Aggregation of Demand Response Units in Power System Modelling

Transcription

1 power systems eehlaboratory Antonakopoulos Christos Aggregation of Demand Response Units in Power System Modelling Semester Thesis PSL159 EEH Power Systems Laboratory Swiss Federal Institute of Technology (ETH) Zurich Examiner: Prof. Dr. Göran Andersson Supervisor: Theodor Borsche, Dr. Andreas Ulbig Zurich, August 31, 215

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3 Abstract The integration of Renewable Energy Sources (RES) over the last decades has been increasing at a great extent. In order to ensure that high penetrations scenarios of RES are feasible without curtailing the energy, demand side management must be extended in a smart way which will enable to control several loads accordingly to the supply from RES. Large scale power simulations are considered an important tool in order to assess the feasibility of high RES scenarios in a power system through Demand Response Management (DRM). In this thesis, detailed modelling for several loads and their extensions are provided and analysis is made about the degree of aggregation that is acceptable. Comparisons over several evaluation quantities are made between detailed aggregated and approximating models for different load units. Firstly, an introduction is made to the formulation of our optimization problem and to Power Node Framework [1] which ensures that each unit like conventional generation, hydro pump storages, intermittent generators or controllable demand can be modelled in detail and accurately. Consequently, for each of the aforementioned units Power Node representations are derived and special emphasis is given to controllable demand which consists of electric water heaters, heat pumps and their extensions. Analysis between the different extensions is performed and their differences are described. Sensitivity Analysis is performed in order to investigate how many power nodes in the detailed aggregated modelling are adequate so as to ensure a continuous consumption profile of our load units that accordingly lead to lower total costs in the whole system. Finally, two different approximating models are constructed and are compared with the detailed aggregated models by means of total costs, total curtailed energy and similarity over their power dispatch profile. The clear advantage for the approximating models concerning their computational cost becomes evident through obvious comparisons. iii

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5 Acknowledgements First of all, I would like to thank Prof. Göran Andersson for giving me the opportunity to write my thesis in Power System Laboratory in such an interesting area. Additionally, I wish to thank my supervisors Dr. Andreas Ulbig and Theodor Borsche for all the support during this project. I would like to particularly thank Theodor Borsche for the beneficial conversations we have had above several aspects in this project and for the fact that he has always been available and eager to discuss with me different concepts and proposals about the general area of this thesis. Finally, I wish to express my gratitude to my parents and to all my friends for their understanding and unlimited support throughout this project. v

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7 Contents List of Acronyms ix 1 Introduction Research Goals Structure of thesis Basic Tools Economic Dispatch in Power Systems Economic Dispatch for our optimization problem Power Nodes Modelling Framework Modelling of Power System Units Conventional Generation Further Extensions Cost Function Pumped Hydro Storage Renewable Sources Conventional Load Electric Water Heaters First Implementation Second Implementation Approximating Models Heat Pumps Modelling of a Heat Pump Approximating Models Dispatch Simulator Introduction to Software Environment Full Horizon vs Receding Horizon Optimization Definition of Full Horizon Implementation Results and Comments Water Heaters Sensitivity to aggregation level vii

8 viii CONTENTS Approximating model results Heat Pumps Sensitivity to Aggregation Level Approximating model results Conclusions and Future Research Conclusions Future Research A Reverse Heat Pump 85

9 List of Acronyms DRM ED DC TSO SoC FIT MPC RPC MIP PV RES VPP WH HP Demand Response Management Economic Dispatch Direct Current Transmission System Operator State of Charge Feed in Tarrif Model Predictive Control Receding Horizon Control Mixed Integer Program Photovoltaic Renewable Energy Sources Virtual Power Plant Water Heaters Heat Pumps ix

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11 Chapter 1 Introduction In the last decades, more and more emerging issues pose challenge to the traditional power system operation. In order to deal with the numerous threats in our climate, several attempts are made so as to reduce green house emissions. It is more than evident that renewable sources are going to play an even more important role as it is planned to replace fossil fuels at a great extent. Several studies have pointed out that from the added potentials of wind and solar energy in all countries it is sufficient to meet the world s demand [2]. One important question that arises from this situation is whether the whole grid can efficiently deal with this new situation in matters of supply and stability. The increasing integration of RES in a conventional power system is a great challenge that more and more people today are starting to understand. The main characteristic of RES is their intermittency which practically concerns their variable output over time. Even if there has been a great progress in forecasting, it is an undeniable fact that forecasts will never become too accurate and as a result the output can never become known in advance. On the contrary, the conventional power system includes both controllable generation units and inflexible demand. The challenge of Transmission System Operators in each country is to ensure that demand matches supply at every instant. To follow this challenge, there are different options depending on the flexibility that one power systems presents. 1. Expansion of Transmission Grid 2. Promotion of construction of additional storage power plants 3. Exploitation of further flexibilities inside a power system While the first two options demand high costs and as a result are considered cost-ineffective, the third option seems the most efficient way. But what does really flexibility mean in a system? Flexibility includes, fast 1

12 2 CHAPTER 1. INTRODUCTION ramping generation units like gas turbines that are able to ramp up/down their output quickly enough in order to ensure that the production is equal to consumption, curtailment of RES which means that in extreme occasions TSO can curtail an amount of RES in case of high RES penetration and finally demand-side management [3]. The latter nowadays is drawing increasing attention and can be defined as any resource that has the capability to change or reduce the electricity consumption at a given time. The mode to change the electricity consumption can be instantaneous or pre-scheduled. Since DR is a demand side resource, in contrast to supply side resource, the key players of DR resources are those who consume, not supply, electricity. Typically, they are represented by residential, commercial and industrial customers of electricity. DR is becoming an integral part of the power system and market operational practice. Application of a DR program can provide better manageability to system operators, optimizing their position, and maximizing the revenue opportunities for DR providers. The inclusion of DR in conjunction with renewable energy, distributed generation will provide benefits to optimize the use of these resources and as a conclusion improve the efficiency and stability of the system operation. Further research has tried to search for new strategies for a power system to work efficiently while there is a high penetration of RES [4]. More precisely, several methods have been proposed in order to coordinate efficiently with minimum costs conventional generation units, intermittent units and storage devices at the same time. In [5], a multi-step optimization was proposed whose goal was to maximize the renewable infeeds and minimize the use of fast-ramping conventional units. The most important factor which enables us to search for different optimization techniques is power system simulations. Through a detailed simulation it can be concluded whether a power system is flexible enough to accept a high RES penetration. For those simulations a special area of control, which is called Model Predictive Control [6], has recently emerged and is considered as the most appropriate way of controlling a power system and assessing the degree of its flexibility. Philip Jonas in his master thesis [5], used MPC in order to coordinate the dispatch of controllable generation units, storage devices and flexible loads with ultimate goal to deal with the intermittency of RES. Furthermore there have been several other attempts [4] that have used MPC having as common characteristic that all their simulation had a large computational effort due their large scale power system models. As a result, in order to find the optimal way to coordinate the dispatch of each power system first, the optimal tool must be found which reduces computational cost while at the same time preserves a certain degree of desired accuracy. To the knowledge of the author, Deml was the first to investigate several ways to approximate different units in order to achieve the minimum computational burden. His master thesis [7] has given the

