Modelling Wind-IntegratedHydro-Thermal Power Systems

Gunnar Geir Petursson

University of Iceland

December 3, 2012

Introduction

� An essential part of an energy company’s operation is to predictthe energy production potential.

� Optimal operation forecasts for multi-reservoir systems is acomputationally demanding problem due to the stochastic natureof inflow and to the multiple ways that demand can be met at anygiven time.

� The highly stochastic nature of wind further complicates theproblem.

2 of 16

Introduction

� An essential part of an energy company’s operation is to predictthe energy production potential.

� Optimal operation forecasts for multi-reservoir systems is acomputationally demanding problem due to the stochastic natureof inflow and to the multiple ways that demand can be met at anygiven time.

� The highly stochastic nature of wind further complicates theproblem.

2 of 16

Introduction

� An essential part of an energy company’s operation is to predictthe energy production potential.

� Optimal operation forecasts for multi-reservoir systems is acomputationally demanding problem due to the stochastic natureof inflow and to the multiple ways that demand can be met at anygiven time.

� The highly stochastic nature of wind further complicates theproblem.

2 of 16

A Purely Thermal System

� Optimal operation minimizes fuel cost (c) subject to generation (g)meeting demand (d) at each time t:

zt = MinJ∑

j=1

c(j)gt(j) subject toJ∑

j=1

gt(j) = dt and gt ≤ g

� Generation capacity at stage t + 1 does not depend on stage t.

� Decision variables: gt(j), j = 1, ..., J and t = 1, ...,T .

3 of 16

A Purely Thermal System

� Optimal operation minimizes fuel cost (c) subject to generation (g)meeting demand (d) at each time t:

zt = MinJ∑

j=1

c(j)gt(j) subject toJ∑

j=1

gt(j) = dt and gt ≤ g

� Generation capacity at stage t + 1 does not depend on stage t.

� Decision variables: gt(j), j = 1, ..., J and t = 1, ...,T .

3 of 16

A Purely Thermal System

� Optimal operation minimizes fuel cost (c) subject to generation (g)meeting demand (d) at each time t:

zt = MinJ∑

j=1

c(j)gt(j) subject toJ∑

j=1

gt(j) = dt and gt ≤ g

� Generation capacity at stage t + 1 does not depend on stage t.

� Decision variables: gt(j), j = 1, ..., J and t = 1, ...,T .3 of 16

Hydro-Thermal System

The generation/loadconstraint becomes

I∑i=1

ρ(i)ut(i)+J∑

j=1

gt(j) = dt

where ut ≤ u and gt ≤ gand ρ(i) is the productioncoefficient.

� If at , ut and st denote inflow, turbined outflow and spillrespectively, conservation of water requires

vt+1(i) = vt(i) + at(i)− ut(i)− st(i) +∑

m∈U(i)

[ut(m) + st(m)]

U(i) is a set of up-river hydro-plants.

4 of 16

Hydro-Thermal System

The generation/loadconstraint becomes

I∑i=1

ρ(i)ut(i)+J∑

j=1

gt(j) = dt

where ut ≤ u and gt ≤ gand ρ(i) is the productioncoefficient.

� If at , ut and st denote inflow, turbined outflow and spillrespectively, conservation of water requires

vt+1(i) = vt(i) + at(i)− ut(i)− st(i) +∑

m∈U(i)

[ut(m) + st(m)]

U(i) is a set of up-river hydro-plants.

4 of 16

Hydro-Thermal System

The generation/loadconstraint becomes

I∑i=1

ρ(i)ut(i)+J∑

j=1

gt(j) = dt

where ut ≤ u and gt ≤ gand ρ(i) is the productioncoefficient.

� If at , ut and st denote inflow, turbined outflow and spillrespectively, conservation of water requires

vt+1(i) = vt(i) + at(i)− ut(i)− st(i) +∑

m∈U(i)

[ut(m) + st(m)]

U(i) is a set of up-river hydro-plants.4 of 16

A Hydro-Thermal System (continued)

Since reservoir inflows are limited, hydro generation is coupled in time.

