Modeling and Optimization of Renewables: Applying the Energy Hub ...

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conf192a462

Modeling and Optimization of Renewables: Applying the Energy Hub Approach M. Schulze, L. Friedrich, and M. Gautschi

Index Terms—Distributed generation, multi-generation, network flow, power distribution, renewable energy integration.

I. INTRODUCTION Distributed generation is one of the strategies to cope with the rising energy demand and the diversification of energy sources. The quality and security of supply should be further improved or at least maintained at today’s standard. Simulations of the power supply grid, should therefore, help to show deficiencies. The “Vision of Future Energy Networks” project launched at ETH Zurich tries to master this task with an unusual “Energy Hub” concept. To be free from the burdens of forecasts and scenarios, a Greenfield approach is used to approximately forecast future conditions. The focus lies on the geographical distribution of energy generation and consumption, without being bound to power lines. Since energy is not only electricity, all kinds of energy carriers are taken into account for modeling and optimization. A multi-energy modeling framework was established, evaluated and tested under real conditions. II. THE ENERGY HUB CONCEPT The Energy Hub Concept models energy flows of different energy carriers on the macroscopic level. A Hub therefore is defined as a local concentrated set of energy converters and/or storages, see Fig. 1. The dimensions of such a Hub can range from a single household up to an entire city model as a single Manuscript send July 15, 2008. M. Schulze, L. Friedrich and M. Gautschi are with the Swiss Federal Institute of Technology Zurich, Zurich, Physikstrasse 3, 8092 Switzerland (phone: +41-44-632-83-73; fax: +41-44-632-12-02; e-mail: {mschulze/ lukasfr/ gautscma}@ethz.ch).

TABLE I SYMBOL DEFINITION

P L

C S E

Ψ

TC

F R T Φ

TB

THC TSC

Description

Introduction

Hub input power vector Hub output power vector Converter coupling matrix Storage coupling matrix Vector of storage energy derivates System marginal costs Total costs related to P-vector Node input Vector Renewables input vector Feed-in vector System marginal benefits Total benefit related to T-vector Total costs of a single Hub Total costs for all Hubs

From the Energy Hub Concept

Symbol

With the Case Study

Equation Chapter 1 Section 1 Abstract—At the authors institute a novel concept, the socalled Energy Hub, was developed to solve optimal power flow (OPF) problems for integrated energy systems with multiple energy carriers. This paper applies the Energy Hub model on an OPF problem considering multiple renewables in a mixedconsumer area. From the experiences of a case study the following modifications of the general model were derived: the proposed representation focuses on renewables by decomposition of the power flow into the energy exchanged with the grid and the locally produced and consumed energy. In addition, a network flow description for a system of Hubs is presented. An example finally illustrates the feasibility of the model on a single Hub including 5 different energy carriers and 4 conversion elements.

Hub. At a certain moment in time or during a given period the power going in and out of an Energy Hub is considered. Table I shows the acronyms and symbols used to explain the concept. An Energy Hub, is from a mathematical perspective, more or less a two-port network. The power input P supplied from the grid is transformed via storage E and converter elements C to feed the load L , as is shown in Fig. 1, where lower case Greek letters indicate different energy carriers.The relationship between input and output for converter elements only can be obtained with L = C ⋅ P . By adding the storage coupling matrix S one arrives at the complete Energy Hub Model: ⎡P⎤  = C -S L = C⋅P −S⋅E (1) ⎢ Ε ⎥ ⎣ ⎦

[

]

which was first mentioned in [1], the fully developed concept was published in [2].

