EMK:Simplifying River Chains for Dynamic Programming - Revision history

Sisep: /* Maximum Flows */

2012-12-03T21:05:30Z

Maximum Flows

Sisep: /* Using the Simplified River Chains */

2012-12-03T21:01:08Z

Using the Simplified River Chains

Sisep at 20:56, 3 December 2012

2012-12-03T20:56:57Z

Sisep: /* Allocation of Spills */

2012-12-03T20:55:14Z

Allocation of Spills

Sisep: /* Allocation of Spills */

2012-12-03T20:54:25Z

Allocation of Spills

Sisep: /* Allocation of Spills */

2012-12-03T20:53:45Z

Allocation of Spills

Sisep: /* River Chain Simplification */

2012-12-03T20:32:19Z

River Chain Simplification

Sisep: /* River Chain Simplification */

2012-12-03T20:31:21Z

River Chain Simplification

Sisep: /* River Chain Simplification */

2012-12-03T20:27:54Z

River Chain Simplification

Sisep: /* River Chain Simplification */

2012-12-03T20:16:15Z

River Chain Simplification

← Older revision		Revision as of 21:05, 3 December 2012
Line 154:		Line 154:

	This solution will have an effect when an outage is applied to only one of the stations. Rather than having to spill past this station, only the total capacity of the sub-system will be changed. This should be kept in mind when applying outages.		This solution will have an effect when an outage is applied to only one of the stations. Rather than having to spill past this station, only the total capacity of the sub-system will be changed. This should be kept in mind when applying outages.
		+
		+
		+
		+	[[EMK:Technical Bulletins \| Back to Technical Bulletins]]

← Older revision		Revision as of 21:01, 3 December 2012
Line 148:		Line 148:

	== Using the Simplified River Chains ==		== Using the Simplified River Chains ==
		+	In the short term river chain optimisation modelling, the primary advantage of simplification is that it avoids slow DP solutions. However a simplified model will result in faster solutions for any operations on the river chain. Currently the river chain is simplified for the DP and for all other operations. This simplification needs to be applied carefully to ensure solutions where the relationship in equation 1 does not hold are not valuable operational solutions for the system.
		+
		+	=== Maximum Flows ===
		+	In ''EMarket'' the nodes in a chain generally have both a generating channel and a spill channel between them. If the maximum flow in one channel in the chain is set differently from the rest it is possible that that station may reach capacity before the rest. This can not be modelled easily in the simplified system. To avoid this difficulty the average maximum flow figure for f<sub>0</sub> to f<sub>n</sub> can be used as the maximum flow figure for the release from SB. This will not differ to much from the full dynamics of the system, provided all maximum flows are roughly the same.
		+
		+	This solution will have an effect when an outage is applied to only one of the stations. Rather than having to spill past this station, only the total capacity of the sub-system will be changed. This should be kept in mind when applying outages.

