HYENA (HYdro ENergy simulAtor) Some Issues in a Simulation Model of a Hydro Thermal Power System
HYENA (HYdro ENergy simulAtor) Some Issues in a Simulation Model of a Hydro Thermal Power System. Skuli Johannsson, Annad veldi ehf, Reykjavik, Iceland skuli@veldi.is and Elias B Eliasson The National Power Company, Iceland , elias@lv.is 01-April-2005 1. Introduction This report is a continuation of the report: “Simulation Model of Transmission Constrained Hydrothermal Power System” Skuli Johannsson, Annad veldi ehf, Reykjavik, Iceland and Elias B Eliasson, The National Power Company, Iceland, 01-August-2003. Revised 09-Dec-2003. The report can be found at http://www.veldi.is/ HYENA is based on a mathematical iteration process as follows: Read and prepare model data for Power system, Water series, Load demand etc for all operational years and water years. Start Iteration o Strategic part. Simplified description of System. Redefine load in water areas with information from last Iteration as: Demand + Losses – Transmission into Water Area + Transmission from Water Area + Selling activity on the Spot Market – Buying activity on the Spot Market. Calculation of water values for water areas taking into account local reserve power in water area. o Tactical part. Simulation of the Power System base on calculated water values for water areas, detailed description of the Production System, Transmission System and Load Demand in Substations. o Analyse results. If convergence then leave Repeat Print out Results End
The convergenge of the
iteration process is very
efficient so usually three
iterations give satis-
factory result. The chart
shows typical five 1200
Price feasibility
iterations, first four using 1000
LP and the last using
LP/NLP. NLP gives
better result in this case. 400
Figure 1 HYENA LP/NLP Iterations LP/NLP
LP
LP
LP
LP
The iteration process
used in HYENA does
not calculate exact
Ite ration
optimum solution which
would take exponential time and computer resources. Instead the process finds
an approximate solution trading optimality for efficiency. Such solutions, often
loosely called heuristic methods or heuristics, seek to obtain good near-optimal
solutions at relatively low computational costs sacrificing guarantee of optimality.
The mathematical method used in HYENA, that iteratively improves the solution
towards optimality, is in the literature referred to as metaheuristic. Example of this
kind of methodology is the emerging technology of Ant Colony Optimization and
Genetic Algorithms. From Mathematical Programming Glossary: “This is a
general framework for heuristics in solving hard problems. The idea of ‘meta’ is
that of level.”
The biggest disadvantage of this approach is in sensitivity analysis but by tuning the HYENA model properly it has proven not to be a major problem. All kinds of sensitivity analysis can be performed with satisfactory and stable results.
As an advantage with HYENA one may argue that simulations are considered closer to reality than optimization with perfect foresight.
2. Hyena model Options Following are the most important options available in HYENA and they can be mixed in any combination. Optimization technique Linear Programming LP Mixed Integer Linear Programming NLP Nonlinear Programming Mixed Integer Nonlinear Programming Constraints Without Transmission Lines Without Transmission/FlowGates Constraints With Transmission Constraints in Lines With Transfer constraints in FlowGates Losses in transmission lines o Nonlinear or Linear Losses Time Unit Week or Day If NLP is used say for 5 iterations then in the first four iterations LP is used and in the last iteration the model first calculates an approximate solution before starting th NLP calculation. LP is 5-15 times faster than NLP depending on size of the model. Using the model for say 1 operational year, 51 water years, weekly time units 3 iterations, Linear Programming, and 30-40 hydro units takes ca 1 minute. The chart shows typical results from HYENA for a small reservoir in a spotmarket environment. The figure shows swarm of reservoir curves of 50 water years with day as a time unit and using NLP. The thick water value curves give indication on when to buy or sell on the spot market. Between the curves there is neither buying nor selling, the system is in a waiting state.
Figure 2 Figure 3 on the other hand shows a swarm of reservoir curves for a large reservoir in a non-spotmarket environment, 51 water years and 5 operational years with a time unit of 1 day and using NLP. The reservoir is assumed starting operation in 01.09.2006. Figure 3 Detailed information as in figure 3 gives important guidelines on optimal timing, filling and subsequent operation of reservoirs in hydro power systems.
3. Operational years and water years There are two setups in HYENA, the static setup and the dynamic setup
In the Static setup we have 1 operational year i.e. 2010 and say 51 water years say 1950-2000. We simulate over the 51 water year with the same demand (2010) every year. We start with specific reservoir content in the beginning of the first water year and the reservoir content in the end of one water year is the reservoir content in the beginning of next water year.