Fig. 1: The Energy Hub Model

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conf192a462 III. MODELING AND OPTIMIZATION FOR RENEWABLE ENERGY A. Background In the last few years many aspects of the Energy Hub have been investigated. Special focus was placed on the theoretical potential of the concept due to its open modeling framework [3], fewer restrictions on technologies [4] and the wide range of energy carriers [5]. First practical test was a case study with a municipal utility in Switzerland started in the summer of 2007 [6]. In the investigation area are four different energy carrier located: electricity, natural gas, fuel oil and district heating. Electricity is available in the entire area, but the other energy carrier spread specifically on the land use. Commercial areas, light industry, residential areas, a research facility and a large hospital are present within about 2 km2. Depending on their characteristics 11 Hubs were chosen to describe the area. Due to the proximity of the Hubs ohmic losses are disregarded and with the Greenfield approach the focus is put on the local distribution of energy generation and consumption. Some difficulties occurred during the modeling process. The Energy Hub model allows energy flows from input (left) to output (right) and vice versa. With a wide spread renewable energy production one would get feedback from output to input in the most of the Hubs, so this is rather typical than uncommon. The Energy Hub model should not only describe the mathematical coherences, also the illustration is the purpose. The renewable energies should be treated as an input, but they are, unlike the normal P -vector, not unknown variables for the optimization. The power from renewable energy sources is either stochastically (wind, solar) or from an adjustable plant (hydro, bio-gas) given. So why not separating input and output in a general manner if it improves the understanding and simplifies the visualization? B. Adjusting the model Based on the Energy Hub concept, some additional vectors were defined: • T as an output allows the Hub to feed a surplus of energy back into the grid • R comprises all local energy production at the Hub location, mostly regenerative • F the energy flow from the grid to a certain Hub The direction of the energy flow within the Hub is now determined from input to output, from left to right. Important is a change in philosophy, that one now has a network of interconnected nodes, instead of a network of Hubs themselves. Hubs are now connected via the F -vector to a corresponding node. This will be described in chapter V, “System of Hubs and Nodes”, like shown similar in [8]. Notice in Fig. 2 the new elements now part of the concept, and recognize the modified Hub equation as follows:

( L + T ) = C ⋅ ( P + R ) − SE = [C

-S

⎡P + R ⎤ ⎥. ⎣ E ⎦

]⎢

(2)

Fig. 2: Energy Hub with focus on renewable energy

C. Optimization The objective function F ( x ) to be minimized represents a combination of minimizing the cost (3) and maximizing the benefit of each Hub H , with a set of inequalities h ( x ) . Minimize F ( P, T,THC )

(L + T) − C ⋅ ( P + R ) = 0 h ( P, T ) ≤ 0

subject to

(3) (4)

The lower boundary for the input and output vectors is zero, since the direction of the energy flow is fixed. The sum over all physical connections for a certain energy carrier into a Hub enables the definition of upper boundaries for F and P . Note that F could be less than zero, which indicates a power flow from the hub to the grid. For calculating the total costs for all Hubs TSC , the individual Hub costs THC are summed (5). Costs and benefits are partial derivates of the Hub costs (6). N

TSC = ∑ THCi

∀i ∈ H

i =1

THCi = TCi − TBi

Ψ=

∂THC ∂P

(5)

∀i

Φ=

∂THC ∂T

(6)

Cost optimization is mostly the aim, therefore units for TSC , THC , TC and TB are monetary-unit (mu). The energy flow in P and T is always calculated in per-unit (pu), so the system marginal costs Ψ and benefits Φ result in mu/pu. Because the Energy Hub is operated here on a more local scale, at each Hub one assumes identical prices for the energy carriers. Otherwise Ψ and Φ are matrixes with: ⎡ψ , φα ,i =1 " ψ , φα ,i = N ⎤ ⎢ ⎥ Ψ,Φ = ⎢ # % # (7) ⎥ ⎢ψ , φω ,i =1 " ψ , φω ,i = N ⎥ ⎣ ⎦.

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conf192a462 IV. SYSTEM OF HUBS AND NODES In an enclosed region, Energy Hubs H can interact via a system of nodes N . Energy conversion is only possible within the Hubs for a multi-carrier multi-energy system. Between Hubs a network of interconnections is therefore used to transport energy. Not every node is necessarily connected to a Hub, as in (8). Not more than one Hub can be attached to a certain node (9).

H ⊆ N :⇔ ∀i, j ( i, j ∈ H → i, j ∈ N )

(8)

∀N i , j := Hi , j ≤ 1

(9)

Nodes without Hub in (10) are strategically positioned elements of the network. Considered from the Greenfield approach, there is no need for such elements. This is a tribute to the experience in the case study [6], where strategical nodes have been established.

