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An optimization method for multi-area combined heat and power production with power transmission network
Applied Energy 168 (2016) 248–256
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
An optimization method for multi-area combined heat and power production with power transmission network Elnaz Abdollahi ⇑, Haichao Wang, Risto Lahdelma Department of Energy Technology, Aalto University, School of Engineering, P.O. BOX 14100, FI-00076, Aalto, Finland
h i g h l i g h t s Decomposition-based optimization method for CHP production and power transmission problems. Minimization of CHP production cost to meet the local heat demand and identify the optimal power transmission between areas. Solving a small sample problem using the developed models to show the solution approach. Method allows using fast specialized algorithms to solve the sub-problems of multiple areas. Fast solution of hourly problems is useful in solving long-term problems.
a r t i c l e
i n f o
Article history: Received 23 July 2015 Received in revised form 22 December 2015 Accepted 22 January 2016
Keywords: Combined heat and power (CHP) Power transmission Linear programming (LP) Optimization Energy efficiency
a b s t r a c t This paper presents an efficient decomposition-based optimization method to optimize the hourly combined heat and power (CHP) production and power transmission between multiple areas. The combined production and power transmission problem is decomposed into local CHP production models and into a power transmission model. The CHP production models are formulated as linear programming (LP) models and solved using a parametric analysis technique to determine the local production cost as a function of power transmitted into or out from each area. To obtain the overall optimum, the power transmission problem is then formulated in terms of the parametric curves as a network flow problem, and solved using a special network Simplex algorithm. The decomposition method has been tested with different sized artificial problems. The method can be used in situations where it is necessary to solve a large number of hourly production and transmission problems efficiently. As an example, the method can be used as part of long-term planning and simulation of CHP systems in different cities or countries connected by a common power market. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Combined heat and power (CHP) is the most efficient way to deliver power, heating and cooling [1]. Based on some statistic data for 2012, CHP electricity generation for the 28 member states of the EU was 373.3 TW h, and heat production was 854.5 TW h [2]. The importance of CHP in district heating system also has been pointed out [3,4]. In Finland, the share of CHP electricity was about 34% and district heating 74% in 2014 [5]. CHP can utilize different types of fuels such as waste, biomass and also traditional fossil fuels [6,7]. Through the heat utilization, efficiency of CHP can be more than 90% and therefore offers dramatic savings of energy ranging between 15% and 40% in comparison with conventional power plants and heat only boilers [1]. Environmental benefits of ⇑ Corresponding author. Tel.: +358 (0)50 383 5075. E-mail address: [email protected] (E. Abdollahi). http://dx.doi.org/10.1016/j.apenergy.2016.01.067 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
CHP in comparison with other conventional plants are due to reduction in emissions [8]. The role of CHP was investigated in a study based on zero emissions scenario [9]. In 2014, the European Commission presented a new framework for climate and energy policy for 2030. Its targets comprise increasing renewable energy utilization, improving energy efficiency, and reducing greenhouse emissions. The targets depend on economic analysis measuring how to achieve decarburization cost-effectively by 2050 [10]. The optimal operation of local CHP production can be determined by an optimization model. The objective is to minimize the total cost of energy production while satisfying demand and operational constraints. The complexity arises from the coupling of power and heat production to match both heat and power demand. The problem will be more complicated, when power transmission between multiple areas is also considered. As a result, faster methods and solvers are needed for optimizing for large scale energy production problems.
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Nomenclature
Abbreviations CHP combined heat and power LP linear programming NP2 efficient network simplex algorithm CHPED combined heat and power economic dispatch MILP mixed integer linear programming CPU time central processing unit time (s) Symbols A set of arcs b additional constraints coupling heat and power production (MW h) C cost (€) ca transmission cost (€/MW h) cj ; pj ; qj production cost (€), power generation (MW h), and heat production (MW h) at characteristic point j ccond condensing power price (€/MW h) heat only boiler price (€/MW h) chob cl marginal cost of production (slope of line segment) (€/MW h) di demand in node i (MW h) destðaÞ destination node of arc a H matric for heat and power production (MW h) J set of extreme characteristic points of all CHP plants in a production area Ju set of extreme characteristic points for CHP plant u 2 U m number of production arcs N set of the nodes (production areas)
The complexity of optimization problems depends strongly on the shape of the objective function and constraints. For example, minimization of convex objective function subject to convex constraints is in general easier than solving non-convex problems. Several models for convex and non-convex problems have been developed to solve combined heat and power economic dispatch (CHPED) problems. For convex CHPED problems, linear models have been proposed and solved by using specialized Simplex algorithms [11–13]. Non-convex CHPED problems may arise from a non-convex characteristic operating region or the need to optimize unit commitment (when to shut down and start up plants). A nonconvex CHPED operating region can be divided into convex subregions and formulated as a mixed integer linear programming (MILP) model, or solved using the Lagrangian relaxation technique [14–18]. To determine the unit commitment of CHP plants, a unit decommitment algorithm was developed and the solution quality was compared with a generic unit decommitment algorithm by using realistic test data [14]. Reference [19] presents a thorough comparison of 37 software tools for different energy applications, ranging from single building models to national energy systems. In particular, the study evaluated the suitability of the tools for integrating renewables into energy systems. Six of the tools were applicable for modeling CHP plants. The BCHP Screening Tool analyzes the combined heating, cooling, and power systems of commercial buildings, excluding large scale heat, power or transport sectors [20]. COMPOSE and EnergyPRO can be used for technoeconomic analysis of single projects [21,22]. BALMOREL can simulate the CHP sector in a multi-national geographical area [23]. EnergyPLAN and SIVAL are tools for national or regional energy systems [24,25]. Among them only EnergyPRO and EnergyPLAN can simulate 100% renewable energy systems. They have been
n orgðaÞ P Q U ua x xj xhob xcond xmax hob xmax cond y ya
number of areas origin node of arc a power demand (MW h) heat demand (MW h) set of CHP plants transmission line capacity (MW h) decision variable to encode convex combination variables used to encode convex combination of operating region heat production of heat only boiler (MW h) power production of condensing power plant (MW h) maximum capacity for heat only boiler production (MW h) maximum capacity condensing power plant production (MW h) power transmission (MW h) power flowing from origin node to destination node (MW h)
Superscripts and subscripts a arc (transmission line between areas) cond condensing power plant production hob heat only boiler production i; k indices for nodes (production areas) in network l line segment index p; q power and heat products prod production trans transmission
used to verify the results of linear programming (LP) models for renewable energy systems with energy storages [26–28]. Table 1 presents an overview of different scale energy systems, which have been solved using different algorithms. A review over research on the CHPED problem was presented in [33]. The high efficiency and profitability of CHP production can be further improved by utilization of power transmission network. Accurate modeling of energy production together with power transmission network in a large scale problem requires a sophisticated optimization model. Efficient solution of CHPED problems is important because a long-term planning model includes thousands of hourly models, and rapid re-optimization is needed when the market situation changes. In this paper, the hourly multi-area CHP production and power transmission problem is formulated as an LP model, but decomposed and solved as separate local CHP production models and an overall model for the power transmission problem. In the local CHP problems, the produced power can be sold to the grid at market price, but heat must be produced to meet the local demand for district heating or specific industrial processes. The local CHP production is modeled as LP problems and solved using a special parametric LP algorithm. This will yield the local production costs as a function of the power production. Then the local production cost curves are encoded as a network flow problem into an overall production and power transmission problem. A specialized primal network simplex algorithm NP2 is used to solve the network flow model. The overall optimization model minimizes the combined production costs in all areas and identifies the optimal balance between local production and power transmissions into or out from each area. The solution technique should be fast enough to solve even large hourly optimization problems with many CHP