13 1.1. RESEARCH GOALS 3 author of this semester thesis the appropriate motives in order to reply to the question of what degree of aggregation is acceptable in order to obtain desired analysis insights for optimal power system simulations. 1.1 Research Goals This semester thesis focuses completely on Asset Aggregation and more specifically deals with units that appear in the consumer side. Aggregation in production side was analysed in a deep insight in Stephan s master thesis. The present thesis uses several concepts which are already referred and plans to extend them in order to investigate several ways of aggregation in power system modelling where several load units are combined into one aggregated unit. It uses the theoretical concepts of the Power Nodes Framework which was created in Power System Laboratory [1]. At first, models are constructed which consider one Power Node of each asset. In addition those models are converted to aggregated models where the same asset consists of several similar units. The sensitivity to aggregation level is examined and conclusions about the degree of aggregation that is needed are presented. Finally, approximating models are used and extended for several possible loads. The latter are compared to the detailed aggregated models and conclusions arise regarding their accuracy and their computational cost. The aforementioned models focus mainly on electric water heaters and heat pumps which are the 2 basic categories of loads that can lead to efficient demand side management. Most of the loads can be modelled based on the principles of those two groups. All models are used and extended in order to investigate several possibilities of optimal usage from the TSO whose goal is to keep curtailment of RES at minimum level in order to achieve the greatest economic efficiency. Finally, in this thesis Model Predictive Control [8] is used and a predictive economic dispatch algorithm is utilized as an optimization strategy that ensures that costs are always the minimum possible. The power grid is considered to be a single-bus electricity network where power imbalances are immediately balanced through appropriate constraints without taking into consideration any losses since, as stated above, goal of this thesis is to investigate how detailed the simulation ought to be in order to ensure the adequate degree of accuracy. Furthermore, the prediction of the available load demand and renewable infeeds are assumed to be perfect or in other words the optimization is considered to be deterministic with accurate inputs. 1.2 Structure of thesis The structure of the semester thesis is the following:

14 4 CHAPTER 1. INTRODUCTION Chapter 2: In Chapter 2, all the tools that are used for power system simulations are analysed in depth. Several information about the formulation of optimisation problem is given and more precisely the economic dispatch in power systems is presented. In addition, the Power Nodes Framework which is an indispensable part of this thesis is analysed. Chapter 3: In Chapter 3, all available assets that are used in this thesis are analysed. Several detailed models for controllable generation units, storage technologies, controllable renewable generation and demand response units are constructed. Special emphasis is given in the load units where several different models both for water heaters and heat pumps are presented. Those models include both aggregated models with detailed design and approximating models with several assumptions when that is possible. Chapter 4: In Chapter 4, the dispatch simulator that is used is presented. Principles of both full horizon implementation and receding horizon implementation are presented and their differences are defined clearly. Furthermore, several parameters such as the marginal costs of the used assets and the factors of the optimization problem are described and the evaluated quantities of all simulations are defined. Chapter 5 In Chapter 5, the results of all simulations are presented and comments about them are presented. All implications that refer to aggregation of demand response units are analysed. Comparisons between detailed aggregation methods and approximating models are presented. Chapter 6 In Chapter 6, the most important conclusions of this thesis are presented and further concepts for future research in this area are proposed.

15 Chapter 2 Basic Tools 2.1 Economic Dispatch in Power Systems Economic dispatch is the process which coordinates both efficiently and profitably the usage of several assets in a power system such as conventional generation units like a coal-fired power plant, intermittent generator sources like a wind turbine, storage units like batteries and load units which can be either flexible or not. Ultimate goal of the economic dispatch is to balance at every instance supply with demand using the most efficient and cost-effective assets of the power system. Based on [5], the economic dispatch problem for a copperplate model as presented in this thesis could be implemented through optimal discrete control as follows: minimize u N f(x k, u k, ω k ) (2.1) k= subject to x k+1 = A x k + B u k (2.2) g(x k, u k, u k+1, ω k ) = (2.3) h(x k, u k, u k+1, ω k ) (2.4) x = x init (2.5) k = 1,..., N (2.6) where x R N+1, u R N and ω R N. This optimization problem is solved for a finite prediction horizon N due to various aspects. This implementation offers high flexibility, due to the look-ahead strategy, since storages such as batteries or hydro power plants can be operated optimally. More specifically by knowing in advance when there will be an excess of renewable sources, storage plants can be operated in the most efficient way to ensure that RES will not be curtailed in case of a surplus. In addition all ramping constraints of conventional generation 5

16 6 CHAPTER 2. BASIC TOOLS units can be depicted and observed more easily during a longer period. This method is defined as Model Predictive Control (MPC) and includes a linear dynamic model that describes in each step k the route of state x k. The initial step of the state is x and is equal to a given initial condition x init. Goal of the MPC is to search for the optimal decision variables u subject to both the equality and inequality constraints that are given in order to minimize the objective function. In this part, it must be added that the above mentioned problem refers to the Receding Horizon Control in which the first control inputs are applied to the system in order to add a feedback on our system and to avoid optimize over the whole horizon. A distinction between Receding Horizon Control and Full Horizon Implementation is made in Chapter 4. The aforementioned optimization problem in relation to the power system can be described as follows: 1. Variable x k defines in each step k the storage level of a hydro plant or a battery. By adding all states of the power system for a prediction horizon N, input vector x is obtained. 2. Variable u k defines in each step k all the decision variables of the power unit dispatch including the intermittent renewable sources whether there is the option of curtailment. 3. Variable ω k represents all the disturbances that may arise either in forecasts of renewable sources or load demand. This variable enables to take into account the stochastic and variable over time nature of the renewable sources since forecasts can never be 1% accurate before hand. 4. Equation (2.3) embodies an additional set of equality constraints into a power system. The most important equality constraint is the grid balance into a power system which ensures that total demand is equal to total supply at every time step k. 5. Equation (2.4) includes the set of inequality constraints into a power system. For instance, all generation units are strictly bound in concerns of their generation output at each step k based on their total capacity. Through the aforementioned variables in each step k, the dynamic (2.2) which represents the state of the storage assets in a power system is updated and this is regarded as an extra constraint in the whole problem. MPC desires to minimize the function f(x k, u k, ω k ) by means of reducing the total costs at a minimum level through efficient usage of every asset in step k. Total costs include generation costs, curtailment costs and load shedding costs.

