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This paper analyzes a pricetaker hydro generating company which participates simultaneously in dayahead energy and ancillary services markets. An approach for deriving marginal cost curves for energy and ancillary services is proposed, taking into consideration price uncertainty and opportunity cost of water, which can later be used to determine hourly bid curves. The proposed approach combines an hourly conditional valueatrisk, probability of occurrence of automatic generation control states and an opportunity cost of water to determine energy and ancillary services marginal cost curves. The proposed approach is in a linear constraint form and is easy to implement in optimization problems. A stochastic model of the hydroeconomic river basin is presented, based on the actual Vinodol hydropower system in Croatia, with a complex threedimensional relationship between the power produced, the discharged water, and the head of associated reservoir.
This paper formulates a hydroeconomic river basin model (HERBM) of a pricetaking hydro generating company (GENCO, Zagreb, Croatia) which participates in a simultaneous energy and ancillary services dayahead markets (DAMs) [
Modeling of ancillary services is specifically addressed in this research using simple linear formulation where automatic generation control (AGC) state (ramp up, ramp down, no ramp) is considered as a random variable. Detailed ancillary services models are important in order to encourage competitiveness and introduction of new ancillary services providers which consequently results in higher penetration of renewable energy sources (RES) particularly wind [
In a shortterm planning most of the parameters are usually considered as known and the resulting models are deterministic [
Relationships between the head of the associated reservoir(s), the discharged water and the generated power is considered. Models describing these relationships are available in [
In this chapter, hourly water marginal costs are determined. The profit maximization objective function is formulated. Hourly CVaR as a risk measure is formulated. Ancillary services model is formulated with AGC state as a discrete random variable. Probability of occurrence of AGC states are later used as a rule for redistributing the water marginal cost along the energy, the regulation and the 10 min spinning reserve. Afterwards the energy, regulation and 10 min spinning reserve marginal costs are used for determining an hourly bidding curves.
The continuity equation of the hydro reservoirs is formulated as
For unit consistency, it should be noted the time periods of 1 h are considered. The τ
The performance curves (
The slope ρ is defined by HPP conversion capabilities (MWs/m^{3}). The five performance curves are associated for the five water contents intervals. Activation of the corresponding performance curve is done by the approach presented in [
The performance curves are activated according to the four binary variables
Depending on the binary variables
In this work the hydro GENCO participates in the simultaneous energy and ancillary service DAM. Markets are considered competitive enough to assume the analyzed GENCO is a pricetaker. The 10 min spinning reserve and regulation markets are considered as the ancillary service markets. In DAM considered here the independent system operator (ISO) will commit GENCOs in a similar way as ISO did in the vertically integrated structure using security constrained unit commitment (SCUC). Suppliers submit their bids to supply the forecasted daily inelastic demand and ISO uses the bidin costs submitted by GENCOs for each generating unit (multipart bids that reflect startup costs, minimum generation costs, capacity offered, running costs,
Besides participating in energy DAM, GENCO participates in the 10 min spinning reserve DAM and the regulation DAM. Ancillary services markets are countrydependent and a review of these markets designs is available in [
When GENCO commits in the 10 min spinning reserve DAM the following AGC states may occur:
The operating range of HPP when committed in the regulation and the 10 min spinning reserve DAM is defined with
For the regulation capacity
Expected electricity produced when committed in the simultaneous energy, regulation and spinning reserve DAM in a time period
Multiplying the ancillary service capacities
The probability matrix
Electricity produced for the energy market
Electricity produced for the regulation service
Electricity produced for the spinning reserve service
Expected electricity produced is equal to the performance curve
Ramp rate limits
The maximum regulation capacity that can be offered is calculated as the MSR times five min
The set Ω will represent future states of knowledge, and the single element
The discrete random variable π as shown in
The probability of particular daily price scenario of market
Finite number of future states are observed, ω∈Ω ⊆ ℕ. The random variable
An easy way to incorporate a risk into linear model is to use CVaR [
CVaR is a coherent measure of risk, meaning CVaR has properties of convexity, monotonicity, closedness, positive homogeneity, subadditivity and translation invariance [
Both αVaR and αCVaR can be calculated by solving an elementary optimization problem of convex type in one dimension. For this purpose the special function
When maximizing
Generally, when the daily CVaR is used, the risk will average out throughout the day, and this is not desirable as, especially in a case of a major contingency in particular hour–this can result in unpredictable values of the daily CVaR. Consequently, the hourly αVaR(
Since objective function is already defined, the special function is thus brought in the optimization problem in a form of the constraint
When implementing the constraint
The linear formulation of
GENCO's objective function is maximal expected profit
GENCO's profit in a time step
Objective function is an expected profit of the planning horizon
Shadow prices of the HERBM constrains are analyzed
The shadow prices
The Shadow price of used water
The shadow price of used water ψ(
Since the water shadow price ψ(
The pairs
Water shadow price of one additional (m^{3}) of water in particular hour, and one (m^{3}) reduced from all other hours, as said represents marginal cost of producing one more (m^{3}) or (MWh) so it is particularly convenient to use for creating a bid curve. An ancillary service DAM is another opportunity for GENCO to increase profit and it generates additional opportunity costs which are reflected as an increase of water shadow prices.