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Table 1 Overview of different scale energy systems. Case number
(1)
(2)
(3)
(4)
(5)
Scale Units
Small 1 conventional power 2 CHPs 1 heat only [29]
Small 1 conventional power 3 CHPs 1 heat only [30]
Small 4 conventional power 2 CHPs 1 heat only [31]
Large 13 conventional power units 6 CHPs 5 heat only units [32]
Large 26 conventional power units 12 CHPs 10 heat only units [32]
Proposed by
plants in several areas so that it can be used as a sub-routine for solving long-term planning problems consisting of many hourly models in sequence. Potential use for the algorithm is to solve for example a yearly planning model with 8760 h, or even models for multiple years. In the following, Section 2 presents the overall convex CHP production and power transmission problem formulated as a generic LP model. Section 3 formulates the local CHP production model in terms of extreme characteristic point of CHP plants and presents how the model can be solved using parametric analysis. Section 4 presents the power transmission problem and how the parametric analysis results of the local models can be encoded into an overall network problem. Section 5 presents the numerical results including the solution of a small four-area sample problem, and tests with different size larger generated test problems. The network problems are solved both with a dedicated network solver and compared against an efficient LP solver. Section 6 contains the conclusions and proposes further research. 2. Combined CHP production and power transmission problem The objective of the hourly CHP production and power transmission problem is to minimize the overall operational costs. Assuming that the unit commitment is fixed and the characteristic operating regions of plants are convex, this problem can be defined as a general LP model as follows [34]:
min
X
Ci xi þ
i
XX C ik yik i
ð1Þ
k–i
s:t: Hqi xi ¼ Q i X X yki yik ¼ Pi Hpi xi þ k–i
ð2Þ ð3Þ
k–i
Hpq i xi ¼ b i
ð4Þ
xi ; yik P 0
ð5Þ
Here the indices i and k loop over all heat and power production areas. The decision variables for each area i are represented by vector xi, and corresponding coefficients of objective function are included in vector Ci. The amount of power traded out of area i and transmitted into area k is denoted by decision variables yik with the corresponding transmission costs Cik. The objective function (1) minimizes the production costs (Cixi) and the power transmission costs (Cikyik). The matrices H with superscripts p, q or pq refer to the production of power, heat, or both. Constraints (2) guarantee that heat demand Qi is matched by local heat production in each area. Constraints (3) require that the power demand Pi must be met in the area by produced power plus the power transmitted into the area and subtracted by power transmitted out of the area. Constraints (4) may include additional technology-specific constraints and also constraints for coupling local heat and power production. In the absence of CHP technology, such coupling constraints are non-existent. Transmission losses are not included in this formulation. One way to deal with them is to treat transmission losses as costs due to need of producing additional power to replace the loss. Losses are then considered in the Cik coefficients.
In the following this model is decomposed into separate local CHP production models for each area and an overall model to consider the power transmissions between areas.
3. Local CHP production model In the following, the hourly operation of a convex CHP plant is modeled in terms of extreme characteristic points. Then, an hourly local CHP system model is composed. The local CHP system model can contain an arbitrary number of CHP plants, heat-only boilers and condensing power plants. Finally the parametric analysis technique is presented. Parametric analysis is used to determine how the local CHP system can interact with the other areas through power transmission.
3.1. Convex CHP plant model The operation region of a CHP plant allows it to produce different combinations of heat and power with corresponding production costs (mainly fuel costs). The operating region can be viewed as a surface in 3D space corresponding to coordinates for produced power and heat (p, q), and associated production cost (c). To apply LP modeling, the CHP plant characteristic is assumed to be convex. A typical convex CHP plant is illustrated in Fig. 1. The projection of the 3D surface on the 2D plane (p, q) shows the range in which the production of heat and power can be adjusted. It can be seen that the 2D region is a convex polygon in the (p, q) plane. Convexity of a region means that any line segment connecting two arbitrary points of the region is also entirely inside the region. Also the production cost must be a convex function of the power and heat production. This means that the cost surface is either flat or curves upwards within the region. Many types of CHP plants have convex operating regions and production costs. For non-convex plants, the operating region can be partitioned into convex sub-regions, and MILP can be applied to solve the problem [15]. A convex model
Fig. 1. Feasible operating region of a convex CHP plant.
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can also be assumed if it is possible to run a non-convex CHP plant for fractions of an hour in different modes. The operation of a convex CHP plant is modeled through linear constraints as a convex combination of the extreme points (qi, pi, ci) of the characteristic surface:
C¼
X j2J
j2J u
X q j xj
ð6Þ
j2J u
X xj ¼ 1 j2J u
xj P 0 Here xj are used for expressing the characteristic operating region as a convex combination (weighted average with non-negative weights) of the extreme points, and Ju is the index set for the extreme points of the plant characteristic. The above model formulation allows running the plant anywhere inside the 3D polyhedron spanned by the extreme points. However, minimization of costs always produces an optimal solution which lies on the lower envelope surface [16]. The lower envelope (plotted in Fig 1) corresponds to minimum cost CHP operation for every heat and power production combination, including part load operation. One of the characteristic points corresponds to maximum load operation. Operation at the other characteristic points and in the area between the points corresponds to part load operation. This modeling technique applies also for separate heat and power production units, such as condensing power plants and heat-only boilers. Heat-only units can be modeled using pj = 0 and power-only units with qj = 0. The modeling technique applies also to demand side management components ðpj ; qj ; cj 6 0Þ and heat pumps converting power to heat ðpj P 0; qj 6 0Þ . Ramp constraints for the power production of the plant can be represented as additional constraints
P0 P 6
X pj xj 6 P0 þ Pþ
in the model, it is also possible to modify the characteristic operating region so that parts outside the feasible ramping region are cut off. Fig. 2 illustrates cutting off parts of the characteristic region by ramp constraints. 3.2. Local CHP system model
c j xj
Xu P¼ pj xj Q¼
251
ð7Þ
Fig. 3 illustrates the local CHP system. The system can consist of any number of energy production units, but for simplicity the figure shows only one CHP plant, one condensing power plant and one heat only boiler. Heat production must satisfy the demand exactly, and power production must meet the demand plus the net amount of power transmitted out from the area. The following model minimizes the overall costs of local production in one area:
min C ¼
X cj xj þ ccond xcond þ chob xhob
s:t: X qj xj þ xhob ¼ Q
ð9Þ
j2J
X pj xj þ xcond ¼ Pprod
ð10Þ
j2J
X xj ¼ 1 u 2 U
ð11Þ
j2J u
xj P 0 j 2 J u ; u 2 U max 0 6 xhob 6 xmax hob ; 0 6 xcond 6 xcond
ð12Þ ð13Þ
The objective function (8) minimizes the production costs consisting of fuel costs for the CHP plants (cjxj), condensing power plant (ccondxcond) and heat only boiler (chobxhob). As explained in the previous section, heat-only boilers and condensing power plants could be modeled as special cases of CHP; however for illustrative purposes one of each is represented explicitly in the model. Constraint (9) is the heat balance where Q is the heat demand and variable xhob is heat only boiler production. Constraint (10) is the power balance constraint in which variable Pprod is the total power production and variable xcond is the power production of condens-
j2J u
where P0 is the power production in the previous hour and P and P+ are maximal allowed down and up ramp for power, correspondingly. Similarly, it is possible to consider ramp constraints for heat production. Instead of including the ramp constraints (7) explicitly
Fig. 2. Cutting off parts of the characteristic operating region by ramp constraints.
ð8Þ
j2J
Fig. 3. Schematic of the local energy production problem.