In the Dynamic setup we have >1 operational years i.e. 2004-2014. The Power System is simulated by the following schema:
Table 1
Dynamic setup of Operational years and Water years
No 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
1 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960
2 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961
3 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962
4 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963
48 1997 1998 1999 2000 1950 1951 1952 1953 1954 1955 1956
49 1998 1999 2000 1950 1951 1952 1953 1954 1955 1956 1957
50 1999 2000 1950 1951 1952 1953 1954 1955 1956 1957 1958
51 2000 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959
We start with specific reservoir content in the beginning of the first operational year 2004 and start with water year 1950. The reservoir content in the end of the year is the reservoir content in the beginning of operational year 2005 and water year 1951. We move like that horizontally until we come to the end of the row. Next we begin in second row with operational year 2004 and start with water year 1951 and continue horizontally throughout the rest of the schema. Water years are cycled so after the last water year 2000 comes again the first water year 1950.
In the Static setup is equivalent to moving down one column starting with the first water year 1950.
4. Strategic part Water values are calculated in the strategic part. First the Power System is subdivided into water areas. In each water area all hydro and geothermal stations are modelled together as a single power station with a single upstream reservoir. Substation with reserve stations and electricity market are placed within water areas. See figure 4.
Water values are expressed as a function of reservoir content in the water area
and time.
(vwa , t )
Figure 4. One Reservoir model of
Water area in a Power System
Calculation of water values is explained in
figure 5. The vertical axis is divided into intervals and vt is the reservoir content in one point. The single reservoir system is simulated one interval forward to time t+1 by the
R regulated inflow Reservoir U unregulated Q inflow
following model:
Hydro
Qt1 Qt Rt (w min(w,Ut ))
Thermal
Qt is reservoir content in GWh, Rt is regulated
Total market w
inflow in GWh/week and Ut is unregulated
inflow in GWh/week. α is aversion factor when drawdown of reservoir results in lower head at
Firm Power Transmission
Demand
to other
Subsystems
Spot Market Exchange
power stations. This constant is an input
parameter and can be determined by experiment. It is usually on the interval 0,6
– 1,0. No variable head means α = 1,0.
In calculating water values at vt we simulate the system for all instances of river flow say 51 and we can have three possible outcomes according to fig 4:
Figure 5. Calculation of Water Values
V1,t+1
Reservoir full
1. Reservoir full Q1,t+1 > Qmax then water value is = 0.
V3,t+1 vt
2. Reservoir empty Q1,t+1 < 0 then water value is equal to the variable cost of the
0 t-1 t
t Reservoir emty V2,t+1
last reserve unit (or
curtailment price or spot market price etc) used to lift the reservoir level
over 0.
3. Reservoir between empty and full then water value is the same as the water value at v3,t+1 calculated by interpolation.
New water value at vt is calculated by taking the average over all water years.
The whole procedure goes backward in time week by week or day by day and upwards from Q = 0 to Q = Qmax in increments of say 2%. Usually 3 rounds in the water value schema results in a satisfactory convergence.
5. Tactical part
The Multiobjective function for every stage t of the planning period T is:
s,t1 u j,t
jHRe s v j
0 u j,t
jHROR
k s j,t
jH All
c j g j,t
jTermal
c j S j,t
jSell
min
c j Bj,t
jBuy
c j g j,t 2
jShort
Lj,t Lj,t
jTrans
vk ,t1
k (x) s,t1(x) dx
k0
a Sa
k
Turbined water (Plants with Reservoirs) Turbined water (Run-of-the-River) Spilled water Thermal Selling on the Spot Market Buying on the Spot Market Power Shortage Transmission Reservoirs Spinning Reserve requirements
Continuity equation for electricity in substations:
i (vt ) ui,t is
g j,t
jT ,S
B j,t js
S j,t jS
Ln ns
.2
Ln a Ln b Ln c
ns
ws,t
Hydro Production Thermal Production and Shortage Buying on the Spot Market Selling on the Spot Market Transmission from substation Transmission to substation Load in substation
Water balance for every hydro power station; continuity equation for water:
vi,t1 vi,t ui,t si,t ri,t um,t sm,t
(3)
mUi
Limits on reservoir storage: vimin vi,t vimax (4) (Rule Curves, fish conservation, environmental constraints etc) Limits on turbined water: uimin ui,t uimax (5) (Maintenance, environmental constraints etc) Limits on spilled water: simin si,t simax (6) (Environmental constraints etc) Limits on thermal generation: 0 g j,t g jmax (7) (Limits on curtailment)
Curtailment of secondary energy: Max 50% every year Max 40% every 4 consecutive years Max 20% every 20 consecutive years
Fairness between I customers in curtailment of secondary energy.
max g j,t Either I or 0 (integer variable) jCurt g j
max
max
Transmission line capacity:
Ln Ln Ln Ln
(8)
Limits on transmission capacity in Flow Gates (FG):
fn Ln fn Ln La
nFG
nFG
(9)
(1 fn ) Ln (1 fn ) Ln La
nFG
nFG
Spinning reserve requirement in hydro and geothermal plants for each area:
i (vt ) ui,t Sa (1 a ) i (vt ) uimax (10)
ia
ia
Note: The definitions above refer to the HYENA Option: Mixed Integer Nonlinear Programming, with Transfer constraints in transmission lines and flowgates and Nonlinear losses. All other options in chapter 2 are subsets of this general option.
