GNEN , GNRN , GSES , GSRS , GERE = 0
Howe & Humphries
Utility Cost Optimization
June 2013
20 / 38
Modeling with Palisade’s Tools
@Risk, Evolver, and VBA
I
We implement all this in a Microsoft Excel spreadsheet model, using
Palisade’s Decision Tools Suite (6.0).
I
@Risk is used for the stochastic modeling and analysis.
I
Evolver is used for the constrained optimization.
I
Let’s see these worksheets.
Howe & Humphries
Utility Cost Optimization
June 2013
21 / 38
Modeling with Palisade’s Tools
@Risk, Evolver, and VBA
I
We implement all this in a Microsoft Excel spreadsheet model, using
Palisade’s Decision Tools Suite (6.0).
I
@Risk is used for the stochastic modeling and analysis.
I
Evolver is used for the constrained optimization.
I
Let’s see these worksheets.
I
Challenge: Evolver requires static inputs and RiskOptimizer requires
static constraints
Howe & Humphries
Utility Cost Optimization
June 2013
21 / 38
Modeling with Palisade’s Tools
@Risk, Evolver, and VBA
I
We implement all this in a Microsoft Excel spreadsheet model, using
Palisade’s Decision Tools Suite (6.0).
I
@Risk is used for the stochastic modeling and analysis.
I
Evolver is used for the constrained optimization.
I
Let’s see these worksheets.
I
Challenge: Evolver requires static inputs and RiskOptimizer requires
static constraints
I
Solution: Create a hybrid solution with VBA to bridge between @Risk
and Evolver
Howe & Humphries
Utility Cost Optimization
June 2013
21 / 38
Modeling with Palisade’s Tools
I
next: optimization efficiency
The modeling algorithm we designed with @Risk is:
i) At the beginning of the simulation, initialize the Evolver model and set
convergence criteria
ii) Simulate generation capacities for the Merit Order on the generation
worksheet
iii) Copy Merit Order to model worksheet
iv) Execute the Evolver model to optimize the total system cost
v) Record results and go back to Step ii)
I
The @Risk model measures the empirical distributions of several
optimized quantities:
I
I
I
I
I
I
It also records information about the optimization model:
I
I
I
I
Regional Generating Capacity (M∗d ) P
Total System Generating Capacity ( ∗ M∗d )
d
Regional Unit Cost (C
P∗ ) d
Total System Cost ( ∗ C∗ G∗ )
Power Flow Across Boundaries (G∗∗ )
Convergence Failure
Total Number Trials & Efficiency
Time Required
Let’s see a short demonstration.
Howe & Humphries
Utility Cost Optimization
June 2013
22 / 38
Modeling with Palisade’s Tools
Optimization Efficiency
I
Evolver has two optimization engines to choose from: the Genetic
Algorithm and the OptQuest engine.
I
Manually selecting the GA led to very poor results, so we let Evolver
pick OptQuest.
I
Actually, we run Evolver twice, to let it better search the solution
space: the first run uses a maximum of 400 trials, while the second
goes for a maximum of 600: 1, 000 in total.
I
With every trial, Evolver compares the current best to the best from
200 trials previous. If the improvement in Total System Cost is less
than 0.001, it stops.
I
Running the @Risk model for 1, 000 simulations takes less than 3
hours.
I
This could be decreased by further tuning of the Evolver parameters.
I
The next two slides show the empirical distributions of optimization
efficiency measures.
Howe & Humphries
Utility Cost Optimization
June 2013
23 / 38
Modeling with Palisade’s Tools
Howe & Humphries
Optimization Efficiency
Utility Cost Optimization
June 2013
24 / 38
Modeling with Palisade’s Tools
Howe & Humphries
next: what we now know
Utility Cost Optimization
June 2013
25 / 38
Model Results
I
I
What We Now Know
Mean total system generating capacity is 65, 830MW
90% confidence interval (64, 141, 69, 180)
Howe & Humphries
Utility Cost Optimization
June 2013
26 / 38
Model Results
I
I
What We Now Know
Mean total system cost is £2, 290, 786
90% confidence interval (1, 734, 205, 2, 771, 690)
Howe & Humphries
Utility Cost Optimization
June 2013
27 / 38
Model Results
What We Now Know
I
Mean unit cost (per MW) = Scotland: £8.74, Republic of Ireland:
£37.93, England & Wales: £38.89, North Ireland: £70 .82 .
I
The next slide has the the empirical distribution of power flows into
North Ireland from Scotland.