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ing power plant in the area. Constraints (11) and (12) are the convexity constraints for the xj variables of each CHP plant. Constraints (13) are the lower and upper bounds for heat-only boiler and condensing power plant production. Possible ramp constraints are in this representation considered implicitly by modifying the original characteristic regions to respect the ramp constraints.
the supply and demand of some commodity (such as power) at nodes [35]. Balancing is done by transferring specific amounts of the commodity between nodes either directly or via other nodes while respecting capacity limits for transmissions. The network is defined as a tuple (N, A) in which N is the set of nodes and A is the set of directed arcs connecting nodes. The problem is formulated as
3.3. Parametric analysis of the model
min C 0 þ The model (8-13) is solved to get the production costs for every possible power production amount Pprod using parametric LP analysis. Parametric analysis determines the optimal production cost of the CHP system (including CHP plants and separate heat and power units) as a function of power generation in each area, C⁄ = C(Pprod). Because the model is an LP model, the resulting cost function is a piecewise linear convex function. Each line segment corresponds to a basic LP solution. When the basic solution changes, the marginal cost (slope of line segment, see Fig. 6 for examples) changes. The production cost functions can thus be represented as a sequence of (C, Pprod) coordinate pairs, ½ðC 0 ; P 0prod Þ; . . . ; ðC L ; P Lprod Þ l 0 L with P l1 prod 6 P prod . Here P prod and P prod are the minimal and maximal possible power production in the area. Due to degeneration of l basic LP solutions, it is possible that P l1 prod ¼ P prod for some points. Such duplicate points can be removed from the sequence to guarantee strict inequality among P-coordinates. The marginal cost (slope) for line segment l is then
cl ¼ ðC l C l1 Þ=ðPlprod Pl1 prod Þ
ð14Þ
The lines segments determine the cost and capacity of production arcs, used later in the power transmission model. The amount of power transmitted out of or into the area is related to the power production Pprod according to
Pprod ¼ ptrans þ P
ð15Þ
Here P is the power demand within the area and ptrans is the power transmitted out of the area. Transfer into the area is expressed as negative values of ptrans. This variable links the production model with the power transmission model.
X ca ya
ð16Þ
a2A
s:t: X a2AjorgðaÞ¼i
ya
X
ya ¼ di ; i 2 N
ð17Þ
a2AjdestðaÞ¼i
0 6 ya 6 ua ; ða 2 AÞ
ð18Þ
Here the variable ya is the flow through arc a from origin node org (a) to destination node dest(a). The transmission cost is ca and the capacity of transmission line is ua. The objective function (16) minimizes the overall costs of the network and also includes a constant term C0 that represents costs that are independent of transmitted amount. Balance constraints (17) require that the total flow out of the node subtracted by the flow into the node must equal di which can be the source (+) or sink () at the node. The sum of all sources must equal the sum of all sinks to make the model feasible. The optimal power transmission problem can be naturally represented as this kind of network problem. Each area corresponds to a node and each transmission line corresponds to an arc with specified unit cost and capacity. This formulation allows any number of arcs (zero or more) between each pair of nodes. Zero arcs between a pair of nodes represents missing transmission line between two areas. Multiple arcs between a pair of nodes can represent multiple parallel transmission lines between areas, but allows also using a piecewise linear approximation for non-linear (convex) transmission cost for a single transmission line. Multiple arcs are also needed to represent the parametric CHP production cost functions as described in the next section.
4.2. Encoding the combined CHP production and power transmission problem
4. Power transmission model Power transmission between areas allows minimizing the overall production costs, since at each hour, power can be produced in the areas having the lowest marginal production costs and transferred to the areas where production cost is higher. Meanwhile, the transfer capacities and transmission costs should be considered when finding the optimal solution to the problem. Optimal transfer of power between areas can be easily represented as a capacitated minimum cost transshipment network flow problem. Also the piecewise linear optimal production cost functions of local CHP production (derived using parametric analysis in previous section) can be represented as a network flow problem. The idea is to combine the network-encoded production problem with the transmission model into an overall network problem. The advantage of this approach is that very efficient solution algorithms exist for the network problem [35]. In the following, we first define the capacitated transshipment network flow problem and then show how the parametric functions of local CHP production can be encoded into the overall network problem. 4.1. Capacitated transshipment problem The general capacitated transshipment problem is a network flow problem where the objective is to balance at minimum cost
To encode the combined CHP production and power transmission problem as a network problem, we augment the network model for the transmission system with new arcs to represent the production costs. The area nodes are defined as sink nodes with P di = Pi. An artificial source node is created with di = iPi. Then, to represent local production, arcs are created from the artificial source node to the area nodes. The costs and capacities of these production arcs are defined by slopes cl and lengths of the line segments ðP lprod Pl1 prod Þ determined by the parametric analysis. The constant term C0 in the objective function (16) is set equal to the sum of the cost of minimal power production in each area. Encoding of the model is illustrated in Section 5. To see that the two-phase approach produces the same optimum as a single overall model, observe that it represents the costs and constraints of the original problem accurately. Firstly, for the transmission network, the costs and constraints are represented directly in the network model. Secondly, for each local CHP model, parametric analysis extracts the optimal production costs as a function of power production. Because the parametric curves are computed based on CHP system models, all production constraints are respected and the corresponding production costs are computed accurately. Non-optimal operating modes of local CHP systems are excluded from the encoded model, but this does not worsen the overall optimum.
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consumption. All transmission lines between areas have 10 MW h transmission capacity and 1 €/MW h transmission cost.
5. Results In the following, the two-phase solution method is first presented using a small sample problem. Then the performance of solving larger network problems is evaluated using artificially generated test problems of different size. The LP2 linear programming package [11] is used for solving local production models using parametric LP analysis, and the NP2 network simplex algorithm [34] is used to solve the encoded transshipment problem.
5.1. Optimization of a sample problem with 4 areas A small sample problem comprising of 4 areas is used to demonstrate the operation of the developed two-phase solution approach. For each area, structurally the same production facilities consisting of three CHP plants, one condensing power plant, and one heat only boiler are used. The schematic of the production facilities in each area is presented in Fig. 4. The fuel consumed by CHP 1, CHP 2, and the condensing power plant is biomass (wood chips). CHP 3 and the heat only boiler are fed by natural gas.