17 2.2. ECONOMIC DISPATCH FOR OUR OPTIMIZATION PROBLEM Economic Dispatch for our optimization problem The optimization that is applied in this thesis focuses on the detailed modelling of assets in the demand side. Special variables are needed which are able to depict the turning on of a particular load unit like an electric water heater. In other words, variables which can account for the non-linear effects of those units are indispensable for the detailed modelling in this thesis. Our problem is extended by means of a set of discrete variables which can account for all non-linear effects in the power system. This kind of optimization is defined as Mixed Integer Optimization (MIP) since in the problem formulation both continuous and discrete variables are included and can be represented as follows: min z f(z) (2.7) s.t z Z (2.8) where z = {z c, z d } and z c R, z d N. The domain Z can be described as the feasible set that entails a set of inequality constraints and a set of equality constraints. The formulation of the problem can accordingly be represented as: minimize z f(z) (2.9) subject to A ineq,c z b ineq,c (2.1) A ineq,d z b ineq,d (2.11) A eq,c z = b eq,c (2.12) A eq,d z = b eq,d (2.13) where A ieq,c R m n, A ieq,d R m n, A eq,c R m n, A ieq,d R m n and b ineq,c R m, b ineq,d R m, b eq,c R m, b eq,d R m. An important characteristic of optimization problems in economic dispatch problems is convexity. Many complex problems can be solved accurately with low computational cost when convexity is exploited. In (2.7), f(z) represents the total costs of the final problem. Total costs can be usually modelled as a quadratic cost function, where the quadratic costs mainly refer to conventional generation units. As a result, (2.7) can be rewritten as: f(z) = z T Hz + Qz + c (2.14) where H R n n, Q R n and c is constant which generally represents the fixed costs of all assets and does not influence the solution of the problem

18 8 CHAPTER 2. BASIC TOOLS since it just shifts the curve so for the following part of this thesis will be neglected. The variables and parameters that are used in the problem formulation will be further explained in Chapter Power Nodes Modelling Framework Power Nodes Modelling Framework is an innovative concept which is based on the idea that any power source or sink connected to the electric power system requires the conversion of some form of energy into electric power and vice versa. It allows a system-level consideration of power grids which include controlled and non-controlled generators, energy storages and flexible or not flexible loads. Every Power Node can be considered as a storage unit with storage capacity C and normalized storage level x 1. Figure 2.1 illustrates how the Power Node is embedded between the demand & supply process domain on the left side and the grid domain on the right side. Figure 2.1: Visualization of three domain concept [1] On the demand & supply side the provided and demanded energies are lumped into an external process defined as ξ. When energy is demanded it is denoted by ξ <, while a supply of energy is indicated by ξ >. The power generation is described by the variable u gen with efficiency η gen, while u load describes the electrical power grid injection from the consumption respectively with efficiency η load. An additional variable ω is used to represent enforced energy losses such as load shedding or curtailment of intermittent generators when there is no alternative measure to ensure security of supply and grid stability. In case C > holds true, a decoupling between the external process ξ and the two grid-related exchanges u gen and u load takes place. Finally, variable v can determine the storage losses of the Power

19 2.3. POWER NODES MODELLING FRAMEWORK 9 Node which are state dependent and positive as v(x). Through the latter term, internal dependencies can be taken into account which refer to heat storage loads and their relation with the environment. The aforementioned described Power Node can be represented by the following Figure 2.2. Figure 2.2: Visualization of a single Power Node [1] Finally, the dynamic equation of one Power Node i N = {1,..., N} in the continuous field based on the explained power flows can be represented as follows: C ẋ = η load u load η 1 gen u gen + ξ ω v (2.15) s.t. x min x x max 1 (2.16a) u min gen u gen u max gen (2.16b) u min load u load u max load (2.16c) η ω (2.16d) η ω (2.16e) v (2.16f) The above-mentioned constraints can be interpreted as follows: Constraints (2.16a) show that the state of charge (SoC) of one storage Power Node is bound. Constraints (2.16b) & (2.16c) imply that the grid power injections are non-negative and bound.

20 1 CHAPTER 2. BASIC TOOLS Constraints (2.16d) demand that the external process must have the same sign with the curtailment ω. Constraints (2.16e) define that the enforced losses can not exceed the supply/demand process. Constraints (2.16f) imply that storage losses are non-negative. The Power Node Framework is a rather flexible tool due to its ability to be extended depending every time on the asset that is needed to be represented. For instance, ramping constraints can be added which refer to the difference between two consecutive steps for a conventional generation unit that might need predefined time to increase or reduce its output by a certain level. In the end of this chapter, it will be extended in that way so that the flexible load units can be controlled. The next step of our implementation necessitates the conversion of the aforementioned equation to discrete time since all application are controlled through digital controllers. As a result, optimal control is applied in digital systems and that enables at the same time the detailed modelling of the load demand response units in this thesis. Based on the equations (2.16) after the discretization of the system the new equations can be summarized as follows: C x k+1 = C x k + T (η load u load,k η 1 gen u gen,k + ξ k ω k v k ) (2.17) s.t. x min x k x max 1 (2.18a) u min gen u gen,k u max gen (2.18b) u min load u load,k u max load (2.18c) η k ω k (2.18d) η k ω k (2.18e) v k (2.18f) where T refers to the sampling time of the optimization problem. In this thesis, T is considered equal to 1 since one step is equivalent to one hour. In addition, storage losses will be neglected since it is not on the scope of this thesis to give a deep insight into concepts related to losses and this explains also the choice of 1-bus system neglecting all the transmission losses. Finally, the Power Node Framework must be extended in order to account for the non-linear effects in the demand side which can facilitate the detailed modelling of the corresponding assets. Since, discrete variables are being used

21 2.3. POWER NODES MODELLING FRAMEWORK 11 in order to represent the non-linear effects of the assets some modifications must me made which result in the following equations: P gen = P gen (U gen, d) (2.19) P load = P load (U load, d) (2.2) where U gen R p, U load R l and d R k with p, k, l, N. The latter equations refer to the power injections to the grid side so in case the equations that refer to the Power Node side are looked into, efficiencies η gen and η load respectively must also be included.

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23 Chapter 3 Modelling of Power System Units In this chapter every asset of the overall portfolio will be modelled. Detailed modelling will be performed for the demand response units which is the part of special interest in this thesis. Overall, equations, constraints and cost functions will be derived for conventional generators, a hydro pump storage units, controllable intermittent generators, conventional load and various demand response units. All equations are based on the Power Node Framework and in the extensions that have been made in the end of the previous chapter. 3.1 Conventional Generation Conventional Generation plays an important role in this thesis by means of flexibility since it must be able to supply adequate power when there is a shortage of renewable sources and at the same time the hydro pump storage level is low. As a result, a nuclear power plant or a coal-fired plant that need both an amount of predefined hours to reach their maximum/minimum output would not be a realistic choice. On the contrary, a gas fired power plant will be selected for the power node of conventional generation that is flexible enough and is considered as a peaking power plant, able to ramp up/down in the time that is needed to ensure that balance between supply and demand in the grid will always hold. For this Power Node, constraints which refer to the maximum/minimum output of the units will be formulated. Furthermore, ramping constraints will be implemented which restrict the change of the power output between time steps k and k + 1 by a certain limit even on the occasion of a flexible unit. The above mentioned constraints can be represented respectively through the following equations: u gen u max gen (3.1) 13