@@ Line 16: / Line 16: @@
 Instead of modelling each and every reservoir separately on this chain we will apply a constraint which requires all reservoirs to be filled to the same proportion of their capacity.  So at any time:
-. S<sub>i</sub> = &alpha; Max<sub>i</sub>, for some alpha.
+. '''S<sub>i</sub> = &alpha; Max<sub>i</sub>''', for some alpha.
 The behaviour of this chain is then determined by the inflows, f<sub>IN</sub> and the change in &alpha;. Change in storage &Delta;S<sub>i</sub> is given by change in &alpha;.
-. &Delta;S<sub>i</sub> = [[File:lamda.jpg]] Max<sub>i</sub> ,			[[File:lamda.jpg]] = change in &alpha;
+.''' &Delta;S<sub>i</sub> = [[File:lamda.jpg]] Max<sub>i</sub> ''',			[[File:lamda.jpg]] = change in &alpha;
 Change in storage is also related to flows and inflows:
-.	&Delta;S<sub>0</sub> = I<sub>0</sub> + f<sub>IN</sub> – f<sub>0</sub>
+.	'''&Delta;S<sub>0</sub> = I<sub>0</sub> + f<sub>IN</sub> – f<sub>0</sub>'''
-:&Delta;S<sub>i</sub> = I<sub>i</sub> + f<sub>i</sub> – f<sub>i +1</sub>		,  i > 0
+:'''&Delta;S<sub>i</sub> = I<sub>i</sub> + f<sub>i</sub> – f<sub>i +1</sub>'''		,  i > 0
 Combining 2. and 3. Gives:
@@ Line 51: / Line 51: @@
 In the simplified sub-system flows are given by:
-. f<sub>A</sub> = f<sub>IN</sub> + I<sub>A</sub>
+. '''f<sub>A</sub> = f<sub>IN</sub> + I<sub>A</sub> '''
-:f<sub>B</sub> = f<sub>A</sub> + I<sub>B</sub> - [[File:lamda.jpg]]Max
+:'''f<sub>B</sub> = f<sub>A</sub> + I<sub>B</sub> - [[File:lamda.jpg]]Max'''
-::= I<sub>A</sub> + I<sub>B</sub> + f<sub>IN</sub> - [[File:lamda.jpg]]Max
+::'''= I<sub>A</sub> + I<sub>B</sub> + f<sub>IN</sub> - [[File:lamda.jpg]]Max'''
 Since f<sub>B</sub> must equal f<sub>n</sub> for all inflow, f<sub>IN</sub> and [[File:lamda.jpg]] values, comparing Equations 8 and 4 gives:
@@ Line 63: / Line 63: @@
 Generation is given by:
-. G* = M<sub>A</sub> f<sub>A</sub> + M<sub>B</sub> f<sub>B</sub>
+.''' G* = M<sub>A</sub> f<sub>A</sub> + M<sub>B</sub> f<sub>B</sub>'''
-::=  M<sub>A</sub> I<sub>A</sub> + M<sub>B</sub> (I<sub>A</sub> + I<sub>B</sub>)  +  (M<sub>A</sub> + M<sub>B</sub>) f<sub>IN</sub>  -  M<sub>B</sub> [[File:lamda.jpg]]Max
+::'''=  M<sub>A</sub> I<sub>A</sub> + M<sub>B</sub> (I<sub>A</sub> + I<sub>B</sub>)  +  (M<sub>A</sub> + M<sub>B</sub>) f<sub>IN</sub>  -  M<sub>B</sub> [[File:lamda.jpg]]Max'''
 Since G* must equal G for all inflow, f<sub>IN</sub> and [[File:lamda.jpg]] values comparing Equations 11 and 7 gives:
-. M<sub>A</sub> + M<sub>B</sub> = &sigma;<sub>0</sub>  ( from the f<sub>IN</sub> component )
+.''' M<sub>A</sub> + M<sub>B</sub> = &sigma;<sub>0</sub>'''  ( from the f<sub>IN</sub> component )
 . [[File:SRC E 7.jpg]]  ( from the [[File:lamda.jpg]] component )
@@ Line 88: / Line 88: @@
 Once the results of a simulation with the simplified model are known they can be converted back to results for the original sub-system by calculating [[File:lamda.jpg]] and &alpha;:
-[[File:lamda.jpg]] = ( f<sub>A</sub> + I<sub>A</sub> - f<sub>B</sub> ) / Max
+[[File:lamda.jpg]] = '''( f<sub>A</sub> + I<sub>A</sub> - f<sub>B</sub> ) / Max'''
-&alpha; = S / Max
+'''&alpha; = S / Max'''
 Equations 1 and 4 then give the storages and flows in the original sub-system.
@@ Line 98: / Line 98: @@
 Where
-.	Max = Max<sub>0</sub>
+.	'''Max = Max<sub>0</sub>'''
-.	I = I<sub>0</sub>
+.	'''I = I<sub>0</sub>'''
-.	S = S<sub>0</sub>
+.	'''S = S<sub>0</sub>'''
-.	M = &sigma;0
+.	'''M = &sigma;0'''
 === Allocation of Spills ===