5.1.1. Input data The characteristic points for the three CHP plants are shown in Fig. 5 [26,27]. The figure shows the characteristics in 2D projection on the (p, q) plane without the cost coordinate. Instead of the cost coordinate, the fuel consumption in each point is given. The cost coordinate is computed by multiplying fuel consumption by fuel price. For the three CHP plants and heat only boiler, real data of a city in southern Finland are used. The condensing power plant parameters represent a typical, modern power plant. For CHP 2 and CHP 3, only two characteristic points are available corresponding maximal and minimal power and heat output. For CHP 1, a third characteristic point is obtained based on analysis of its reallife operating history. Operating regions of all three CHP plants are convex, so they can be represented using the LP model presented Section 3.1. Table 2 lists the fuel prices and maximal efficiencies of all plants in addition to the constant parameters including heat and power demand in each area, and the capacity and cost of the transmission between areas. CHP 1 has a very high maximal efficiency due to using flue gas heat recovery. The Rankine cycle is used in CHP 2 and CHP 3. CHP 3 uses an open-cycle gas turbine with heat recovery boiler [21]. The efficiency of each plant at each characteristic point is the ratio of power and heat production to the fuel
5.1.2. Solution of local CHP system models The local energy production model is solved to determine the optimal production costs in the areas as a function of power production using parametric LP analysis implemented in the LP2 linear programming software. Parametric analysis is performed in each area for power production amounts between 0 and 25 MW h. The results are shown in Fig. 6. For larger heat demand the production cost curves are at higher level. This is reasonable, since more fuel is consumed to produce a larger amount of heat. Interestingly, the production costs as function of produced power initially decrease before starting to increase in each area. This is due to the fact that heat production is needed anyway, and if CHP plants must produce a very small amount of power, they cannot satisfy the heat demand either. Instead, the more expensive heat only boiler is used to produce the some of the heat. When power production is higher, the CHP plants can produce also a greater share of the heat, leading to higher overall efficiency and lower production costs. After the initial decrease in production costs, the production costs increase due to applying more expensive condensing power production. In other words, in area 1, CHP plants cannot satisfy all heat demand at point A (marked in Fig. 6). Instead, a heat only boiler must supply 21 MW h additional heat at higher cost than CHP plants. At point B, the CHP plants are able to produce the necessary amount of heat together with 15.125 MW h electricity and the heat only boiler is not needed. Also up to point C, CHP heat is sufficient, which implies very cost efficient operation. Beyond point C, up to point D, more expensive (less efficient) condensing power is gradually needed. In point D condensing power production is 9.072 MW h. Therefore, the production costs increase more rapidly in the range from C to D. In area 4 with the highest heat demand, CHP plants cover most of the heat and all power demand without condensing power production. Therefore, the production costs decrease in the full range of power production. 5.1.3. Encoded production and transmission model Fig. 7 shows the production and power transmission model encoded as a capacitated transshipment problem. Local production arcs from the artificial source node to the area nodes are shown as dashed arcs, and transmission connections by solid arcs. Each parametric line segment in Fig. 6 corresponds to a production arc from the artificial source node to the area node in Fig. 7. For areas 1–3 there are three production arcs and for area 4 there are two. The
Fig. 4. Sample problem schematic plants in each area.
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CHP 2 – biomass
CHP 1 – biomass
CHP 3 – natural gas
Fig. 5. Characteristic points of CHP plants [26,27].
Table 2 Parameters for local production facilities. Fuel price (€/MW h)
Maximal efficiency
Wood 21 Input parameters
Natural gas 40
Area Values
CHP1 CHP2 0.97 0.85 Q: Heat demand (MW h)
CHP3 0.84 P: Power demand (MW h)
1 50
1 5
2 60
3 70
4 80
2 10
3 15
4 20
Condensing power 0.4 Transmission capacity (MW h)
Heat only 0.89 Transmission cost (€/MW h)
10
1
4000 3500
Producon cost (€)
3000 2500
D
A
2000
B
1500
C
1000 500 0
0
5
10
15
20
25
30
Local power producon (MWh) Area 1, Q=50 (MWh)
Area 3, Q=70 (MWh)
Area 2, Q=60 (MWh)
Area 4, Q=80 (MWh)
Fig. 7. Encoding production and power transmission problem as a capacitated transshipment problem.
Fig. 6. Optimal production cost as a function of local power production in different areas.
slope cl of each line segment defines the cost of the production arc, and the range of power production ðPlprod P l1 prod Þ defines the arc capacity. 5.1.4. Optimal production with power transmission The solution of the problem is displayed in Fig. 8. The global optimum value of the objective function (16) is 8771.65€. The local
power production in areas 1 and 2 is 7.927 and 10.543 MW h, which is more than their demand. In areas 3 and 4 the power production is 14.53 and 17 MW h. The result demonstrates that it is optimal to generate more power in areas 1 and 2, where the marginal production costs are lower, and transfer power into areas 3 and 4, where marginal production costs are higher. Area 1 transfers power to both areas 3 and 4, and area 2 transfers power only to area 4. The possible transfers were limited by transfer capacity constraints (10 MW h), but none of them were active in this case.
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Fig. 9 shows the speed of solving the network model using NP2 and LP2. The speed is in seconds of CPU time. The solution time for the local production models is not included. Based on Fig. 9, NP2 is clearly faster. For different sized problems, the average CPU time for NP2 was 0.0222 and 0.1640 for LP2. With larger problems, NP2 is about 7 times faster than LP2. By looking at the test runs with 30 nodes and variable number of production arcs, it can be seen that the CPU time of NP2 increases very moderately by the number of arcs, in contrast to LP2. This indicates that it is possible to solve efficiently problems where the local production capacity is versatile and results in many arcs. Instead, the solution time is more sensitive to the number of areas. Both solvers produce the same global optimum value, which validates the network solver solution. Fig. 8. Network flow problem solution.
6. Conclusion and future research 0.8
NPS
0.7
LP2
CPU me (S)
0.6 0.5 0.4 0.3 0.2 0.1
50*50
30*30
40*40
30*25
30*15
30*20
30*5
30*10
20*20
5*5 10*10
4*4
2*2
3*3
1*1
0
Problem size Fig. 9. CPU time.
5.2. Testing solution speed of larger transshipment network flow problems The presented two-phase algorithm is developed for solving large scale hourly CHP planning models efficiently. Instead of solving one large problem, several smaller problems are solved. Solving several smaller problems is normally faster than solving one large problem. When the problem size increases, the network model may grow faster than the local production models, and the solution speed of the network problem becomes most time critical. Therefore we test in the following how fast different size networks models can be solved using the dedicated NP2 network simplex algorithm. We compare the solution times against the efficient LP2 sparse simplex algorithm. Test runs are performed for a set of artificially generated power transmission problems containing from 1 to 50 nodes. The number of local production arcs corresponding to energy production units in each area also varies from 1 to 50. In the following, the different sized problems are referred to n ⁄ m where n stands for the number of areas and m for the number of production arcs. We have two sets of models. The first set is with n = m 2 {1, 2, 3, 4, 5, 10, 20, 30, 40, 50} and the second set is with n = 30 and m 2 {5, 10, 15, 20, 25, 30}. The latter set with 30 areas corresponds roughly to the number of price areas in the EU, and the number of production arcs represents different levels of aggregation of local production capacity. The tests have been run in a 2.5 GHz Intel Core i5 processor under the Windows7 operating system. In order to reduce random variations effect on CPU (Central Processing Unit) time, each run is repeated three times for each problem, and averages of results are reported.
The economic optimization of hourly large scale multi-area combined heat and power (CHP) production in the presence of power transmission network was investigated in this study. The problem was decomposed into local energy production models and a transshipment network flow model. The local energy production models were solved using parametric LP analysis to compute the optimal production costs as a function of power production. Then, the parametric production functions were encoded together with the transmission network into a transshipment network flow model and solved using the special network simplex algorithm to find out the optimal production amounts and power transfers between areas. A small four-area sample problem was solved based on real CHP characteristics to illustrate the method. The developed two-phase optimization method finds the global optimum to the combined CHP production and power transmission problem. The sample problem illustrates that more power should be produced in areas where production cost is low and transmitted to areas where production cost is higher. To test the efficiency of the approach, artificially generated power transmission problems containing from 1 to 50 nodes were solved and compared with regular LP. For bigger problems, the presented method was about 7 times faster than general LP algorithm. The result also shows that the solution time increases more rapidly for the number of areas than for the number of production arcs. This proves the possibility to solve efficiently the problems the local production capacity is versatile and results in many arcs. Therefore, parallel to the increase in problem size, the network model may grow faster in comparison with the local production models, and the efficiency of time for the solution of network problems becomes more critical. The speed advantage makes it possible to solve long-term CHP planning problems with thousands of hourly models efficiently. Future research involves solving the local production models using Power Simplex, which is a family of efficient special algorithms for local CHP problems. Another challenge is applicability of the presented algorithm to develop an efficient optimization method for long-term CHP production and power transmission problems. To obtain global optimum for the long term planning problem with storages and ramp constraints, it is necessary to iteratively solve the hourly models and, based on marginal price for power in different hours, use storages to shift power forward or backward in time. See [13] for one such algorithm. Acknowledgement This research is funded by STEEM – Sustainable Transition of European Energy Market, which is a project aiming at novel solu-
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tions with improved energy efficiency as one of the Aalto Energy Efficiency Research Programs.