N \ H := {i, j | ( i, j ∈ N ) ∧ ( i, j ∉ H)}

⎡ Fij ,α ⎤ ⎢ ⎥ Fij = ⎢ # ⎥ ⎢ Fij ,ω ⎥ ⎣ ⎦

j > i, ∀E

(12)

Interconnections may have power limitations. These limitations are modeled as upper Fij and lower Fij limit.

Fij ≤ Fij ≤ Fij

(13)

If for a specific energy carrier no network system exists, power is directly fed into the Hub itself and no interconnection flow Fij is defined for this energy carrier. For example, fuel oil has no supply network, but can be fed into a building locally. Therefore no interconnection network for fuel oil is defined, but power is fed directly into Hub.

(10)

A. Definition of Network Flow The network is composed of nodes and interconnections in between. The energy flow from node Ni to Hub Hi equals the net energy flow Fi into the Hub. This net energy flow Fi is characterized by the difference of input power P and feedin power T , an essential part of the Energy Hub model with respect to an increased number of renewables, see also Fig. 2. (11) Fi = Pi − Ti Fig. 3 shows inside the border strip a simplified system of interconnected nodes with Hubs.

Fig. 4: Network flow between two nodes

In Fig. 3 one more node is drawn in. In addition to the network nodes Ni node N x represents the external node. The Hub system is considered as an enclosed region. Everything beyond this region is represented by the external node N x . From this external node, all net power needed by the sum of all Hubs is provided for the network. To every node Ni a connection facing the external node N x can be added. This considers connections for energy carrier without network. Carriers with network mostly have only connections at certain Hubs outwards, like the electrical coupling via transformer stations from the distribution level inside the region to the transmission level outside. Negative power ( FiX < 0 ) indicates that power from the external node is fed into the network. Adding up all nodal flows, a nodal equation (14) can be formulated at every node.

0 = Fi + ∑ Fij + Fix

∀i , j ∈ H; ∀E

(14)

j

B. Network Optimization

Fig. 3: System view of a network of Hubs and nodes

The vector Fij defined in (11) represents the energy exchange between node i and j . In reality two regions are connected through several physical lines for energy carrier E . For the modeling and the optimization, all these lines are represented by only one interconnection between two nodes. The condition j > i in (11) indicates that flows Fij are always directed to higher node numbers (i.e. F12 and not F21 ). A positive flow is defined as the flow from node i to node j , as in Fig. 4.

The goal of the interconnected Energy Hubs simulation is an optimization of the complete system for a time period in the future. This optimization results in a cost minimization problem for every single step in time t . Total system costs TSC ( t ) at time t can be defined as costs for input power P minus rewards for feed-in power T summed over all Hubs.

TSC (t ) = ∑ ( Ψ T (t ) ⋅ Pi (t ) + ΦT (t ) ⋅ Ti (t ) ) = ∑ THCi (t ) (15) i∈H

i∈H

The total system cost of the entire optimization period is then the sum of costs over all times t . A longer period, e.g. a year, will lead to more significant simulation results.

TSC = ∑ TSC (t ) =∑∑ THCi (t ) t

t

i∈H

(16)

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conf192a462 V. EXAMPLE The next step is the implementation of a realistic scenario of the modified Energy Hub model. From an ongoing case study one Hub with several possibilities of energy input-output combinations was chosen. The system, shown in Fig. 5, consists of a wood gasification facility (within R ) connected with a combined heat and power unit (CHP, the lower green symbol) and a purification of wood gas to natural gas (lower, yellow symbol), a burner with natural gas and heating oil as inputs (yellow/ brown), a transformer (upper, green symbol) and storage possibilities (not shown in Fig. 5).

ν Ιng , ν ΙΙng and ν ΙΙΙng represent the dispatch factors at “a” in Fig. 3; ν Ιwg , ν ΙΙwg resp. ν ΙΙΙwg at “b” . They are variable within

defined limits and are optimized in the simulation. The constant efficiency factors are described with the Greek letter η and are fully defined in table III (appendix). k is defined as the natural gas share of the total gas amount (wood gas and natural gas) and is introduced for dealing with only one system matrix C . Hence, 1 − k represents the wood gas amount of the total gas. Both gases can be burnt in the same CHP, with different efficiencies. The simulation is done with MATLAB. The optimization runs with the predefined MATLAB-function fmincon that is minimizing an object function under linear and non-linear constraints. Only a single time-step was simulated, not a period. Therefore storages were not included in the simulation, because they work mostly for multi-period dispatch optimization. More interaction would additionally come from other Hubs in the network; this will be part of future work. The constraints are primarily the hub equation and furthermore the limits of the variable energy forms described in vectors P and T , resp. the dispatch factors. The cost function of the hub is evidence of the object function and reflects the total hub costs. They consist of the external energy resource costs described in P and the benefits of the feedback described in T times their prices: ω