24 14 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS Further Extensions u k+1 gen u k gen u max gen (3.2) In real life cases, flexible conventional generators like a peaking gas power plant are bound to several additional constraints like minimum generation output in case the asset is on, minimum/maximum down-time & minimum/maximum up-time which pose restrictions to the number of the openings and closing of the asset per period. Those constraints can be modelled through additional constraints and we refer the readers of this thesis to [7]. In order to show that binary variables can also be included in conventional generation units, we decide to depict the minimum generation output constraint of the peaking power plant. This constraint implies that the asset during operation demands to have a minimum limit in its output range. This constraint can be represented through the grid injection of our asset as follows: P gen,k = P gen (u gen, d k ) = P min gen d k + u gen,k (3.3) where d k represents the on state of the asset (d k = 1) and Pgen min is the minimum injection the asset can offer during the on state. Due to the minimum generation output constraint, u max gen must be modified and reduced respectively as follows: u max gen = P max gen P min gen (3.4) where Pgen max is the maximum power injection from the asset that can be provided. Equation (3.1) can be additionally rewritten as a linear constraint through variable d k in order to relate the two decision variables d k, u gen,k as follows: u gen,k u max gen d k (3.5) which ensures that the asset will draw the generation output to during the off state. As a final remark, it must be stated that a modelling which includes the direct multiplication of two decision variables like d k u gen,k is avoided because it increases the computational cost and makes several times the problem infeasible because the constraint or corresponding term loses linearity. Additional information about that kind of constraints can be found in [5] Cost Function Total costs for a conventional generation unit are generally divided in fixed costs and variable or marginal costs. The first refer mainly to investments

25 3.1. CONVENTIONAL GENERATION 15 costs, operation & maintenance costs and asset depreciation costs. On the contrary, variable costs are related to the operation of the conventional generation unit and generally embody mainly fuel costs and additionally CO 2 emission costs. As also stated in previous chapters, convexity is an important characteristic for economic dispatch problem that facilitates the solution of real-case complex and large-scale problems. One of the most well known efficient techniques that exploit at the same time convexity is the representation of the conventional generation costs through a quadratic cost function. It is widely used in the domain of operation research in several optimization problems as well as in power system modelling due to its nice mathematical properties. A quadratic cost function for a conventional generation unit can stated as follows: f(p gen ) = ap 2 gen + bp gen + c (3.6) Obviously it can shown that the second derivative of the quadratic cost function is equal to: f (P gen ) = 2a (3.7) which ensures that convexity is existent since the quadratic cost factor a. The above mentioned quadratic cost function in case we also include the minimum generation output constraint is formulated as follows: f(u gen, d k ) = a(u gen + d k P min gen ) 2 + b(u gen + d k P min gen ) + c (3.8) where parameter c represents the fixed costs that just shift the objective function and do not influence the final solution. Furthermore, it must be stated that in the latter equation the start up costs of the present asset are neglected. Generally, during the opening of a thermal plant there is an additional contribution to the total costs. We define this contribution as start up costs and generally they are related to the fuel consumption that one unit performs before being able to operate or differently refers to the necessary amount of energy in order to precondition the system. In this thesis, we assume that start up costs play a minor role to the total costs so we decide to neglect them. Finally, this thesis is focusing on aggregation in demand response units so detailed aggregating modelling is not performed for generation assets. We assume that in our total power portfolio there is only one power node bound with constraints that enable this unit to provide the necessary energy when needed in order to maintain the balance in the power system.

26 16 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS 3.2 Pumped Hydro Storage Pumped hydro plants have been for several years the most mature and cost efficient technology for storing electricity in a large scale. Batteries for example can not be compared with hydro storages since their costs with respect to energy capacity are much higher so their storage capability is restricted. Currently more that 25 hydro pump storages are operating worldwide which account for more than 95% installed storage capacity. During operation, they can either pump energy from a lower to higher basin in order to store the energy or turbine energy from the upper to the lower energy if a general shortage of energy appears. The main features of hydro pump storage plants that make them favourable in use can be summed as follows [9]: 1. High cycle efficiencies, around 7% 85% 2. Large power ratings in the range of 1 1MW 3. Low capital costs 4. Long lifetime renders them secure investments with low risks 5. Low ramping constraints enable them as flexible units to deal with the intermittency of renewable sources Based on the power node framework, the normalized state of charge of a pumped hydro storage (change of level of the water) can be represented by a state variable x k which changes at each time step k. The state dynamics equation of a standard pump hydro storage power plant is given by: x k+1 = x k + T C 1 (η load u load,k η 1 genu gen,k ) (3.9) where C symbolizes the total asset s storage capacity and T the sampling time in the discreet domain respectively. Variable ω k is neglected since natural inflows are neglected and in addition variable ξ k is assumed since this kind of asset can not be characterized by any externally driven process. In the latter equation, u gen,k and u load,k are the decision variables of the present asset and refer to the turbining and to the pumping of the water accordingly. They are bound to minimum and maximum levels depending on their generation/pumping capabilities so the corresponding constraints are: u min gen u gen,k u max gen (3.1) u min load u load,k u max load (3.11) In addition, in several cases an initial constraint is added in order to account for the initial level of the upper basin as well as a final constraint

27 3.3. RENEWABLE SOURCES 17 of the level of water in order to restrict the amount of water consumption during a defined period. Those two constraints can be described as follows: x = x init (3.12) x Nopt = x final (3.13) In order to achieve additional accuracy in the representation of a hydro pump storage, the dependency of the efficiency on the generation output must be included. In this work, this step is not performed since it is beyond of the scope of its final goals. 3.3 Renewable Sources Two further power nodes are modelled which refer to wind and solar energy. In this thesis, renewable sources are not modelled as negative loads as in most traditional ED optimization problems. The controller has the opportunity to curtail renewable infeeds when there is a high penetration of wind or solar energy respectively. This action is penalized in order to be selected only as a last resort since curtailment of free energy coming from renewable sources is considered as an inefficient action and consequently is avoided by all means possible. The variable ω k is used to incorporate this possibility. Since ξ stands for the renewable generator sources it is mandatory according to the power node framework constraint that ω. In addition, the renewable infeeds are given supply profiles and can be described by the following equation: ξ k = ξ drv,k (3.14) Finally, for the case of these assets a cost function must me found. Due to the abundant amounts of wind & solar energy, marginal costs are assumed to be zero since there are no existent fuel costs. The most important contribution in the total costs for this kind of asset arises from curtailment costs. There are several renumeration schemes for the RES, however in most of the countries, like Germany which also during the last decade has increased its RES potential more than any other country in Europe, the feed in tariff (FIT) mechanism dominates. According to this mechanism the owner, of the asset feeds in the grid with energy without depending on the market or balance between supply and demand in the grid. As a result, the transmission system operator of each country is obliged to pay for the amount of energy that comes into the grid. So curtailed energy must be penalized with high costs in this simulation. We assume that curtailment costs define the cost function of renewable sources and can be represented as linear costs based on the following equation:

28 18 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS f k (ω k ) = a ω k (3.15) where a is the linear factor of the curtailment costs. Furthermore the dynamic equation based on the power node framework is given by: ξ k ω k = u gen,k (3.16) where ω k, u gen,k are the decision variables for this power node. Finally, for the aforementioned decision variables we have the following constraints that keep them bound: 3.4 Conventional Load ω k ξ k (3.17) u gen,k ξ k (3.18) In this thesis, it is assumed that the whole load is aggregated into a single power node. Generally, there is the possibility that the load can be shedded through variable ω k but in this thesis we will assume that the controller is not allowed to shed any amount of the load, so ω k =. In addition, this power node does not include any storage capability so no energy can be stored (C = ). Finally, the conventional load is assumed to be derived from a perfect forecast, so it is well known in advance and externally driven. As a result, it can be defined as only an external process with negative sign ξ k = ξ drv,k as follows: P load,k = ξ drv,k (3.19) 3.5 Electric Water Heaters There are several demand response units which can be controlled by different ways in order to optimally deal with high penetration of RES into the grid and as a result keep curtailment costs at lowest possible level. Through demand side management, the maximum amount of energy coming from RES with the current structure of the power system can be searched in order to be used in the total balance of the system. The most common devices that can take part in demand side management are residential devices with inertial storage Ventilation and Air-conditioning (HVAC), heating devices or Plug in Hybrid Electric Vehicles. Through the property of inertial storage of those units, the total consumption in the power system can increase/dicrease just by setting when those loads will be active/inactive. In order to give more motives for residential customers to take part in the demand response, this