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
An optimization method for multi-area combined heat and power production with power transmission network Elnaz Abdollahi ⇑, Haichao Wang, Risto Lahdelma Department of Energy Technology, Aalto University, School of Engineering, P.O. BOX 14100, FI-00076, Aalto, Finland
h i g h l i g h t s Decomposition-based optimization method for CHP production and power transmission problems. Minimization of CHP production cost to meet the local heat demand and identify the optimal power transmission between areas. Solving a small sample problem using the developed models to show the solution approach. Method allows using fast specialized algorithms to solve the sub-problems of multiple areas. Fast solution of hourly problems is useful in solving long-term problems.
a r t i c l e
i n f o
Article history: Received 23 July 2015 Received in revised form 22 December 2015 Accepted 22 January 2016
Keywords: Combined heat and power (CHP) Power transmission Linear programming (LP) Optimization Energy efficiency
a b s t r a c t This paper presents an efficient decomposition-based optimization method to optimize the hourly combined heat and power (CHP) production and power transmission between multiple areas. The combined production and power transmission problem is decomposed into local CHP production models and into a power transmission model. The CHP production models are formulated as linear programming (LP) models and solved using a parametric analysis technique to determine the local production cost as a function of power transmitted into or out from each area. To obtain the overall optimum, the power transmission problem is then formulated in terms of the parametric curves as a network flow problem, and solved using a special network Simplex algorithm. The decomposition method has been tested with different sized artificial problems. The method can be used in situations where it is necessary to solve a large number of hourly production and transmission problems efficiently. As an example, the method can be used as part of long-term planning and simulation of CHP systems in different cities or countries connected by a common power market. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Combined heat and power (CHP) is the most efficient way to deliver power, heating and cooling [1]. Based on some statistic data for 2012, CHP electricity generation for the 28 member states of the EU was 373.3 TW h, and heat production was 854.5 TW h [2]. The importance of CHP in district heating system also has been pointed out [3,4]. In Finland, the share of CHP electricity was about 34% and district heating 74% in 2014 [5]. CHP can utilize different types of fuels such as waste, biomass and also traditional fossil fuels [6,7]. Through the heat utilization, efficiency of CHP can be more than 90% and therefore offers dramatic savings of energy ranging between 15% and 40% in comparison with conventional power plants and heat only boilers [1]. Environmental benefits of ⇑ Corresponding author. Tel.: +358 (0)50 383 5075. E-mail address: [email protected] (E. Abdollahi). http://dx.doi.org/10.1016/j.apenergy.2016.01.067 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
CHP in comparison with other conventional plants are due to reduction in emissions [8]. The role of CHP was investigated in a study based on zero emissions scenario [9]. In 2014, the European Commission presented a new framework for climate and energy policy for 2030. Its targets comprise increasing renewable energy utilization, improving energy efficiency, and reducing greenhouse emissions. The targets depend on economic analysis measuring how to achieve decarburization cost-effectively by 2050 [10]. The optimal operation of local CHP production can be determined by an optimization model. The objective is to minimize the total cost of energy production while satisfying demand and operational constraints. The complexity arises from the coupling of power and heat production to match both heat and power demand. The problem will be more complicated, when power transmission between multiple areas is also considered. As a result, faster methods and solvers are needed for optimizing for large scale energy production problems.
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249
Nomenclature
Abbreviations CHP combined heat and power LP linear programming NP2 efficient network simplex algorithm CHPED combined heat and power economic dispatch MILP mixed integer linear programming CPU time central processing unit time (s) Symbols A set of arcs b additional constraints coupling heat and power production (MW h) C cost (€) ca transmission cost (€/MW h) cj ; pj ; qj production cost (€), power generation (MW h), and heat production (MW h) at characteristic point j ccond condensing power price (€/MW h) heat only boiler price (€/MW h) chob cl marginal cost of production (slope of line segment) (€/MW h) di demand in node i (MW h) destðaÞ destination node of arc a H matric for heat and power production (MW h) J set of extreme characteristic points of all CHP plants in a production area Ju set of extreme characteristic points for CHP plant u 2 U m number of production arcs N set of the nodes (production areas)
The complexity of optimization problems depends strongly on the shape of the objective function and constraints. For example, minimization of convex objective function subject to convex constraints is in general easier than solving non-convex problems. Several models for convex and non-convex problems have been developed to solve combined heat and power economic dispatch (CHPED) problems. For convex CHPED problems, linear models have been proposed and solved by using specialized Simplex algorithms [11–13]. Non-convex CHPED problems may arise from a non-convex characteristic operating region or the need to optimize unit commitment (when to shut down and start up plants). A nonconvex CHPED operating region can be divided into convex subregions and formulated as a mixed integer linear programming (MILP) model, or solved using the Lagrangian relaxation technique [14–18]. To determine the unit commitment of CHP plants, a unit decommitment algorithm was developed and the solution quality was compared with a generic unit decommitment algorithm by using realistic test data [14]. Reference [19] presents a thorough comparison of 37 software tools for different energy applications, ranging from single building models to national energy systems. In particular, the study evaluated the suitability of the tools for integrating renewables into energy systems. Six of the tools were applicable for modeling CHP plants. The BCHP Screening Tool analyzes the combined heating, cooling, and power systems of commercial buildings, excluding large scale heat, power or transport sectors [20]. COMPOSE and EnergyPRO can be used for technoeconomic analysis of single projects [21,22]. BALMOREL can simulate the CHP sector in a multi-national geographical area [23]. EnergyPLAN and SIVAL are tools for national or regional energy systems [24,25]. Among them only EnergyPRO and EnergyPLAN can simulate 100% renewable energy systems. They have been
n orgðaÞ P Q U ua x xj xhob xcond xmax hob xmax cond y ya
number of areas origin node of arc a power demand (MW h) heat demand (MW h) set of CHP plants transmission line capacity (MW h) decision variable to encode convex combination variables used to encode convex combination of operating region heat production of heat only boiler (MW h) power production of condensing power plant (MW h) maximum capacity for heat only boiler production (MW h) maximum capacity condensing power plant production (MW h) power transmission (MW h) power flowing from origin node to destination node (MW h)
Superscripts and subscripts a arc (transmission line between areas) cond condensing power plant production hob heat only boiler production i; k indices for nodes (production areas) in network l line segment index p; q power and heat products prod production trans transmission
used to verify the results of linear programming (LP) models for renewable energy systems with energy storages [26–28]. Table 1 presents an overview of different scale energy systems, which have been solved using different algorithms. A review over research on the CHPED problem was presented in [33]. The high efficiency and profitability of CHP production can be further improved by utilization of power transmission network. Accurate modeling of energy production together with power transmission network in a large scale problem requires a sophisticated optimization model. Efficient solution of CHPED problems is important because a long-term planning model includes thousands of hourly models, and rapid re-optimization is needed when the market situation changes. In this paper, the hourly multi-area CHP production and power transmission problem is formulated as an LP model, but decomposed and solved as separate local CHP production models and an overall model for the power transmission problem. In the local CHP problems, the produced power can be sold to the grid at market price, but heat must be produced to meet the local demand for district heating or specific industrial processes. The local CHP production is modeled as LP problems and solved using a special parametric LP algorithm. This will yield the local production costs as a function of the power production. Then the local production cost curves are encoded as a network flow problem into an overall production and power transmission problem. A specialized primal network simplex algorithm NP2 is used to solve the network flow model. The overall optimization model minimizes the combined production costs in all areas and identifies the optimal balance between local production and power transmissions into or out from each area. The solution technique should be fast enough to solve even large hourly optimization problems with many CHP