ω

ε =α

ε =α

THC = ∑ψ εT ⋅Pε − ∑ φεT ⋅ Tε Fig. 5: Example configuration modeled as an Energy Hub

Dispatch factors ν are introduced for natural gas at “a” and at “b” for wood gas in Fig. 5. The P -vector represents the inputs from the external sources with electricity, natural gas and fuel oil. The R -vector reflects the internal sources, the renewable energies, with solar and wind electricity, wood gas and heat from the gasification process. The output vectors, T and L , represent the usable energies: electricity, natural gas, heat and district heating for other Hubs. According to the definition of the Energy Hub model, P and T describe the variable vectors that are optimized, and R resp. L the fixed vectors that are predetermined. The couplings and coherencies of the given system are described in the system matrix C which is defined as follows:

⎛ηel −el ⎜ C=⎜ 0 ⎜ 0 ⎝

c12 0 0 ⎞ ⎟ c22 0 0 ⎟ c32 η fo −dh / he ηhe−dh / he ⎟⎠

(17)

From this follows:

THC = ψ αT ⋅ Pα + ψ βT ⋅ Pβ + ψ γT ⋅ Pγ

The simulations are performed with a certain configuration of the hub described in table II. In the first simulation the price of electricity varies, and in the second the price of natural gas varies. The units are fictional, but have realistic relations to each other. Hence they are called per units (pu) and monetary units (mu). This sensitivity analysis should result in hard changes for the input, as long as the efficiencies are kept constant. From the equation system for input, output and conversion, the solution hyperplane reflects the variability of energy flows (ng, wg) inside the Hub. A future version will use non-linear functions instead of constant efficiencies. This changes the shape of the cost-related hyperplane of the solution and yields to a smoothed surface. TABLE II CONFIGURATION OF THE EXAMPLE HUB

c12 = ν Ιng ⋅η ng − el ⋅ k + ν Ιwg ⋅η wg − el ⋅ (1 − k )

Loads (in pu)

c22 = ν ΙΙng ⋅1 ⋅ k + ν ΙΙwg ⋅η wg − ng ⋅ (1 − k ) CHP wg PUR + (ν ΙΙΙwg ⋅η wg − dh / he + ν Ιwg ⋅η wg − dh / he + ν ΙΙ ⋅η wg − dh / he )

⋅(1 − k )

(20)

−φαT ⋅ Tα − φβT ⋅ Tβ − φδT ⋅ Tδ

With the parameters:

ng CHP c32 = (ν ΙΙΙ ⋅ηng − dh / he +ν Ιng ⋅η ng − dh / he ) ⋅ k

(19)

(18)

Prices (in mu/pu)

Inputs (in pu)

Electric

500

Electric

var / 5.7

Wind el.

50

Nat. gas

100

Nat. gas

3.1 / var

Solar el.

50

Dist. heat

200

Fuel oil

2.9

Wood gas

100

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The system is optimized for each price step. The optimized system variables can then be plotted dependent on the varying prices. On the left Figs. 6-8 (a) show the results by varying the electricity price, and on the right Figs. 6-8 (b) by varying the natural gas price.

VI. CONCLUSIONS AND OUTLOOK This paper presented an application of the Energy Hub concept as an modeling framework for OPF problems with integrated energy systems and multiple energy carriers. With special focus on the integration of renewable energy into the distribution level extensions to the Energy Hub model are proposed and illustrated within an example. By assigning separate variables for the power taken from the grid, the locally produced renewable energy, the infeed to the grid and the load demand, the power flow of renewables becomes better apparent. ACKNOWLEDGEMENTS

Fig. 6: Multi-carrier optimal dispatch of the P-vector for (a) the variation of the electricity price and (b) the variation of the natural gas price