The approach for pricing energy and ancillary services is depicted on
For the needs of this research HPS Vinodol (Croatia) is modeled, which consists of: four reservoirs, one pump station (PS), two pump storage hydropower plants (PSP) and one HPP (
The prices are formulated according to
The energy and ancillary services prices are obtained from New York DAM [
Observed GENCO owns HPS Vinodol. For clarity and computational effectiveness HPP Vinodol profit is maximized while rest of HPS is used for that objective. There is no significant lose in accuracy when having in mind that in GENCO's profit, HPP Vinodol profit participates with 95% or more. Probabilities are considered stationary,
The perfectly inelastic (vertical) energy marginal cost curves here considered as bid curves are obtained for the ordinate values 38 MWh and 57 MWh (
For the regulation DAM (
For spinning reserve DAM (
The water shadow prices have relatively low deviation compared to DAM prices and in this research use of the simplified approach with the constant water shadow prices over the planning horizon is justified. The hourly water shadow prices also have relatively low mean value which is a result of having more than enough water with relative relationships maintained so conclusions can be given (e.g., ratio between energy and ancillary services bid prices is similar to the ratio between energy and ancillary services DAM prices). The results were not obtained for lesser amount of water to avoid long computation time. Further bid curve analysis is omitted in this work since its slope, ordinate and aspics position is highly determined by input parameters.
When the function is nonincreasing in some intervals two period moving average approach is used for determining the nondecreasing function as depicted on
As the profit tolerance rises from $0 to $1380 for the each hour (
The obtained results confirm a risk aversion principle, generally GENCO should avoid hours with high price deviation as can be seen for the 13th and 15th hour in
The approach presented in this work assesses the problem which the pricetaker hydro generating company faces when participating in the simultaneous dayahead energy and ancillary services markets and which results in the riskconstrained approach for pricing energy and ancillary services.
The results of this case study justify the use of a simplified approach where water shadow price was considered constant over planning horizon. It was also shown that perfectly inelastic bid curves are appropriate for a pricetaker hydro generating company. It was shown that reducing risk of financial losses costs, and that these cost can be easily measured.
Since the research was done on relatively small number of scenarios, further work should be done to implement computational efficient algorithms which will tackle dimensionality issue and improve computational time. For higher exactness lognormal probability distribution should be used which will pronounce benefits of this riskconstrained approach.
This work has been supported by the project “Sustainable Energy and Environment in the Western Balkans—SEEWB” inside the HERD (Higher Education, Research and Development) Energy Programme, funded by the Norwegian Ministry of Foreign Affairs and the European Community Seventh Framework Programme under grant No. 285939 (ACROSS).
The authors declare no conflict of interest.
Set of indices of the steps of the optimization period, planning horizon,
Set of indices of the reservoirs, plants,
Set of indices of the perf. curves
Set of upstream reservoirs of plant
Set of indices of the blocks of the piecewise linearization of the unit performance curve
Set of indices representing future states of knowledge, it is a set of scenarios that can occur, Ω = {1,2,…,Ω_{max}},
Set of indices of the profit tolerances
Conversion factor equal to 3600 (m^{3}·s·m^{−3}·h^{−1}).
Maximal content of the reservoir
Minimal content of the reservoir
Initial water content of the reservoir
Final water content of the reservoir
Forecasted natural water inflow of the reservoir
Forecasted price of realtime electricity market in time step
Forecasted price of day ahead electricity market in time step
Forecasted price of day ahead regulation market in time step
Forecasted price of dayahead 10 minute spinning reserve market in time step
Minimum water discharge of plant
Maximum water discharge of plant
Maximum water discharge of block
Ecological minimum of plant
Ancillary services probability matrix of size Ω
Probability of price scenario ω.