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Table 1 Overview of different scale energy systems. Case number
(1)
(2)
(3)
(4)
(5)
Scale Units
Small 1 conventional power 2 CHPs 1 heat only [29]
Small 1 conventional power 3 CHPs 1 heat only [30]
Small 4 conventional power 2 CHPs 1 heat only [31]
Large 13 conventional power units 6 CHPs 5 heat only units [32]
Large 26 conventional power units 12 CHPs 10 heat only units [32]
Proposed by
plants in several areas so that it can be used as a sub-routine for solving long-term planning problems consisting of many hourly models in sequence. Potential use for the algorithm is to solve for example a yearly planning model with 8760 h, or even models for multiple years. In the following, Section 2 presents the overall convex CHP production and power transmission problem formulated as a generic LP model. Section 3 formulates the local CHP production model in terms of extreme characteristic point of CHP plants and presents how the model can be solved using parametric analysis. Section 4 presents the power transmission problem and how the parametric analysis results of the local models can be encoded into an overall network problem. Section 5 presents the numerical results including the solution of a small four-area sample problem, and tests with different size larger generated test problems. The network problems are solved both with a dedicated network solver and compared against an efficient LP solver. Section 6 contains the conclusions and proposes further research. 2. Combined CHP production and power transmission problem The objective of the hourly CHP production and power transmission problem is to minimize the overall operational costs. Assuming that the unit commitment is fixed and the characteristic operating regions of plants are convex, this problem can be defined as a general LP model as follows [34]:
min
X
Ci xi þ
i
XX C ik yik i
ð1Þ
k–i
s:t: Hqi xi ¼ Q i X X yki yik ¼ Pi Hpi xi þ k–i
ð2Þ ð3Þ
k–i
Hpq i xi ¼ b i
ð4Þ
xi ; yik P 0
ð5Þ
Here the indices i and k loop over all heat and power production areas. The decision variables for each area i are represented by vector xi, and corresponding coefficients of objective function are included in vector Ci. The amount of power traded out of area i and transmitted into area k is denoted by decision variables yik with the corresponding transmission costs Cik. The objective function (1) minimizes the production costs (Cixi) and the power transmission costs (Cikyik). The matrices H with superscripts p, q or pq refer to the production of power, heat, or both. Constraints (2) guarantee that heat demand Qi is matched by local heat production in each area. Constraints (3) require that the power demand Pi must be met in the area by produced power plus the power transmitted into the area and subtracted by power transmitted out of the area. Constraints (4) may include additional technology-specific constraints and also constraints for coupling local heat and power production. In the absence of CHP technology, such coupling constraints are non-existent. Transmission losses are not included in this formulation. One way to deal with them is to treat transmission losses as costs due to need of producing additional power to replace the loss. Losses are then considered in the Cik coefficients.
In the following this model is decomposed into separate local CHP production models for each area and an overall model to consider the power transmissions between areas.
3. Local CHP production model In the following, the hourly operation of a convex CHP plant is modeled in terms of extreme characteristic points. Then, an hourly local CHP system model is composed. The local CHP system model can contain an arbitrary number of CHP plants, heat-only boilers and condensing power plants. Finally the parametric analysis technique is presented. Parametric analysis is used to determine how the local CHP system can interact with the other areas through power transmission.
3.1. Convex CHP plant model The operation region of a CHP plant allows it to produce different combinations of heat and power with corresponding production costs (mainly fuel costs). The operating region can be viewed as a surface in 3D space corresponding to coordinates for produced power and heat (p, q), and associated production cost (c). To apply LP modeling, the CHP plant characteristic is assumed to be convex. A typical convex CHP plant is illustrated in Fig. 1. The projection of the 3D surface on the 2D plane (p, q) shows the range in which the production of heat and power can be adjusted. It can be seen that the 2D region is a convex polygon in the (p, q) plane. Convexity of a region means that any line segment connecting two arbitrary points of the region is also entirely inside the region. Also the production cost must be a convex function of the power and heat production. This means that the cost surface is either flat or curves upwards within the region. Many types of CHP plants have convex operating regions and production costs. For non-convex plants, the operating region can be partitioned into convex sub-regions, and MILP can be applied to solve the problem [15]. A convex model
Fig. 1. Feasible operating region of a convex CHP plant.
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can also be assumed if it is possible to run a non-convex CHP plant for fractions of an hour in different modes. The operation of a convex CHP plant is modeled through linear constraints as a convex combination of the extreme points (qi, pi, ci) of the characteristic surface:
C¼
X j2J
j2J u
X q j xj
ð6Þ
j2J u
X xj ¼ 1 j2J u
xj P 0 Here xj are used for expressing the characteristic operating region as a convex combination (weighted average with non-negative weights) of the extreme points, and Ju is the index set for the extreme points of the plant characteristic. The above model formulation allows running the plant anywhere inside the 3D polyhedron spanned by the extreme points. However, minimization of costs always produces an optimal solution which lies on the lower envelope surface [16]. The lower envelope (plotted in Fig 1) corresponds to minimum cost CHP operation for every heat and power production combination, including part load operation. One of the characteristic points corresponds to maximum load operation. Operation at the other characteristic points and in the area between the points corresponds to part load operation. This modeling technique applies also for separate heat and power production units, such as condensing power plants and heat-only boilers. Heat-only units can be modeled using pj = 0 and power-only units with qj = 0. The modeling technique applies also to demand side management components ðpj ; qj ; cj 6 0Þ and heat pumps converting power to heat ðpj P 0; qj 6 0Þ . Ramp constraints for the power production of the plant can be represented as additional constraints
P0 P 6
X pj xj 6 P0 þ Pþ
in the model, it is also possible to modify the characteristic operating region so that parts outside the feasible ramping region are cut off. Fig. 2 illustrates cutting off parts of the characteristic region by ramp constraints. 3.2. Local CHP system model
c j xj
Xu P¼ pj xj Q¼
251
ð7Þ
Fig. 3 illustrates the local CHP system. The system can consist of any number of energy production units, but for simplicity the figure shows only one CHP plant, one condensing power plant and one heat only boiler. Heat production must satisfy the demand exactly, and power production must meet the demand plus the net amount of power transmitted out from the area. The following model minimizes the overall costs of local production in one area:
min C ¼
X cj xj þ ccond xcond þ chob xhob
s:t: X qj xj þ xhob ¼ Q
ð9Þ
j2J
X pj xj þ xcond ¼ Pprod
ð10Þ
j2J
X xj ¼ 1 u 2 U
ð11Þ
j2J u
xj P 0 j 2 J u ; u 2 U max 0 6 xhob 6 xmax hob ; 0 6 xcond 6 xcond
ð12Þ ð13Þ
The objective function (8) minimizes the production costs consisting of fuel costs for the CHP plants (cjxj), condensing power plant (ccondxcond) and heat only boiler (chobxhob). As explained in the previous section, heat-only boilers and condensing power plants could be modeled as special cases of CHP; however for illustrative purposes one of each is represented explicitly in the model. Constraint (9) is the heat balance where Q is the heat demand and variable xhob is heat only boiler production. Constraint (10) is the power balance constraint in which variable Pprod is the total power production and variable xcond is the power production of condens-
j2J u
where P0 is the power production in the previous hour and P and P+ are maximal allowed down and up ramp for power, correspondingly. Similarly, it is possible to consider ramp constraints for heat production. Instead of including the ramp constraints (7) explicitly
Fig. 2. Cutting off parts of the characteristic operating region by ramp constraints.