The authors would like to thank the Regionalwerke AG Baden and the Agency of Energy - City of Baden for the prosperous cooperation. Also a special thank goes to ABB, Areva, Siemens and the Swiss Federal Office of Energy for the support of the VoFEN-project and finally to all group members and Prof. K. Fröhlich. APPENDIX TABLE III EFFICIENCIES OF THE EXAMPLE HUB

Fig. 7: Distribution of natural gas for (a) the variation of the electricity price and (b) the variation of the natural gas price; the sum is equal to the natural gas curve in Fig. 6

Fig. 8: Distribution of wood gas for (a) the variation of the electricity price and (b) the variation of the natural gas price; the sum is equal to the wood gas curve in Fig. 6

Figure 6 shows discrete changes of the system behavior and one can determine marginal prices for different hub configurations. The ‘step’ behavior arises from the efficiency factors that are assumed as constant, just as they were predicted to be. The fmincon-algorithm took between 50 and 1500 iteration steps for each time step, even with the constant efficiencies. For the variation of the electricity price in Figs. 68 (a), a mean change around 5.7 mu/pu is visible. Up to a price of 5.6 mu/pu 80 pu of natural gas from wood gas purification and 20 pu of natural gas from the natural gas input are used to satisfy the load of 100 pu natural gas. A higher electricity price, beginning at 5.85 pu forces more natural gas to be converted in the CHP. The additional byproduct of heat from the CHP substitutes the fuel oil input. A contrary behavior can be studied in Figs. 6-8 (b), when the natural gas price is varied. Altogether the example confirms the validity of the Energy Hub concept. It is now possible to solve more realistic Hub configurations, like in [2] and [7].

Symbol

Value

Conversion

Element

η el − el

1.00

Electricity to electricity

Transformer

ηng −dh / he

0.90

Burner

η fo − dh / he

0.85

ηwg −dh / he

0.80

Natural gas to district heating or heat Fuel oil to district heating or heat Wood gas to district heating or heat Natural gas to electricity

ηng −el

0.30

ηwg −el

Burner Burner CHP

0.27

Wood gas to electricity

CHP

η

CHP ng − dh / he

0.40

CHP

η

CHP wg − dh / he

0.38

Natural gas to district heating or heat Wood gas to district heating or heat Wood gas to natural gas

ηwg −ng η

0.80

PUR wg − dh / he

0.08

ηhe − dh / he

1.00

Wood gas to district heating or heat Heat to district heating or heat

CHP Purifier Purifier - none-

REFERENCES [1] [2] [3]

[4] [5]

P. Favre-Perrod, M. Geidl, B. Klöckl and G. Koeppel, “A Vision of Future Energy Networks”, presented at the IEEE PES Inaugural Conference and Exposition in Africa, Durban, South Africa, 2005. M. Geidl, “Integrated Modeling and Optimization of Multi-Carrier Energy Systems”, Ph.D. dissertation, ETH Diss. 17141, 2007. B. Klöckl, P. Stricker, and G. Koeppel, “On the properties of stochastic power sources in combination with local energy storage”, CIGRÉ Symposium on Power Systems with Dispersed Generation, Athens, Greece, 13-16 April, 2005. G. Koeppel and M. Korpås, “Increasing the Network In-feed Accuracy of Wind Turbines with Energy Storage Devices”, presented at the 6th International World Energy System Conference, Torino, Italy, 2006. P. Favre-Perrod, A. Hyde, and P. Menke, “A Framework for the study of multi-energy networks”, 2nd SmartGrids Technology Plattform General Assembly, Bad Staffelstein, Germany, November 2007.

conf192a462 [6] [7]

[8]

M. Schulze and A. Hillers, “Energy Hubs für die urbane Energieversorgung“, presented at the 10th Symposium Energieinnovation, Graz, Austria, February 13-15, 2008. A. Hajimiragha, C. Canizares, M. Fowler, M. Geidl and G. Andersson, “Optimal Energy Flow of Integrated Energy Systems with Hydrogen Economy Considerations”, Bulk Power System Dynamics and Control Conference, Charleston, USA, 2007. F. S. Hillier and G. J. Lieberman, “Introduction to Operations Research”, 8th ed., McGraw-Hill, New York, 2005, ch. 2.

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