� We minimize the sum ofimmediate and future costs (α):

zt = MinJ∑

j=1

c(j)gt(j)+αt+1(vt+1)

v is a vector of reservoir levels.

Long-term Hydro Scheduling based on Stochastic Models

EPSOM’98, Zurich, September 23-25, 1998Page PEREIRA-4

In turn, the future cost function - FCF - is associated with the expected thermal generationexpenses from stage t+1 to the end of the planning period. We see that the FCF decreases withfinal storage, as more water becomes available for future use. The FCF is calculated by simulating system operation in the future for different starting valuesof initial storage and calculating the operating costs. The simulation horizon depends on thesystem storage capacity. If the capacity is relatively small, as in the Spanish or Norwegiansystem, the impact of a decision is diluted in several months. If the capacity is substantial, as inthe Brazilian system, the simulation horizon may reach five years. This simulation is made morecomplex by the variability of inflows to reservoirs, which fluctuate seasonally, regionally, andfrom year to year. In addition, inflow forecasts are generally inaccurate, in particular wheninflow comes from rainfall, not snowmelt. As a consequence, FCF calculation has to be carriedout on a probabilistic basis, i.e. using a large number of hydrological scenarios (dry, mediumand wet years etc.), as illustrated in Figure 2.3.

1 2 3 4 time

spillage

rationing

replacesthermalgeneration

max. storage

Figure 2.3 - FCF Calculation

In contrast with thermal plants, which have direct operating costs, hydro plants have an indirectopportunity cost, associated to savings in displaced thermal generation now or in the future. 2.2.3 Water Values The optimal use of stored water corresponds to the point that minimizes the sum of immediateand future costs. As shown in Figure 2.4, this is also where the derivatives of ICF and FCF withrespect to storage become equal. These derivatives are known as water values.

ICF

FCF

final storage

watervalue

ICF + FCF

optimaldecision

Figure 2.4 - Optimal Hydro Scheduling

The optimal hydro dispatch is at the point which equalizes immediate and future water values.

� Decision variables: ut and gt , i = 1, ...I , j = 1, ..., J andt = 1, ...,T .

5 of 16

A Hydro-Thermal System (continued)

Since reservoir inflows are limited, hydro generation is coupled in time.

� We minimize the sum ofimmediate and future costs (α):

zt = MinJ∑

j=1

c(j)gt(j)+αt+1(vt+1)

v is a vector of reservoir levels.

Long-term Hydro Scheduling based on Stochastic Models

EPSOM’98, Zurich, September 23-25, 1998Page PEREIRA-4

In turn, the future cost function - FCF - is associated with the expected thermal generationexpenses from stage t+1 to the end of the planning period. We see that the FCF decreases withfinal storage, as more water becomes available for future use. The FCF is calculated by simulating system operation in the future for different starting valuesof initial storage and calculating the operating costs. The simulation horizon depends on thesystem storage capacity. If the capacity is relatively small, as in the Spanish or Norwegiansystem, the impact of a decision is diluted in several months. If the capacity is substantial, as inthe Brazilian system, the simulation horizon may reach five years. This simulation is made morecomplex by the variability of inflows to reservoirs, which fluctuate seasonally, regionally, andfrom year to year. In addition, inflow forecasts are generally inaccurate, in particular wheninflow comes from rainfall, not snowmelt. As a consequence, FCF calculation has to be carriedout on a probabilistic basis, i.e. using a large number of hydrological scenarios (dry, mediumand wet years etc.), as illustrated in Figure 2.3.

1 2 3 4 time

spillage

rationing

replacesthermalgeneration

max. storage

Figure 2.3 - FCF Calculation

In contrast with thermal plants, which have direct operating costs, hydro plants have an indirectopportunity cost, associated to savings in displaced thermal generation now or in the future. 2.2.3 Water Values The optimal use of stored water corresponds to the point that minimizes the sum of immediateand future costs. As shown in Figure 2.4, this is also where the derivatives of ICF and FCF withrespect to storage become equal. These derivatives are known as water values.

ICF

FCF

final storage

watervalue

ICF + FCF

optimaldecision

Figure 2.4 - Optimal Hydro Scheduling

The optimal hydro dispatch is at the point which equalizes immediate and future water values.