ð8Þ
j2J
Fig. 3. Schematic of the local energy production problem.
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ing power plant in the area. Constraints (11) and (12) are the convexity constraints for the xj variables of each CHP plant. Constraints (13) are the lower and upper bounds for heat-only boiler and condensing power plant production. Possible ramp constraints are in this representation considered implicitly by modifying the original characteristic regions to respect the ramp constraints.
the supply and demand of some commodity (such as power) at nodes [35]. Balancing is done by transferring specific amounts of the commodity between nodes either directly or via other nodes while respecting capacity limits for transmissions. The network is defined as a tuple (N, A) in which N is the set of nodes and A is the set of directed arcs connecting nodes. The problem is formulated as
3.3. Parametric analysis of the model
min C 0 þ The model (8-13) is solved to get the production costs for every possible power production amount Pprod using parametric LP analysis. Parametric analysis determines the optimal production cost of the CHP system (including CHP plants and separate heat and power units) as a function of power generation in each area, C⁄ = C(Pprod). Because the model is an LP model, the resulting cost function is a piecewise linear convex function. Each line segment corresponds to a basic LP solution. When the basic solution changes, the marginal cost (slope of line segment, see Fig. 6 for examples) changes. The production cost functions can thus be represented as a sequence of (C, Pprod) coordinate pairs, ½ðC 0 ; P 0prod Þ; . . . ; ðC L ; P Lprod Þ l 0 L with P l1 prod 6 P prod . Here P prod and P prod are the minimal and maximal possible power production in the area. Due to degeneration of l basic LP solutions, it is possible that P l1 prod ¼ P prod for some points. Such duplicate points can be removed from the sequence to guarantee strict inequality among P-coordinates. The marginal cost (slope) for line segment l is then
cl ¼ ðC l C l1 Þ=ðPlprod Pl1 prod Þ
ð14Þ
The lines segments determine the cost and capacity of production arcs, used later in the power transmission model. The amount of power transmitted out of or into the area is related to the power production Pprod according to
Pprod ¼ ptrans þ P
ð15Þ
Here P is the power demand within the area and ptrans is the power transmitted out of the area. Transfer into the area is expressed as negative values of ptrans. This variable links the production model with the power transmission model.
X ca ya
ð16Þ
a2A
s:t: X a2AjorgðaÞ¼i
ya
X
ya ¼ di ; i 2 N
ð17Þ
a2AjdestðaÞ¼i
0 6 ya 6 ua ; ða 2 AÞ
ð18Þ
Here the variable ya is the flow through arc a from origin node org (a) to destination node dest(a). The transmission cost is ca and the capacity of transmission line is ua. The objective function (16) minimizes the overall costs of the network and also includes a constant term C0 that represents costs that are independent of transmitted amount. Balance constraints (17) require that the total flow out of the node subtracted by the flow into the node must equal di which can be the source (+) or sink () at the node. The sum of all sources must equal the sum of all sinks to make the model feasible. The optimal power transmission problem can be naturally represented as this kind of network problem. Each area corresponds to a node and each transmission line corresponds to an arc with specified unit cost and capacity. This formulation allows any number of arcs (zero or more) between each pair of nodes. Zero arcs between a pair of nodes represents missing transmission line between two areas. Multiple arcs between a pair of nodes can represent multiple parallel transmission lines between areas, but allows also using a piecewise linear approximation for non-linear (convex) transmission cost for a single transmission line. Multiple arcs are also needed to represent the parametric CHP production cost functions as described in the next section.
4.2. Encoding the combined CHP production and power transmission problem
4. Power transmission model Power transmission between areas allows minimizing the overall production costs, since at each hour, power can be produced in the areas having the lowest marginal production costs and transferred to the areas where production cost is higher. Meanwhile, the transfer capacities and transmission costs should be considered when finding the optimal solution to the problem. Optimal transfer of power between areas can be easily represented as a capacitated minimum cost transshipment network flow problem. Also the piecewise linear optimal production cost functions of local CHP production (derived using parametric analysis in previous section) can be represented as a network flow problem. The idea is to combine the network-encoded production problem with the transmission model into an overall network problem. The advantage of this approach is that very efficient solution algorithms exist for the network problem [35]. In the following, we first define the capacitated transshipment network flow problem and then show how the parametric functions of local CHP production can be encoded into the overall network problem. 4.1. Capacitated transshipment problem The general capacitated transshipment problem is a network flow problem where the objective is to balance at minimum cost
To encode the combined CHP production and power transmission problem as a network problem, we augment the network model for the transmission system with new arcs to represent the production costs. The area nodes are defined as sink nodes with P di = Pi. An artificial source node is created with di = iPi. Then, to represent local production, arcs are created from the artificial source node to the area nodes. The costs and capacities of these production arcs are defined by slopes cl and lengths of the line segments ðP lprod Pl1 prod Þ determined by the parametric analysis. The constant term C0 in the objective function (16) is set equal to the sum of the cost of minimal power production in each area. Encoding of the model is illustrated in Section 5. To see that the two-phase approach produces the same optimum as a single overall model, observe that it represents the costs and constraints of the original problem accurately. Firstly, for the transmission network, the costs and constraints are represented directly in the network model. Secondly, for each local CHP model, parametric analysis extracts the optimal production costs as a function of power production. Because the parametric curves are computed based on CHP system models, all production constraints are respected and the corresponding production costs are computed accurately. Non-optimal operating modes of local CHP systems are excluded from the encoded model, but this does not worsen the overall optimum.
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consumption. All transmission lines between areas have 10 MW h transmission capacity and 1 €/MW h transmission cost.
5. Results In the following, the two-phase solution method is first presented using a small sample problem. Then the performance of solving larger network problems is evaluated using artificially generated test problems of different size. The LP2 linear programming package [11] is used for solving local production models using parametric LP analysis, and the NP2 network simplex algorithm [34] is used to solve the encoded transshipment problem.
5.1. Optimization of a sample problem with 4 areas A small sample problem comprising of 4 areas is used to demonstrate the operation of the developed two-phase solution approach. For each area, structurally the same production facilities consisting of three CHP plants, one condensing power plant, and one heat only boiler are used. The schematic of the production facilities in each area is presented in Fig. 4. The fuel consumed by CHP 1, CHP 2, and the condensing power plant is biomass (wood chips). CHP 3 and the heat only boiler are fed by natural gas.