@@ Line 111: / Line 111: @@
 . '''f<sub>A</sub> = f <sup>g</sup><sub>A</sub> + f <sup>s</sup><sub>A</sub> '''
-. f<sub>B</sub> = f <sup>g</sup><sub>B</sub> + f <sup>s</sup><sub>B</sub>
+. '''f<sub>B</sub> = f <sup>g</sup><sub>B</sub> + f <sup>s</sup><sub>B</sub>'''
 where f <sup>g</sup> is the flow used for generation and f <sup>s</sup> is the spill. The potential generation is:
-. G* <sup>P</sup> = M<sub>A</sub> f<sub>A</sub> + M<sub>B</sub> f<sub>B</sub>
+. '''G* <sup>P</sup> = M<sub>A</sub> f<sub>A</sub> + M<sub>B</sub> f<sub>B</sub>'''
 The real generation is given by
-. G* <sup>R</sup> = M<sub>A</sub> f <sup>g</sup> <sub>A</sub> + M<sub>B</sub> f <sup>g</sup> <sub>B</sub>
+. '''G* <sup>R</sup> = M<sub>A</sub> f <sup>g</sup> <sub>A</sub> + M<sub>B</sub> f <sup>g</sup> <sub>B</sub>'''
 If the resulting flows for the original sub-system are also divided into spill and generation flows:
-. f<sub>i</sub> = f <sup>g</sup><sub>i</sub> + f <sup>s</sup><sub>i</sub>
+. '''f<sub>i</sub> = f <sup>g</sup><sub>i</sub> + f <sup>s</sup><sub>i</sub>'''
 G* <sup>P</sup> will be equal to the potential generation of the f<sub>i</sub> flows:
-. G* <sup>P</sup> = G <sup>P</sup> = &Sigma;<sub>k</sub> = <sub>0.. n</sub> f<sub>k</sub> M<sub>k</sub>
+. '''G* <sup>P</sup> = G <sup>P</sup> = &Sigma;<sub>k</sub> = <sub>0.. n</sub> f<sub>k</sub> M<sub>k</sub>'''
 What we need as well as this is for the real generation figures to be equal:
-. G* <sup>R</sup> = G <sup>R</sup>
+. '''G* <sup>R</sup> = G <sup>R</sup>'''
 If we force all spill to be the same proportion of the total flow in the solution.  i.e.
@@ Line 139: / Line 139: @@
 Then real generation in the original sub-system will be
-. G<sup>R</sup> = &mu; &Sigma;<sub>k = 0..n</sub> f<sub>k</sub> M<sub>k</sub>
+. '''G<sup>R</sup> = &mu; &Sigma;<sub>k = 0..n</sub> f<sub>k</sub> M<sub>k</sub>'''
 Combining equations 27, 28 and 30 give:
-. &mu; = G* <sup>R</sup> / G* <sup>P</sup>
+. '''&mu; = G* <sup>R</sup> / G* <sup>P</sup>'''
 This value for &mu; will provide spill and release figures for all nodes in the original sub-system, which match both real and potential generation.
 == Using the Simplified River Chains ==

@@ Line 109: / Line 109: @@
 The flow figures calculated above will match both outflow and generation, if all flows are used for generation.  If however the flows are divided into generating and spill flow figures then the match will only be with ''potential'' generation, or the generation that would have occurred had all channels been generating with the given power conversion factors.  It is necessary than to allocate the total f<sub>A</sub> and f<sub>B</sub> figures into f<sub>i</sub> figures for both spill and generation. If
-. f<sub>A</sub> = f <sup>g</sup><sub>A</sub> + f <sup>s</sup><sub>A</sub>
+. '''f<sub>A</sub> = f <sup>g</sup><sub>A</sub> + f <sup>s</sup><sub>A</sub> '''
 . f<sub>B</sub> = f <sup>g</sup><sub>B</sub> + f <sup>s</sup><sub>B</sub>

← Older revision		Revision as of 20:53, 3 December 2012
Line 107:		Line 107:

	=== Allocation of Spills ===		=== Allocation of Spills ===
		+	The flow figures calculated above will match both outflow and generation, if all flows are used for generation. If however the flows are divided into generating and spill flow figures then the match will only be with ''potential'' generation, or the generation that would have occurred had all channels been generating with the given power conversion factors. It is necessary than to allocate the total f<sub>A</sub> and f<sub>B</sub> figures into f<sub>i</sub> figures for both spill and generation. If
		+
		+	22. f<sub>A</sub> = f <sup>g</sup><sub>A</sub> + f <sup>s</sup><sub>A</sub>
		+
		+	23. f<sub>B</sub> = f <sup>g</sup><sub>B</sub> + f <sup>s</sup><sub>B</sub>
		+
		+	where f <sup>g</sup> is the flow used for generation and f <sup>s</sup> is the spill. The potential generation is:
		+
		+	24. G* <sup>P</sup> = M<sub>A</sub> f<sub>A</sub> + M<sub>B</sub> f<sub>B</sub>
		+
		+	The real generation is given by
		+
		+	25. G* <sup>R</sup> = M<sub>A</sub> f <sup>g</sup> <sub>A</sub> + M<sub>B</sub> f <sup>g</sup> <sub>B</sub>
		+
		+	If the resulting flows for the original sub-system are also divided into spill and generation flows:
		+
		+	26. f<sub>i</sub> = f <sup>g</sup><sub>i</sub> + f <sup>s</sup><sub>i</sub>
		+
		+	G* <sup>P</sup> will be equal to the potential generation of the f<sub>i</sub> flows:
		+
		+	27. G* <sup>P</sup> = G <sup>P</sup> = Σ<sub>k</sub> = <sub>0.. n</sub> f<sub>k</sub> M<sub>k</sub>
		+
		+	What we need as well as this is for the real generation figures to be equal:
		+
		+	28. G* <sup>R</sup> = G <sup>R</sup>
		+
		+	If we force all spill to be the same proportion of the total flow in the solution. i.e.
		+
		+	29. c
		+
		+	Then real generation in the original sub-system will be
		+
		+	30. G<sup>R</sup> = μ Σ<sub>k = 0..n</sub> f<sub>k</sub> M<sub>k</sub>
		+
		+	Combining equations 27, 28 and 30 give:
		+
		+	29. μ = G* <sup>R</sup> / G* <sup>P</sup>
		+
		+	This value for μ will provide spill and release figures for all nodes in the original sub-system, which match both real and potential generation.
		+
		+	== Using the Simplified River Chains ==