29 3.5. ELECTRIC WATER HEATERS 19 shift in consumption is preferable to be performed without influencing the end-user s convenience. The idea, on which demand side managements is based, is the existence of Virtual Power Plant (VPP) in which several units are aggregated and controlled. This leads to an adequate alteration of load demand in order to ensure high penetration of RES and maximum possible avoidance of curtailed renewable energy. Heating devices seem to contain the maximum potential for altering the total consumption especially in north countries in Europe where cold temperatures dominate like Germany. One of most well-known heating devices that will be modelled in detail in this thesis is electric water heaters (WH). Since several years, there is already a well-known technique named Pulse-Width-Modulation (PWM) which controls several clusters of WH that are all lumped into the same bus of the power system [1]. This bus contains all the WH of a neighbourhood and through a signal can activate them all together at the same time in order to start the heating process. This technique was particularly used for shifting the consumption from high levels during the day to low levels during the night. Basic disadvantage is that this alteration is not continuous since all the WH are receiving the same signal at the same time. In addition, there is no bidirectional communication to know the state of WH since high costs for telecommunication infrastructure is needed. The situation, in the last decade has changed since through the day even at noon hours a surplus of energy can be observed coming mainly from the operation of PV and additionally by wind turbines so new challenges are appearing. The goal is through a VPP to control all the aggregated clusters of WH with a continuous way. Modelling In order to model the electric water heaters, no state variable x will be used since state estimation for those units is not always a straightforward process. Two different implementations will be presented both of which will use only constraints through binary variables d k that will define either the event of the turning on of the WH or the on-state. Both implementations in this thesis are modelled based on a contract that demands an opening of T on hours in a given time frame T horizon First Implementation In the first implementation, it is assumed that power node of WH must consume energy constantly for T on in a given time frame Tboiler horizon. The binary variable d k will symbolize the event of the turning on of the unit and will not be a sign for the on/off state of the unit. In the reverse case, several combinations between minimum and maximum up time constraints should be performed that lead to high computational costs due to the complex

30 2 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS mixed-integer optimization structure. Hence in this implementation, d k = 1 means that in step k the unit will turn on and will consume energy consecutively for the following N on steps. As a result, each step the state of the demand unit depends on the previous N on 1 steps. The water heater s consumption can be represented accurately by k N on+1 P load,k = u avg load d n (3.2) where u avg load is the constant value that the constant value of energy that the power node will consume consecutively for T on steps. This implementation can be shown clearly through a real-case scenario by Figure 3.1. n=k

31 3.5. ELECTRIC WATER HEATERS 21 1 Water Heater Opening Binary Time step(1 hour) Water Heater Power Power(MWh) Time step(1 hour) Figure 3.1: Arbitrary Opening and Power of WH Power Node with mandatory consecutive consumption Here it is assumed that the WH opens in time step 11 with T on = 4, Tboiler horizon = 24 and u avg load = 1MW. Through this implementation, T on is implicitly included into the WH load equation while at the same time the complexity of the problem does not increase since we do not add a large number of new constraints. Compared to the occasion in which both minimum and maximum up time constraints, less computational cost and more simplicity in our code is achieved. Based on the contract, there must be exactly one opening of the WH power node per Tboiler horizon. In order to achieve this, an additional constraint must be added which ensures that only one opening will take place every. At this point, based on [7] two different approaches for the representation of this constraint will be explained and their differences will be developed. Tboiler horizon

32 22 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS Rolling Window Implementation In this approach, there is window with length equal to N horizon that rolls at each time step k. It manages to cover all possible scenarios in the optimization problem and can be represented by the following equation: k+n horizon 1 n=k k = 1,, N opt N horizon + 1 This approach can be illustrated by Figure 3.2. d n = 1 (3.21) Figure 3.2: Visualization of rolling window approach [7] The basic advantage of this approach is that it achieves accuracy since it covers all the possible cases during the optimization including the boundary limits of the window. On the contrary, it demands more computational time since for every power node of WH that is added N opt N horizon constraints are needed, regarding that generally N opt N horizon primarily for a full horizon implementation which will be mainly used in this thesis. Fixed Window Approach This approach is a simplification of the rolling window approach, by means of reducing enormously the number of constraints which respectively leads to a dicrease of the total computational cost of the optimization economic dispatch problem. At this case, the window has accordingly a fixed length N horizon but does not roll over this period so it does not consider all the possible occasions. This approach can be described as follows:

33 3.5. ELECTRIC WATER HEATERS 23 k+n horizon 1 n=k d n = 1 (3.22) k { mod(1, N horizon ) = 1}, k = 1,, N opt N horizon + 1 This approach can be illustrated respectively by Figure 3.3. Figure 3.3: Visualization of fixed window approach [7] In every N horizon frame this constraint is only activated at a single step and this can reduce the computation cost exponentially. However, if no other restriction is added there are several crucial scenarios that are not taken into account that can alter the final result substantially from the optimal possible solution. More precisely, it does not take into account the values in the boundaries of two consecutive periods of length N horizon. At the transition of two intervals, consecutive openings of the same asset in a time frame smaller than N on are allowed which can lead to negative consumption. As a result, an additional constraint must be added that ensures that no opening will take place in a time period shorter or equal to N on. This constraint can be implemented as follows: k+n on 1 n=k k = 1,, N opt N on + 1 d n 1 (3.23) Due to this additional constraint, the number of constraints has increased however N opt N on so the total sum presents a significantly less computational cost than the rolling window approach. Furthermore, the rolling window approach has an additional crucial disadvantage for this kind of asset. The energy of total load consumption is increasing each time the WH is turning on since the energy is not rebalanced into the power system. As a result, rolling window implementation would just turn the WH as rarely

34 24 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS as possible without taking into account the condition of the system. It was tested in this thesis and it was proved that the only factor that influenced the turning on of the WH was N opt, so this approach was not implemented for the WH load power node Second Implementation In the second implementation, it is assumed accordingly that each power node of WH must consume energy for N on steps in a given time frame Nboiler horizon, however there is no restriction that the consumption will occur in consecutive time steps. In this case, the binary variable d k symbolizes the state of the WH so d k = 1 means that the WH consumes energy at time step k while for the next value it is not known in advance if the WH will continue to consume energy. As a result, at each step the state of the demand unit does not depend on the previous N on 1 steps like in the first implementation. The water heater s consumption can be represented in this case accurately by the following equation: P load,k = u avg load d k (3.24) where u avg load is like in the first case the constant value of energy that the power node will consume for T on steps per T horizon. This implementation can also be shown clearly through a real-case scenario by Figure 3.4.