5.1.1. Input data The characteristic points for the three CHP plants are shown in Fig. 5 [26,27]. The figure shows the characteristics in 2D projection on the (p, q) plane without the cost coordinate. Instead of the cost coordinate, the fuel consumption in each point is given. The cost coordinate is computed by multiplying fuel consumption by fuel price. For the three CHP plants and heat only boiler, real data of a city in southern Finland are used. The condensing power plant parameters represent a typical, modern power plant. For CHP 2 and CHP 3, only two characteristic points are available corresponding maximal and minimal power and heat output. For CHP 1, a third characteristic point is obtained based on analysis of its reallife operating history. Operating regions of all three CHP plants are convex, so they can be represented using the LP model presented Section 3.1. Table 2 lists the fuel prices and maximal efficiencies of all plants in addition to the constant parameters including heat and power demand in each area, and the capacity and cost of the transmission between areas. CHP 1 has a very high maximal efficiency due to using flue gas heat recovery. The Rankine cycle is used in CHP 2 and CHP 3. CHP 3 uses an open-cycle gas turbine with heat recovery boiler [21]. The efficiency of each plant at each characteristic point is the ratio of power and heat production to the fuel
5.1.2. Solution of local CHP system models The local energy production model is solved to determine the optimal production costs in the areas as a function of power production using parametric LP analysis implemented in the LP2 linear programming software. Parametric analysis is performed in each area for power production amounts between 0 and 25 MW h. The results are shown in Fig. 6. For larger heat demand the production cost curves are at higher level. This is reasonable, since more fuel is consumed to produce a larger amount of heat. Interestingly, the production costs as function of produced power initially decrease before starting to increase in each area. This is due to the fact that heat production is needed anyway, and if CHP plants must produce a very small amount of power, they cannot satisfy the heat demand either. Instead, the more expensive heat only boiler is used to produce the some of the heat. When power production is higher, the CHP plants can produce also a greater share of the heat, leading to higher overall efficiency and lower production costs. After the initial decrease in production costs, the production costs increase due to applying more expensive condensing power production. In other words, in area 1, CHP plants cannot satisfy all heat demand at point A (marked in Fig. 6). Instead, a heat only boiler must supply 21 MW h additional heat at higher cost than CHP plants. At point B, the CHP plants are able to produce the necessary amount of heat together with 15.125 MW h electricity and the heat only boiler is not needed. Also up to point C, CHP heat is sufficient, which implies very cost efficient operation. Beyond point C, up to point D, more expensive (less efficient) condensing power is gradually needed. In point D condensing power production is 9.072 MW h. Therefore, the production costs increase more rapidly in the range from C to D. In area 4 with the highest heat demand, CHP plants cover most of the heat and all power demand without condensing power production. Therefore, the production costs decrease in the full range of power production. 5.1.3. Encoded production and transmission model Fig. 7 shows the production and power transmission model encoded as a capacitated transshipment problem. Local production arcs from the artificial source node to the area nodes are shown as dashed arcs, and transmission connections by solid arcs. Each parametric line segment in Fig. 6 corresponds to a production arc from the artificial source node to the area node in Fig. 7. For areas 1–3 there are three production arcs and for area 4 there are two. The
Fig. 4. Sample problem schematic plants in each area.
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CHP 2 – biomass
CHP 1 – biomass
CHP 3 – natural gas
Fig. 5. Characteristic points of CHP plants [26,27].
Table 2 Parameters for local production facilities. Fuel price (€/MW h)
Maximal efficiency
Wood 21 Input parameters
Natural gas 40
Area Values
CHP1 CHP2 0.97 0.85 Q: Heat demand (MW h)
CHP3 0.84 P: Power demand (MW h)
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Heat only 0.89 Transmission cost (€/MW h)
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Area 3, Q=70 (MWh)
Area 2, Q=60 (MWh)
Area 4, Q=80 (MWh)
Fig. 7. Encoding production and power transmission problem as a capacitated transshipment problem.
Fig. 6. Optimal production cost as a function of local power production in different areas.
slope cl of each line segment defines the cost of the production arc, and the range of power production ðPlprod P l1 prod Þ defines the arc capacity. 5.1.4. Optimal production with power transmission The solution of the problem is displayed in Fig. 8. The global optimum value of the objective function (16) is 8771.65€. The local
power production in areas 1 and 2 is 7.927 and 10.543 MW h, which is more than their demand. In areas 3 and 4 the power production is 14.53 and 17 MW h. The result demonstrates that it is optimal to generate more power in areas 1 and 2, where the marginal production costs are lower, and transfer power into areas 3 and 4, where marginal production costs are higher. Area 1 transfers power to both areas 3 and 4, and area 2 transfers power only to area 4. The possible transfers were limited by transfer capacity constraints (10 MW h), but none of them were active in this case.
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Fig. 9 shows the speed of solving the network model using NP2 and LP2. The speed is in seconds of CPU time. The solution time for the local production models is not included. Based on Fig. 9, NP2 is clearly faster. For different sized problems, the average CPU time for NP2 was 0.0222 and 0.1640 for LP2. With larger problems, NP2 is about 7 times faster than LP2. By looking at the test runs with 30 nodes and variable number of production arcs, it can be seen that the CPU time of NP2 increases very moderately by the number of arcs, in contrast to LP2. This indicates that it is possible to solve efficiently problems where the local production capacity is versatile and results in many arcs. Instead, the solution time is more sensitive to the number of areas. Both solvers produce the same global optimum value, which validates the network solver solution. Fig. 8. Network flow problem solution.
6. Conclusion and future research 0.8
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5.2. Testing solution speed of larger transshipment network flow problems The presented two-phase algorithm is developed for solving large scale hourly CHP planning models efficiently. Instead of solving one large problem, several smaller problems are solved. Solving several smaller problems is normally faster than solving one large problem. When the problem size increases, the network model may grow faster than the local production models, and the solution speed of the network problem becomes most time critical. Therefore we test in the following how fast different size networks models can be solved using the dedicated NP2 network simplex algorithm. We compare the solution times against the efficient LP2 sparse simplex algorithm. Test runs are performed for a set of artificially generated power transmission problems containing from 1 to 50 nodes. The number of local production arcs corresponding to energy production units in each area also varies from 1 to 50. In the following, the different sized problems are referred to n ⁄ m where n stands for the number of areas and m for the number of production arcs. We have two sets of models. The first set is with n = m 2 {1, 2, 3, 4, 5, 10, 20, 30, 40, 50} and the second set is with n = 30 and m 2 {5, 10, 15, 20, 25, 30}. The latter set with 30 areas corresponds roughly to the number of price areas in the EU, and the number of production arcs represents different levels of aggregation of local production capacity. The tests have been run in a 2.5 GHz Intel Core i5 processor under the Windows7 operating system. In order to reduce random variations effect on CPU (Central Processing Unit) time, each run is repeated three times for each problem, and averages of results are reported.
The economic optimization of hourly large scale multi-area combined heat and power (CHP) production in the presence of power transmission network was investigated in this study. The problem was decomposed into local energy production models and a transshipment network flow model. The local energy production models were solved using parametric LP analysis to compute the optimal production costs as a function of power production. Then, the parametric production functions were encoded together with the transmission network into a transshipment network flow model and solved using the special network simplex algorithm to find out the optimal production amounts and power transfers between areas. A small four-area sample problem was solved based on real CHP characteristics to illustrate the method. The developed two-phase optimization method finds the global optimum to the combined CHP production and power transmission problem. The sample problem illustrates that more power should be produced in areas where production cost is low and transmitted to areas where production cost is higher. To test the efficiency of the approach, artificially generated power transmission problems containing from 1 to 50 nodes were solved and compared with regular LP. For bigger problems, the presented method was about 7 times faster than general LP algorithm. The result also shows that the solution time increases more rapidly for the number of areas than for the number of production arcs. This proves the possibility to solve efficiently the problems the local production capacity is versatile and results in many arcs. Therefore, parallel to the increase in problem size, the network model may grow faster in comparison with the local production models, and the efficiency of time for the solution of network problems becomes more critical. The speed advantage makes it possible to solve long-term CHP planning problems with thousands of hourly models efficiently. Future research involves solving the local production models using Power Simplex, which is a family of efficient special algorithms for local CHP problems. Another challenge is applicability of the presented algorithm to develop an efficient optimization method for long-term CHP production and power transmission problems. To obtain global optimum for the long term planning problem with storages and ramp constraints, it is necessary to iteratively solve the hourly models and, based on marginal price for power in different hours, use storages to shift power forward or backward in time. See [13] for one such algorithm. Acknowledgement This research is funded by STEEM – Sustainable Transition of European Energy Market, which is a project aiming at novel solu-
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tions with improved energy efficiency as one of the Aalto Energy Efficiency Research Programs.
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