� Decision variables: ut and gt , i = 1, ...I , j = 1, ..., J andt = 1, ...,T .

5 of 16

A Hydro-Thermal System (continued)

Since reservoir inflows are limited, hydro generation is coupled in time.

� We minimize the sum ofimmediate and future costs (α):

zt = MinJ∑

j=1

c(j)gt(j)+αt+1(vt+1)

v is a vector of reservoir levels.

Long-term Hydro Scheduling based on Stochastic Models

EPSOM’98, Zurich, September 23-25, 1998Page PEREIRA-4

In turn, the future cost function - FCF - is associated with the expected thermal generationexpenses from stage t+1 to the end of the planning period. We see that the FCF decreases withfinal storage, as more water becomes available for future use. The FCF is calculated by simulating system operation in the future for different starting valuesof initial storage and calculating the operating costs. The simulation horizon depends on thesystem storage capacity. If the capacity is relatively small, as in the Spanish or Norwegiansystem, the impact of a decision is diluted in several months. If the capacity is substantial, as inthe Brazilian system, the simulation horizon may reach five years. This simulation is made morecomplex by the variability of inflows to reservoirs, which fluctuate seasonally, regionally, andfrom year to year. In addition, inflow forecasts are generally inaccurate, in particular wheninflow comes from rainfall, not snowmelt. As a consequence, FCF calculation has to be carriedout on a probabilistic basis, i.e. using a large number of hydrological scenarios (dry, mediumand wet years etc.), as illustrated in Figure 2.3.

1 2 3 4 time

spillage

rationing

replacesthermalgeneration

max. storage

Figure 2.3 - FCF Calculation

In contrast with thermal plants, which have direct operating costs, hydro plants have an indirectopportunity cost, associated to savings in displaced thermal generation now or in the future. 2.2.3 Water Values The optimal use of stored water corresponds to the point that minimizes the sum of immediateand future costs. As shown in Figure 2.4, this is also where the derivatives of ICF and FCF withrespect to storage become equal. These derivatives are known as water values.

ICF

FCF

final storage

watervalue

ICF + FCF

optimaldecision

Figure 2.4 - Optimal Hydro Scheduling

The optimal hydro dispatch is at the point which equalizes immediate and future water values.

� Decision variables: ut and gt , i = 1, ...I , j = 1, ..., J andt = 1, ...,T .

5 of 16

A Hydro-Thermal System (continued)

Since reservoir inflows are limited, hydro generation is coupled in time.

� We minimize the sum ofimmediate and future costs (α):

zt = MinJ∑

j=1

c(j)gt(j)+αt+1(vt+1)

v is a vector of reservoir levels.

Long-term Hydro Scheduling based on Stochastic Models

EPSOM’98, Zurich, September 23-25, 1998Page PEREIRA-4

In turn, the future cost function - FCF - is associated with the expected thermal generationexpenses from stage t+1 to the end of the planning period. We see that the FCF decreases withfinal storage, as more water becomes available for future use. The FCF is calculated by simulating system operation in the future for different starting valuesof initial storage and calculating the operating costs. The simulation horizon depends on thesystem storage capacity. If the capacity is relatively small, as in the Spanish or Norwegiansystem, the impact of a decision is diluted in several months. If the capacity is substantial, as inthe Brazilian system, the simulation horizon may reach five years. This simulation is made morecomplex by the variability of inflows to reservoirs, which fluctuate seasonally, regionally, andfrom year to year. In addition, inflow forecasts are generally inaccurate, in particular wheninflow comes from rainfall, not snowmelt. As a consequence, FCF calculation has to be carriedout on a probabilistic basis, i.e. using a large number of hydrological scenarios (dry, mediumand wet years etc.), as illustrated in Figure 2.3.