35 3.5. ELECTRIC WATER HEATERS 25 1 On State of Water Heater Binary Variable Time step(1 hour) Water Heater Power Power(MWh) Time step(1 hour) Figure 3.4: Arbitrary Opening and Power of WH Power Node with not mandatory consecutive consumption Here it is estimated that the WH opens in time steps 18, 22, 23, 24 with T on = 4, Tboiler horizon = 24 and u avg load = 1MW. In this implementation, an additional constraint must be added that binary variable d k must be equal to 1 exactly N on steps per T horizon. This constraint can be implemented through the following equation: k+n horizon 1 n=k d n = N on (3.25) k { mod(1, N horizon ) = 1}, k = 1,, N opt N horizon + 1 In this case, there is no need for using any additional constraints for the transition between two consecutive intervals. The power node of WH is obliged to open exactly N on per time frame T horizon. This might lead to less

36 26 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS computational cost when simulations with many power nodes of WH are performed which proves to be beneficial. Another possibility that the second implementation offers is that the optimal solution for the controller does not have necessarily to arise from consecutive heating of the WH so the optimal solution can be found by obtaining highest possible penetration at nonconsecutive hours. On the other hand, first implementation outweighs the second by means of the inclusion of scenarios in which the optimal solution is for the WH to turn on close to the transition of the two consecutive intervals. In both implementations of WH, fixed window technique will be used since rolling window implementation embodies the problems that were referred in previous section. In this thesis, it is assumed that the contract demands for the WH to be turned on for 4 hours per day so in both implementations we have that T horizon = 24 and T on = Approximating Models In the previous section, two different implementations were derived that are used for representing a single power node of WH. In order to aggregate many clusters of WH in one VPP, the above mentioned procedure is performed with the difference that the binary variables d k from vectors with length N opt are transformed into matrices since now the additional dimension refers to the number of aggregated clusters of WH that are lumped into the VPP. As a result, the binary constraints are severely increasing and especially in the rolling window implementation after a number of clusters of WH, the problem can no longer be solved in acceptable time. In order to keep the computational time as low as possible, we are trying to implement an approximating model which should be able to approximate the aggregation of several WH power nodes by achieving low computational costs and maintaining at the same time an adequate degree of accuracy. Only continuous variables are going to be used, so the optimization is no longer considered as mixed-integer which reduces the execution time significantly. The approach that is going to be followed is called heuristic approach and is based on [7] with some further extensions. Heuristic Approach The idea behind the approach is to approximate the operation pattern of the WH in aggregation in a continuous way. No extra state variable is going to be used, though, an appropriate constraint will be applied that ensures that a certain amount of energy must be consumed over a certain time frame. The maximal power that is consumed over this time frame must be equal to overall consumption of all clusters of WH in aggregation. This situation can be described by the following constraint:

37 3.6. HEAT PUMPS 27 k+n horizon 1 n=k u load,n = C (3.26) where u load,n is the continuous decision variable representing the additional consumption while WH are on and C represents the WH s overall energy consumption per N horizon. One further constraint, that is going to be used and assessed relatively to both implementations in the detailed modelling, refers to the ramping capability of the WH. The ramping in each step for the WH s continuous decision variable u load,n can be computed as: u = u load,(k+1) u load,k k = 1,, N opt 1 where u R Nopt 1. So now the ramping constraint can be implemented as follows: k+n horizon 1 n=k u u max (3.27) k = 1,, N opt N horizon where u max is the maximum ramping permitted per N horizon. This parameter will be defined on the following chapters and general comments will be presented about the role of this constraint relating to both implementations. Finally, there is no contribution to total costs through the demand side management of the WH so f k = at each time step k. 3.6 Heat Pumps After the analysis of the WH, we proceed to the description of a similar load unit that can also participate in demand side management. This unit is named Heat Pump (HP) and contains similarly to the WH an energy reservoir that enables it to consume electricity in different time frames. The substantial advantages of HP against WH are efficiency and environmental friendliness. In addition, the efficiency of HP can outweigh WH s efficiency by a factor of 4. It becomes evident, that this load unit is more crucial for achieving high penetration of RES with minimum costs since most of the countries in the last several decades are replacing water heaters with heat pumps [11]. In this thesis, detailed modelling will be derived for the heat pump and several modifications will be performed to the general model in order to discover the several opportunities that arise in the DSM of this load unit.

38 28 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS It is assumed that all HP are clustered in a VPP and can be controlled via PWM. There is a substantial difference between the modelling of HP and WH. In this case, it is assumed that the HP consumes constantly average energy and can only turned off for a given time period. After the end of the turn off event, the HP will consume higher energy than the average for a small period before it reaches finally to its average value. This pattern can represent with accuracy the heating of a building since while the heat pump is off for some hours the building will cool down so after it turns on, it will need extra consumption at the first period in order to keep the room in the comfort zone. One important remark for the modelling of the heat pump and its several extensions is that in every case, the DR event must be neutral on the overall balance of the system. In other words, the integral of the total energy remains constant and the only thing that happens is a shift in consumption between two different time periods Modelling of a Heat Pump The pattern of the equation of the HP must be formed by three terms, one referring to the constant consumption per time frame, the second responsible for the turning off period which is defined as step-down term and the last one which will account for the higher consumption for the period after the turning on event and will be defined as step up term. This pattern for the power of the HP can be precisely modelled by the following equation: k N P load,k = u avg load off +1 k 2N off +1 uavg load d n + u avg load d n (3.28) n=k n=k N off where N off represents the time period in which the step down term leads to zero consumption. After the end of this period, consumption is doubled for the further N off steps and by that way total balance in the system is achieved through constant integral of energy. Figure 3.5 can depict the pattern of the heat pump assuming that it turns of at time step 5 as follows:

39 3.6. HEAT PUMPS 29 1 Heat pump Closing Binary variable Time step(1 hour) Heat pump Power Power(MWh) Time step(1 hour) Figure 3.5: Illustration of a single turn off of HP Power Node where N off = 4 N horizon = 24 and the consumption of the HP doubles after the turning on as the operation pattern indicates. Patterns with different step up/down lengths In the aforementioned section, it was assumed that the period in which the HP is off is equal to the period with the double consumption. Due to the flexibility of our model, two more patterns were executed and their equations are depicted below:

40 3 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS k N off +1 k (N P load,k = u avg load off +N on )+1 uavg load d n + 2 u avg load d n (3.29) n=k n=k N off k N P load,k = u avg load off +1 uavg load n=k d n uavg load k (N off +N on )+1 n=k N off d n (3.3) In (3.29), the turning off event lasts N off steps while the turning on event lasts N on = N off /2 steps so consumption must double the normal consumption of the corresponding period in (3.28) and this is achieved by factor 2 in the step up term. In (3.3), the turning off event lasts N off steps while the turning on event lasts N on = 2 N off steps so consumption must be half the normal consumption in the corresponding period in (3.28) and this is achieved by factor 1 2 in the step up term. Several different cases can be modelled as long as in every turning off/on event the integral of the total energy that is consumed is constant. Constraints formulation Contrary to the WH case in which one mandatory turn on event per day should take place since the boilers are not consuming energy constantly apart from the N on period, in the case of the HP due to its different modelling more flexibility is available for the controller. Two different implementations have been performed that can be summed as follows: 1. Contract is taken into account that allows maximum 1 turn off event per T horizon. The controller in case that the optimal solution does not demand, can let the heat pump on during the whole T horizon, equal to its average value. 2. Contract is taken into account that allows maximum 2 turn off events per T horizon. The controller in case that the optimal solution does not demand, can let the heat pump on during the whole T horizon, equal to its average value. For the first case, in each time step it must be decided between no action or one turn off event. This case can be implemented just by a single constraint which embodies this pattern and is shown below: k+n horizon 1 n=k d n 1 (3.31)