Howe & Humphries
Utility Cost Optimization
June 2013
28 / 38
Model Results
I
next: investment decision making
In nearly half the simulations, power flow from Scotland into North
Ireland was limited by the constraint. A similar impact of the
constraint between Republic of Ireland and North Ireland was not
observed.
Howe & Humphries
Utility Cost Optimization
June 2013
29 / 38
Model Results
Investment Decision Making
I
North Ireland’s generating capacity is relatively low; it has an
approximately 65% chance of being less than demand.
I
Because of the high cost and this low capacity, North Ireland
generates power for other regions infrequently and only in small
amounts, so we can focus on the Scotland ↔ North Ireland link.
I
We test the scenario where we increase transmission capacity from
Scotland to North Ireland to 2, 000MW.
I
Can we lower total system cost by investing in transmissiong capacity
to allow more cheaper power to flow into North Ireland?
Howe & Humphries
Utility Cost Optimization
June 2013
30 / 38
Model Results
Howe & Humphries
Investment Decision Making
Utility Cost Optimization
June 2013
31 / 38
Model Results
I
Investment Decision Making
In the base model, North Ireland supplied 1, 523MW of its own
demand. With the transmission constraints relaxed, this became
637MW.
Howe & Humphries
Utility Cost Optimization
June 2013
31 / 38
Model Results
I
Investment Decision Making
Here we see the impact of relaxing the flow constraint; in nearly 90%
of the simulations more power than the original constraint flowed
across this boundary.
Howe & Humphries
Utility Cost Optimization
June 2013
32 / 38
Model Results
I
Investment Decision Making
With transmission constraints relaxed, the mean cost dropped by 2%
to £2, 245, 495. The 5th percentile decreased by about 4%, and the
95th by 2%.
Howe & Humphries
Utility Cost Optimization
June 2013
33 / 38
Model Results
Investment Decision Making
I
UK National Grid has been proactive in contracting for and installing
alternative power plants, including Wind, Tidal, and Nuclear.
I
Offshore wind and tidal, specifically, still have much room for growth.
I
What is the effect on total system cost of investing in an
approximately 50% increase in offshore wind and tidal power?
Increase of Offshore wind and tidal generating capacity (MW).
Region
North Ireland
Scotland
England & Wales
Republic of Ireland
Total
Howe & Humphries
Offshore Wind
300
450
3, 185
4, 778
13, 375 20, 063
325
488
17, 185 25, 779
Wave & Tidal
2 → 60MW
3 → 90MW
19 → 570MW 29 → 870MW
3 → 90MW
4 → 120MW
3 → 90MW
4 → 120MW
27 → 810
40 → 1, 200
Utility Cost Optimization
June 2013
34 / 38
Model Results
I
Investment Decision Making
The addition of almost 9, 000MW of alternate power decreased the
mean total system cost by 7.7% while the variability jumped up by
25%; the 5th percentile dropped by 20%.
Howe & Humphries
Utility Cost Optimization
June 2013
35 / 38
Model Results
Investment Decision Making
I
Many utilities have been recently investing in programs to decrease
demand - especially peak demand.
I
This includes rebates for energy efficient smart appliances, education
programs, interruptibility/curtailage agreements, and the Smart Grid.
I
We assume that if overall peak demand is reduced, all regions will be
affected the same, so the relative proportions remain unchanged.
I
What is the effect on total system cost of investing in consumer
programs that lead to a modest decrease of 6.25% in peak demand to
60, 000MW?
Howe & Humphries
Utility Cost Optimization
June 2013
36 / 38
Model Results
I
next: concluding remarks
Decreasing demand by 6.25% dropped the total system cost by 12%
to £2, 019, 410. Variability dropped by 7%, and the 5th and 95th
percentiles dropped by 14 and 11 percent.
Howe & Humphries
Utility Cost Optimization
June 2013
37 / 38
Concluding Remarks
I
I
I
I
I
I
In this case study, we have demonstrates how Palisade’s @Risk and
Evolver products can work together to perform cost optimization for
a system of incompletely-connected power grids.
We began by mathematically defining an optimization problem that
allows us to meet the aggregate demand of all regions under the
transmission constraints while minimizing the total cost.
This was then implemented under the framework of uncertain
generation capacity, so the optimization can take into consideration
the fact that capacity is a stochastic quantity.
The resulting Monte Carlo simulation model allowed us to make
probabilistic statements about regional generating capacity and cost,
total system generating capacity and cost, and inter-regional power
flows.
Finally, we demonstrated how this optimization model can be used to
guide and inform investment decisions.
The results about the impact on power generation costs could be
used as inputs to an investment model which would consider the
capital, contractual, regulatory, and other costs.
Howe & Humphries
Utility Cost Optimization
June 2013
38 / 38