1 2 3 4 time

spillage

rationing

replacesthermalgeneration

max. storage

Figure 2.3 - FCF Calculation

In contrast with thermal plants, which have direct operating costs, hydro plants have an indirectopportunity cost, associated to savings in displaced thermal generation now or in the future. 2.2.3 Water Values The optimal use of stored water corresponds to the point that minimizes the sum of immediateand future costs. As shown in Figure 2.4, this is also where the derivatives of ICF and FCF withrespect to storage become equal. These derivatives are known as water values.

ICF

FCF

final storage

watervalue

ICF + FCF

optimaldecision

Figure 2.4 - Optimal Hydro Scheduling

The optimal hydro dispatch is at the point which equalizes immediate and future water values.

� Decision variables: ut and gt , i = 1, ...I , j = 1, ..., J andt = 1, ...,T .

5 of 16

SDDP infterface by PSR

6 of 16

Cloud interface

7 of 16

A Hydro-Thermal Example from SDDP: Generation

!"#$

!"#$%&'%()$*$$+'%,$-.&/')0$$

%&'()*+,-$()./'-01$23"'4)567'"$89:$;<$$

8 of 16

A Hydro-Thermal Example from SDDP: Volume

0.00  

500.00  

1000.00  

1500.00  

2000.00  

2500.00  

01/2008  

18/2008  

35/2008  

52/2008  

17/2009  

34/2009  

51/2009  

16/2010  

33/2010  

50/2010  

15/2011  

32/2011  

49/2011  

14/2012  

31/2012  

48/2012  

13/2013  

30/2013  

47/2013  

12/2014  

29/2014  

46/2014  

11/2015  

28/2015  

45/2015  

Gl   Volume   Ini1al  storage                                      Bla.P                Aver.                Ini1al  storage                                      Blon.P              Aver.                Ini1al  storage                                      Hag.P                Aver.                Ini1al  storage                                      Hrv.P                Aver.                Ini1al  storage                                      Jav.P                Aver.                Ini1al  storage                                      Kar.P                Aver.                Ini1al  storage                                      Sig.P                Aver.                Ini1al  storage                                      Sul.P                Aver.                Ini1al  storage                                      Thv.P                Aver.                

9 of 16

An Example from SDDP: Turbinable Spill

0.00  

5.00  

10.00  

15.00  

20.00  

25.00  

01/2008  

19/2008  

37/2008  

03/2009  

21/2009  

39/2009  

05/2010  

23/2010  

41/2010  

07/2011  

25/2011  

43/2011  

09/2012  

27/2012  

45/2012  

11/2013  

29/2013  

47/2013  

13/2014  

31/2014  

49/2014  

15/2015  

33/2015  

51/2015  

GWh   Turbinable  Spilled  energy   Turbinable  Spilled  Energy                  Bla.P                Series  001      Turbinable  Spilled  Energy                  Bur.P                Series  001      Turbinable  Spilled  Energy                  Ira.P                Series  001      Turbinable  Spilled  Energy                  Kar.P                Series  001      Turbinable  Spilled  Energy                  Lax.P                Series  001      Turbinable  Spilled  Energy                  Ljo.P                Series  001      Turbinable  Spilled  Energy                  Ste.P                Series  001      Turbinable  Spilled  Energy                  Sul.P                Series  001      

10 of 16

A Wind Itegrated Hydro-Thermal System

� Unlike reservoir inflows, wind must be used when available or elseit is lost (much like run of river plants).

Generation/Load:K∑

k=1

wt(k) +I∑

i=1

ρ(i)ut(i) +J∑

j=1

gt(j) = dt

11 of 16

A Wind Itegrated Hydro-Thermal System

� Unlike reservoir inflows, wind must be used when available or elseit is lost (much like run of river plants).

Generation/Load:K∑

k=1

wt(k) +I∑

i=1

ρ(i)ut(i) +J∑

j=1

gt(j) = dt

11 of 16

Optimization methods

If n reservoirs are discretized into m levels each, computation timeand storage requirements are proportional to mn.Several algorithms have been divised to tackle this so called curse ofdimensionality.

� Very simple approach: Combining all reservoirs (of a system orsubsystem) into one.

� Water value method (in use by Landsvirkjun): Combinedsubsystem reservoirs used to estimate the value of water frommarginal costs using stochastic dynamic programming (SDP).Next, a simulation phase allocates production to demand in thewhole system.