← Older revision		Revision as of 20:32, 3 December 2012
Line 105:		Line 105:

	21. M = σ0		21. M = σ0
		+
		+	=== Allocation of Spills ===

← Older revision		Revision as of 20:31, 3 December 2012
Line 93:		Line 93:

	Equations 1 and 4 then give the storages and flows in the original sub-system.		Equations 1 and 4 then give the storages and flows in the original sub-system.
		+	[[File:SRC Fig 3.jpg\|140px\|thumb\|right\|Figure 3]]
		+	If M<sub>A</sub> = 0 for the system, then Equations 16 and 17 will not solve. This will occur only if there is zero storage in all but the last reservoirs. It is inappropriate to try and simplify this configuration, unless there are no inflows below the first node, in which case the simplified sub-system is shown in figure 3.
		+
		+	Where
		+
		+	18. Max = Max<sub>0</sub>
		+
		+	19. I = I<sub>0</sub>
		+
		+	20. S = S<sub>0</sub>

−	If M~~<sub>A</sub>~~ = 0 for the system, then Equations 16 and 17 will not solve. This will occur only if there is zero storage in all but the last reservoirs. It is inappropriate to try and simplify this configuration, unless there are no inflows below the first node, in which case the simplified sub-system is shown in figure 3.	+	21. M = σ0
−	~~[[File:SRC Fig 3.jpg\|140px\|thumb\|right\|Figure 3]]~~

← Older revision		Revision as of 20:27, 3 December 2012
Line 73:		Line 73:

	14. [[File:SRC E 8.jpg]] ( from the inflows components )		14. [[File:SRC E 8.jpg]] ( from the inflows components )
		+
		+	Combining 12 and 13 gives:
		+
		+	15. [[File:SRC E 9.jpg]]
		+
		+	Combining 14 and 9 gives:
		+
		+	16. [[File:SRC E 10.jpg]]
		+
		+	17. [[File:SRC E 11.jpg]]
		+
		+	Equations 10,13,15 enable a specification of the simplified sub-system to be calculated. Equations 16 and 17 find the inflow figures to use in the system.
		+
		+	Once the results of a simulation with the simplified model are known they can be converted back to results for the original sub-system by calculating [[File:lamda.jpg]] and α:
		+
		+	[[File:lamda.jpg]] = ( f<sub>A</sub> + I<sub>A</sub> - f<sub>B</sub> ) / Max
		+
		+	α = S / Max
		+
		+	Equations 1 and 4 then give the storages and flows in the original sub-system.
		+
		+	If M<sub>A</sub> = 0 for the system, then Equations 16 and 17 will not solve. This will occur only if there is zero storage in all but the last reservoirs. It is inappropriate to try and simplify this configuration, unless there are no inflows below the first node, in which case the simplified sub-system is shown in figure 3.
		+	[[File:SRC Fig 3.jpg\|140px\|thumb\|right\|Figure 3]]

← Older revision		Revision as of 20:16, 3 December 2012
Line 67:		Line 67:

	Since G* must equal G for all inflow, f<sub>IN</sub> and [[File:lamda.jpg]] values comparing Equations 11 and 7 gives:		Since G* must equal G for all inflow, f<sub>IN</sub> and [[File:lamda.jpg]] values comparing Equations 11 and 7 gives:
		+
		+	12. M<sub>A</sub> + M<sub>B</sub> = σ<sub>0</sub> ( from the f<sub>IN</sub> component )
		+
		+	13. [[File:SRC E 7.jpg]] ( from the [[File:lamda.jpg]] component )
		+
		+	14. [[File:SRC E 8.jpg]] ( from the inflows components )