41 3.6. HEAT PUMPS 31 k = 1,, N opt N horizon + 1 On the contrary, for the second case the situation gets more complex since more different possible scenarios appear. Equation (3.31) must now allow the decision between no action, one single turn off or two turn off events so it is updated as follows: k+n horizon 1 n=k k = 1,, N opt N horizon + 1 d n 2 (3.32) However this constraint by itself is not adequate to embody all the possible scenarios. It can be observed that there is a scenario that the HP will turn off during the step down period which will lead to negative consumption, an unrealistic situation. In order to ensure that this will not happen we need an extra plausibility constraint which allows only one step up/down process. This can be represented by the following equation: k+n off 1 n=k k = 1,, N opt N off + 1 d n 1 (3.33) where we take into account the general pattern of the HP where N off refers to both the step up and step down interval since the two intervals have equal duration. In addition, there are some still some possible scenarios that might happen which are not taken into account. One feasible scenario is that the HP will turn off for its second event during its step up period of the first event. Even if the integral of the total energy will stay constant, this is a case with no practical interest or in other words not compatible with real life case scenarios. So in order to restrict this scenario the plausibility constraint must be extended for the total length of step down and step up interval so its becomes: k+2 N off 1 n=k k = 1,, N opt 2 N off + 1 d n 1 (3.34) where practically the only difference is factor 2 if we assume again that N off of step down interval is equal to N on for step up interval in the general pattern of HP. Finally, there is a last possible modification which can be performed in order to give more flexibility to the controller. The controller can decide to

42 32 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS turn off the HP again exactly in the end of first step-down interval which will lead to zero consumption for 2 N off steps and accordingly to higher consumption for 2 N on steps. However at the same time, the possibility for two separate turn off events could be maintained. This more complex pattern in order to be implemented demands two different states of the binary decision variable d, one corresponding to the separate turn off events and another representing the double turn off event. This constraint can be formulated as follows: k+n horizon 1 n=k d n,1 + 2 d n,2 2 (3.35) k = 1,, N opt N horizon + 1 where d n,1 refers to the separate turn off events and d n,2 to the single double turn off event. Equation (3.28) must now incorporate the d n,2 so it is modified as follows: k N P avg load = uavg load off +1 k 2N off +1 uavg load d n,1 + u avg load d n,1 n=k n=k N off k 2N off +1 u avg load d n,2 + u avg n=k k 4N off +1 load n=k 2N off +1 d n,2 (3.36) As a final remark, it is just referred that there is a possibility to oblige the controller to choose between no action or a double turn off event (d n,1 = ). For this case the equation would be modified as follows: k+n horizon 1 n=k k = 1,, N opt N horizon + 1 while accordingly (3.36) becomes in this case: 2 d n,2 2 (3.37) k 2N P avg load = uavg load off +1 uavg load d n,2 + u avg n=k k 4N off +1 load n=k 2N off +1 d n,2 (3.38) Finally as in electric water heater s case a ramping constraint will be introduced which refers to ramping capability of the HP. The ramping in each step for the HP s final power PHP k can be computed as:

43 3.6. HEAT PUMPS 33 P HP,k = P load,k P gen,j u = P HP,(k+1) P HP,k k = 1,, N opt 1 where u R N opt 1. So now the ramping constraint can respectively be implemented as follows: k+n horizon 1 n=k u u max (3.39) k = 1,, N opt N horizon where u max is respectively the maximum ramping permitted per N horizon. This parameter will also be defined on Chapter 5 and general comments will be added about the role of this constraint relating to both implementations. Reverse Heat Pump By reverse heat pump the exact reverse pattern of the classical pattern, which was described for the heat pump load, is defined. This pattern could either account for the operation of a heat pump or generally a load that demands first a boost in its consumption followed by a period with no consumption until it finally reaches its optimal constant value. Only the simple case of single boost event is shown in the below equation since the analysis for two events is similar to that of the classical pattern: k N P load,k = u avg load + off +1 k 2N off +1 uavg load d n u avg load d n (3.4) n=k n=k N off where naturally only the signs from the step up/down terms change from the equation of the typical patter of the HP in (3.28). Accordingly, only one constraint is needed which ensures that either no action or a single boost event will happen per time frame T horizon which is identical to the normal case and was referred in (3.31) Approximating Models In the previous section, several implementations were derived that are used for representing a single power node of HP. In order to aggregate many clusters of HP in one VPP similarly with the WH case, the above mentioned procedures are performed with the obvious difference that the binary variables d k from vectors with length N opt are transformed into matrices

44 34 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS since the additional dimension refers to the number of aggregated clusters of HP that are lumped into the VPP. As a result, the binary constraints are significantly increasing compared to the WH case due the usage of the rolling window implementation which on the contrary of WH case here does not present any problematic behaviour and can achieve both detailed and accurate outcomes. In case of the WH, there was no alternative as it was explained but generally fixed window implementation causes non-linearities in the transitions of the intervals which lead to limited accuracy so it will be avoided in the HP case even if the computational cost increases. In order to keep the computational time as low as possible, we are trying respectively to implement approximating models which should be able to approximate the aggregation of several HP power nodes while maintaining at the same time an adequate degree of accuracy. Only continuous variables are going to be used, so the optimization is no longer considered as mixedinteger which reduces the execution time importantly. The first approach that is going to be followed is called heuristic approach and additionally several extensions will be made depending on the corresponding detailed implementation. Finally, a different approach named pseudo-binary will be analysed. Heuristic Approach The idea behind this approach is to approximate the operation pattern of the HP in aggregation in a continuous way. No extra state variable is going to be used, though, appropriate constraints will be applied that ensure that the classical pattern of the HP with the step up/down intervals will be as closely as possible approximated. First, it must be ensured that a certain amount of energy must be consumed over a certain time frame. This practically corresponds to the step up interval of the detailed modelling and can be represented through a constraint that demands that the amount of maximal power consumed over a defined time frame must be smaller or equal to overall consumption of all clusters of HP in aggregation. This statement can be described by the following constraint: k+n horizon 1 n=k u load,n = C (3.41) k {mod(1, N horizon ) = 1}, k = 1,, N opt N horizon + 1 where u load,n is the continuous decision variable that represents the additional consumption of the several clusters of HP during step up interval and C represents the HP s overall energy capacity per N horizon. Based on the latter equation both the constant term of continuous consumption and the