� SDDP: Stochastic Dual Dynamic Programming uses dualinformation to approximate the future cost function in eachiteration. Terminates once tolerance is reached.

12 of 16

Optimization methods

If n reservoirs are discretized into m levels each, computation timeand storage requirements are proportional to mn.Several algorithms have been divised to tackle this so called curse ofdimensionality.

� Very simple approach: Combining all reservoirs (of a system orsubsystem) into one.

� Water value method (in use by Landsvirkjun): Combinedsubsystem reservoirs used to estimate the value of water frommarginal costs using stochastic dynamic programming (SDP).Next, a simulation phase allocates production to demand in thewhole system.

� SDDP: Stochastic Dual Dynamic Programming uses dualinformation to approximate the future cost function in eachiteration. Terminates once tolerance is reached.

12 of 16

Optimization methods

If n reservoirs are discretized into m levels each, computation timeand storage requirements are proportional to mn.Several algorithms have been divised to tackle this so called curse ofdimensionality.

� Very simple approach: Combining all reservoirs (of a system orsubsystem) into one.

� Water value method (in use by Landsvirkjun): Combinedsubsystem reservoirs used to estimate the value of water frommarginal costs using stochastic dynamic programming (SDP).Next, a simulation phase allocates production to demand in thewhole system.

� SDDP: Stochastic Dual Dynamic Programming uses dualinformation to approximate the future cost function in eachiteration. Terminates once tolerance is reached.

12 of 16

Optimization methods

If n reservoirs are discretized into m levels each, computation timeand storage requirements are proportional to mn.Several algorithms have been divised to tackle this so called curse ofdimensionality.

� Very simple approach: Combining all reservoirs (of a system orsubsystem) into one.

� Water value method (in use by Landsvirkjun): Combinedsubsystem reservoirs used to estimate the value of water frommarginal costs using stochastic dynamic programming (SDP).Next, a simulation phase allocates production to demand in thewhole system.

� SDDP: Stochastic Dual Dynamic Programming uses dualinformation to approximate the future cost function in eachiteration. Terminates once tolerance is reached.

12 of 16

Parallelization

To decrease computation time, parallelization has been used.

� Since future inflow is never known, many measured inflows areused to build a future scenario. The above optimization problemcan be solved for several different inflows using parallel computing.Corresponding wind series also, if they exist.

� Some SDDP calculations are performed on the amazon cloudservice, running as many as 1024 processors in parallel.

� A so called water value method (used by Landsvirkjun) estimatesthe value of water for a subsystem whose reservoirs have all beencombined into one. Different subsystem use separate processingunits.

13 of 16

Parallelization

To decrease computation time, parallelization has been used.

� Since future inflow is never known, many measured inflows areused to build a future scenario. The above optimization problemcan be solved for several different inflows using parallel computing.Corresponding wind series also, if they exist.

� Some SDDP calculations are performed on the amazon cloudservice, running as many as 1024 processors in parallel.

� A so called water value method (used by Landsvirkjun) estimatesthe value of water for a subsystem whose reservoirs have all beencombined into one. Different subsystem use separate processingunits.

13 of 16

Parallelization

To decrease computation time, parallelization has been used.

� Since future inflow is never known, many measured inflows areused to build a future scenario. The above optimization problemcan be solved for several different inflows using parallel computing.Corresponding wind series also, if they exist.

� Some SDDP calculations are performed on the amazon cloudservice, running as many as 1024 processors in parallel.

� A so called water value method (used by Landsvirkjun) estimatesthe value of water for a subsystem whose reservoirs have all beencombined into one. Different subsystem use separate processingunits.

13 of 16

Parallelization

To decrease computation time, parallelization has been used.

� Since future inflow is never known, many measured inflows areused to build a future scenario. The above optimization problemcan be solved for several different inflows using parallel computing.Corresponding wind series also, if they exist.

� Some SDDP calculations are performed on the amazon cloudservice, running as many as 1024 processors in parallel.

� A so called water value method (used by Landsvirkjun) estimatesthe value of water for a subsystem whose reservoirs have all beencombined into one. Different subsystem use separate processingunits.