6. Definitions
t:
time index
T:
planning period
a:
area index
s:
subsystem index
k:
reservoir index
j:
thermal plant index
i:
hydro plant index
m:
index for upstream hydro plants
n:
transmission line index
vi,t
Stored volume at plant i at the beginning of stage t
vi,t+1
Stored volume at plant i at the end of stage t
ri,t
Lateral river flow arriving at power station i in stage t
ui,t
Turbined outflow at power station i in stage t
ui,t
Spilled outflow at power station i in stage t
Ui
Set of hydro plants immediately upstream of plant i
gi,t
Generation of thermal plant j in stage t
cj
Generation cost of thermal, shortage and prices on spot market
Bj
Buying on Spot Market (Defined for peak and off-peak load)
Sj
Selling on Spot Market
c j g j,t j
The immediate thermal operating cost in stage t
Factor in penalty function for turbined water (0,4-1,0)
Penalty for spilling water
Damping factor for transmission (i.e. 0,00003)
k (vt1) k,t1 vk,t1 The future cost represented by: k
k (vt1) The production coefficient of reservoir k [kWh/kl]
s,t1
The water value for subsystem k
[kr/kWh]
vk ,t 1
Reservoir volume at the end of stage t [kl]
:
Discount factor
L n
Transmission in positive direction between subsystems
L n
Transmission in negative direction between subsystems
n
Transmission losses
LFG*
Limit on transmission in Flow Gates.
fn+
=1 if direction of transmission line is the same as direction of Flow
Gate, else=0.
fn-
=1 if direction of transmission line is the opposite to direction of
Flow Gate, else=0.
a
Penalty factor for spinning reserve requirement in area
Sa
Lack of spinning reserve
a
Spinning reserve requirements, i.e. week=0,10 and day=0,075
ws,t
Energy market in a subsystem
wa,t
Energy market linked to a spinning reserve group.
7. Supply and Demand In deregulated power systems market equilibrium is established where the supply (marginal cost) and demand curves meet, which defines competitive quantity on the market and market price as described in figure 6. The demand curve represents that in nearly all markets quantity demanded rises as price falls; buyers wish to buy more at lower price and less at higher price. If the demand curve is vertical price changes will not affect the buying behavior of the consumer.
Competitive
price
Price
Figure 6 Market mechanism
Demand
Supply
Consumer Surplus Producer Surplus
Competitive equilibrium
Quantity
Competitive quantity
The supply curve represents that at lower prices, suppliers are less willing or capable of producing. The slope of the curve can be zero if suppliers are able because of their costs to supply more output at the same cost as with hydro power in Iceland in off critical periods.
Consumer surplus is the difference between the market price and what customers would have been willing to pay and producer surplus is the difference between the market price and at what price producers would have been willing to produce. Social surplus is the sum of consumer and producer surplus.
Figures 7-9 show examples of supply/demand curves and represents actual data calculated in the HYENA Simulation Model.
Figure 7 shows typical supply and demand curve for the South-West area of Iceland which has predominantly hydro power. Curtailment option is both included in the supply and demand curve. Figure 7 Supply and Demand. South-West Iceland Jan-Feb 2011 30
Price
20 Supply
Demand
Shortage
Curtailment NettoTransmission to Area option
HydroProduction at Short run marginal cost
200 400 600 800
Thermal 1200 1400
Quantity MW
Figure 8 shows supply and demand curves for a small power system connected to an active spot market. In this situation the customer is buying from the market and Hydro Power has been diminished to a minimum to collect water in reservoirs. Figure 8
Price
Supply and Demand. Spot Market Situation I 250
Hydro
Curtailment BuyPeak
50 SellPeak BuyOffPeak
SellOffPeak 100 Quantity MW
Supply Demand
Figure 9 shows for the same system a situation with much more Hydro Power Production a part of which is sold on the market (Sell Peak). Figure 9 Supply and Demand. Spot Market Situation II 250
Price
200 Supply Demand 150 Curtailment
100 50
SellPeak BuyOffPeak
BuyPeak
Hydro
SellOffPeak
Quantity MW