Minimum power output of plant
Capacity of plant
Slope of the block
Conversion factor used for converting (m^{3}) to (MWh) for reservoir
Probability of being in Regulationup state in time step
Probability of being in Regulationdown state in time step
Probability of spinning reserve to be activated in time step
Maximum sustain ramp rate of plant
Ramping up limit of plant
Ramping down limit of plant
Difference between maximal values of two neighboring perf. curves of plant
Difference between minimal values of two neighboring perf. curves of plant
Confidence level of a profit probability density function.
Confidence level of a loss probability density function
Shadow price of constraint determining final
Shadow price of water balance equation of
Shadow price of constraint determining maximum
Shadow price of constraint determining minimum
Shadow price of constraint determining maximum
Shadow price of constraint determining minimum
Water content of the reservoir
Average water content of the reservoir
Water discharge of plant
Water discharge of block
Spillage of the reservoir
0/1 variable used for discretization of performance curves and scenario ω,
0/1 variable which is equal to 1 if plant
0/1 variable which is equal to 1 if water discharged by plant
Power output of plant
Power output of plant
Regulation service capacity of plant
10 min spinning reserve of plant
Electricity produced for energy market by plant
Electricity produced for regulation service by plant
Electricity produced for spinning reserve service by plant
Real variable in special function
(
The hypothetical probability density function (PDF) of a (
Algorithm for riskshaping with CVaR.
Hypothetical bid curve for 10 min spinning reserve DAM of hour
The approach for pricing energy and ancillary services.
HPS Vinodol.
Randomly generated daily price scenarios for the energy DAM.
The marginal cost curves for energy production for profit tolerances (
The marginal cost curves for regulation for the profit tolerances for (
The marginal cost curves for spinning reserve for the profit tolerances (
Creating nondecreasing bid curve from inapplicable shadow price curve.
The perfectly inelastic energy bid curves for (
HPS Vinodol facilities parameters.
Križ  0.06 × 10^{6}  3  PS Križ  1.1  0.34 
Lokve  34.8 × 10^{6}  3  PSP Fužine  10/9  4.6/4.8 
Bajer  1.32 × 10^{6}  3  HPP Vinodol  18.6  94.5 
Lepenica  4.26 × 10^{6}  3  PSP Lepenica  6.2/5.3  1.14/1.25 
Descriptive statistics of NYISO prices in ($/MWh).

 

39.8  13.0  9.4  3.0  4.8  0.3  64.6  26.6  9.8  0.8  8.9  2.2  
36.2  11.4  5.3  0.6  4.7  0.5  66.0  31.0  10.9  0.7  10.2  2.0  
34.6  10.9  5.3  0.4  4.2  0.4  67.6  33.2  10.4  1.6  11.2  2.5  
34.5  10.8  4.8  0.4  4.7  0.4  68.4  34.6  11.0  1.7  10.4  2.3  
36.5  10.4  8.9  2.9  4.5  0.4  67.2  30.6  12.9  2.3  13.9  2.6  
41.5  11.7  18.1  6.9  4.6  0.5  68.1  27.8  11.3  2.1  10.1  2.6  
48.2  15.9  13.1  4.7  6.1  1.1  67.4  24.8  10.8  2.4  9.7  2.1  
52.5  17.8  12.1  3.1  8.9  0.7  64.1  21.7  10.3  0.8  9.6  1.1  
55.6  17.8  13.3  3.2  9.0  0.4  58.2  20.1  10.4  1.2  10.1  1.2  
59.2  19.5  10.0  0.0  8.5  0.4  51.5  17.5  11.4  1.1  11.0  1.1  
61.7  21.5  10.0  0.1  9.4  0.6  46.1  14.9  10.7  1.6  6.4  1.3  
62.8  22.9  11.1  1.0  9.9  1.1  43.7  14.5  10.3  1.8  5.0  0.0 
Daily profit and daily risk measures αCVaR and αVaR for confidence level
0  98,193  0  0 
400  97,838  9,600  14,400 
800  98,162  19,100  28,500 
1,380  96,959  33,120  44,037 