13 of 16

Time Scales

� In a hydro-thermal system, modelled over afew years, a time step of 1 week is oftenenough.

� Wind production is more dynamic than otherstate variables and so 1 week averages are notenough. A week could include days of lowwind and high wind, during which thetransmission network is violated.

� One could increase the time-resolution but then the computationtime suffers.

14 of 16

Time Scales

� In a hydro-thermal system, modelled over afew years, a time step of 1 week is oftenenough.

� Wind production is more dynamic than otherstate variables and so 1 week averages are notenough. A week could include days of lowwind and high wind, during which thetransmission network is violated.

� One could increase the time-resolution but then the computationtime suffers.

14 of 16

Time Scales

� In a hydro-thermal system, modelled over afew years, a time step of 1 week is oftenenough.

� Wind production is more dynamic than otherstate variables and so 1 week averages are notenough. A week could include days of lowwind and high wind, during which thetransmission network is violated.

� One could increase the time-resolution but then the computationtime suffers.

14 of 16

Time Scales

� In a hydro-thermal system, modelled over afew years, a time step of 1 week is oftenenough.

� Wind production is more dynamic than otherstate variables and so 1 week averages are notenough. A week could include days of lowwind and high wind, during which thetransmission network is violated.

� One could increase the time-resolution but then the computationtime suffers.

14 of 16

Future work

One way to handle the highly stochastic nature of wind power is toconstruct a short time model within a long term model.

� The long term model would include slowly changing parts,reservoirs and load along with the power system.

� The short time model could handle all high resolution aspects,wind and possibly run of river plants.

� The long term model could determine the load and generation,taking into account the transmission grid.

� The short time model allocates energy in more resolution and iftransmission constraints are violated it can force the long termmodel to reiterate.

15 of 16

Future work

One way to handle the highly stochastic nature of wind power is toconstruct a short time model within a long term model.

� The long term model would include slowly changing parts,reservoirs and load along with the power system.

� The short time model could handle all high resolution aspects,wind and possibly run of river plants.

� The long term model could determine the load and generation,taking into account the transmission grid.

� The short time model allocates energy in more resolution and iftransmission constraints are violated it can force the long termmodel to reiterate.

15 of 16

Future work

One way to handle the highly stochastic nature of wind power is toconstruct a short time model within a long term model.

� The long term model would include slowly changing parts,reservoirs and load along with the power system.

� The short time model could handle all high resolution aspects,wind and possibly run of river plants.

� The long term model could determine the load and generation,taking into account the transmission grid.

� The short time model allocates energy in more resolution and iftransmission constraints are violated it can force the long termmodel to reiterate.

15 of 16

Future work

One way to handle the highly stochastic nature of wind power is toconstruct a short time model within a long term model.

� The long term model would include slowly changing parts,reservoirs and load along with the power system.

� The short time model could handle all high resolution aspects,wind and possibly run of river plants.

� The long term model could determine the load and generation,taking into account the transmission grid.

� The short time model allocates energy in more resolution and iftransmission constraints are violated it can force the long termmodel to reiterate.

15 of 16

Future work

One way to handle the highly stochastic nature of wind power is toconstruct a short time model within a long term model.

� The long term model would include slowly changing parts,reservoirs and load along with the power system.

� The short time model could handle all high resolution aspects,wind and possibly run of river plants.

� The long term model could determine the load and generation,taking into account the transmission grid.

� The short time model allocates energy in more resolution and iftransmission constraints are violated it can force the long termmodel to reiterate.

15 of 16

References

� Johannsson, S., & Eliasson, E. B. (2002). Simulation Model of theHydro-Thermal Power System in Iceland. Report for the nationalpower company of Iceland. (18 pages).

� Labadie, J. W. (2004). Optimal operation of multireservoirsystems: State-of-the-art review. Journal of Water ResourcesPlanning and Management, 130(2), 93–111.

� M V F Pereira. (1989). Optimal stochastic operations schedulingof large hydroelectric systems. Electric Power Energy systems,11(3), 161–169.

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