45 3.6. HEAT PUMPS 35 step up term can be combined and depicted by a single variable P load,k as follows: P load,k = u load,k + u avg load (3.42) where u avg load is the constant consumption and u load,k u avg load. The choice of the subscript load in u load,k and P load,k is not coincidental since through those variables consumption is increasing. Accordingly, in order to represent the step up interval of the classical detailed modelling, a different variable must be introduced which will somehow relate step up and step down interval. Unfortunately, a strong assumption must be made for this interval. In case that we want to compare a detailed modelling with existence of double turn off with a heuristic model then we will assume that the consumed energy always will get shifted every length equal to 2 N off so the equation can be implemented as: P gen,k = u load,(k+2n off ) (3.43) where the subscript gen refers to generation by means of reduction in consumption which happens during the step down interval. Further Analysis of Heuristic Model The assumption that has been made in the latter equation about the modelling of the step interval can be characterized as strong since it is not always the case that the turning off events in detailed modelling will happen consecutively. However, there it is an approximation which has to be made since most of the times the single turn off events will happen closely one another. Recalling that in the detailed modelling there was referred a scenario that the controller is forced to perform a double turn off event consecutively it is expected that this assumption in the heuristic model will fit better. If we want to approximate the detailed modelling of an aggregation in which only a single turn off event can occur then the equation which accounts for the step down interval is modified as follows: P gen,k = u load,(k+n off ) (3.44) where the assumption is in this case quite realistic and satisfactory results are expected. During the detailed modelling in this section, two implementations were defined that involved different lengths of the step down and step up intervals (N off N on ), which have led to different amounts of consumed energy at each interval in order to maintain constant the integral of the consumed energy. This inequality in lengths results in more difficulties to find an adequate approximation for modelling the corresponding P gen,k. It will be

46 36 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS assumed that for (3.29) where N on = N off /2 the step down interval can be represented as follows: P gen,k = u load,(k+n on ) + u load,(k+n off ) (3.45) which for a real case scenario with N off = 4 and N on = 2 satisfactory results were obtained since every step k of the time frame N off corresponds to a single value only of the N on interval because one term will cancel out. To illustrate more clearly the validity of this assumption for the above mentioned given lengths, values of the latter equation are computed with corresponding results as follows: P gen,1 = u load,3 + u load,5 = u load,5 P gen,2 = u load,4 + u load,6 = u load,6 P gen,3 = u load,5 + u load,7 = u load,5 P gen,4 = u load,6 + u load,8 = u load,6 so since step down interval refers to k [1 4] and step up to [5 6], the respective terms of u load,k with k 5,6 should be cancelled. For (3.3) where N on = 2N off unfortunately it was not achieved to find a logic assumption that can be explained for the reverse case. Finally as a rough approximations it was estimated that: P gen,k = 1 ( ) u 2 load,(k+n on ) + u load,(k+n off ) (3.46) which takes the average of the two interval s length. Then assuming that N off = 2 and N on = 4 we could take accordingly that: P gen,1 = 1 2 (u load,3 + u load,5 ) P gen,2 = 1 2 (u load,4 + u load,6 ) P gen,3 = 1 2 (u load,5 + u load,7 ) = 1 2 (u load,5) P gen,4 = 1 2 (u load,6 + u load,8 ) = 1 2 (u load,6) from which it can be concluded that the results for the step down interval will not be so accurate since P gen,3 and P gen,4 are not cancelled but still remain non-zero which leads to an undesired increase of the step down interval. Finally, for the reverse heat pump which includes one single boosting event an accurate heuristic model can be derived according to classical heuristic model. For this case, P gen,k represents the step up interval since

47 3.6. HEAT PUMPS 37 consumption is further increased from the constant level during this interval. On the contrary, P load,k represents the step down interval since consumption is driven to zero after the end of the step up interval. Constant factor u avg load can be incorporated in variable P gen,k. Based on this explanation, the equations can be described as follows: P gen,k = u load,k + u avg load (3.47) P load,(k+4) = u load,k (3.48) The only constraint that is needed is a single that gives the flexibility to the controller whether a boosting event or no action will happen per T horizon. It is not stated in this section since it is exactly the same with the classical case in (3.31). Pseudo-Binary Approach In this section, a different approach named pseudo-binary will be analysed for the general pattern of the HP where N off = N on. As in the heuristic approach only continuous variables are used but the difference is that this approach is based on detailed binary model approach. In this approach, at each step the whole pattern of the HP is constructed. This can be clearly illustrated by the following equation: P HP load,k = uavg load k N off +1 n=k k (N off +N on )+1 Pn contr + Pn contr (3.49) k = 1,, N opt n=k N off It can be recalled that the latter is similar to the form of (3.28) which explains the name pseudo-binary. In this approach, though a continuous variable the exact pattern from the detailed analysis is followed. As in the detailed modelling, P contr must be assumed positive and restricted per time frame T horizon in order to ensure that the integral of the consumed energy is constant. This can be shown by the following constraints: k+n horizon 1 n=k P contr n u avg load (3.5) k{mod (1, N horizon ) = 1}, k = 1,, N opt N horizon + 1 P contr k (3.51) k = 1,, N opt

48 38 CHAPTER 3. MODELLING OF POWER SYSTEM UNITS The aforementioned equations and constraints regard the typical pattern of HP with maximum one turn off event. This approach can similarly be extended for the case of N off N on or the case of the reverse HP and as a result it will not be shown here.

49 Chapter 4 Dispatch Simulator In this chapter, an introduction to the software environment of this thesis will be done. In addition, analysis between full horizon and receding horizon will be performed making it clear when one is used against the other. Finally, some details will be given for the cost factors and parameters values for our optimization problem. 4.1 Introduction to Software Environment The dispatch simulator of this thesis is implemented in Matlab R214b. As solver of the optimization problem, CPLEX Version 12.6 from IBM is used. This solver is called within the Matlab environment since there are appropriate tools that can easily connect them. Cplex displays a variety of advantages against other solvers and is regarded as one of the most efficient solvers in linear programming problems (LP), quadratic programming problems (QP) and mixed integer programming problems (MIP) which can either be linear (MILP) or quadratic (MIQP). It manages high performance for large-scale realistic optimization problems while at the same time compared to the classical incorporated solvers of Matlab, proves to be more reliable and fast. It is generally an appropriate solver for modelling power systems which lead to problems with thousands buses and constraints. The basic drawback is that it is considered a commercial solver with an expensive fee in order to obtain the license. For this thesis, it was used free of charge due to ETH University which cooperates with IBM and provides licenses to students for academic purposes. Finally, in conjunction to Matlab, the modelling language Yalmip is used which is provided as a free toolbox [12]. This language is characterized for its simplicity to formulate optimization problems since the optimization problem can be coded just a single time in high-level form. It is linked with several external solvers as Cplex and is able to solve the problem efficiently and with minimum possible computational costs. 39

50 4 CHAPTER 4. DISPATCH SIMULATOR 4.2 Full Horizon vs Receding Horizon Optimization There are two kinds of possible optimization when we refer to model predictive control. The basic difference between full horizon (FH) and receding horizon optimization (RHC) refers to the length of the relation between N opt which refers to length in which the problem is optimized and N sim which is the total length of the simulation that finally the problem has to be solved. In case N opt = N sim, at each step k the problem is optimized from the beginning based on the its data and does not take into account history values in other steps. On the contrary, in case N opt N sim the problem is optimized over an horizon N opt where generally stands that N opt N sim. As a result, at each step k the optimal control problem is computed regarding only the next N opt steps. From this optimization problem, only the optimal values of the decision variables which refer to the first step are kept and applied to the system. At each time step k, a window with length N opt recedes and considers only the optimized values from the first step. This process continues until it shifts accordingly to N sim step. The process that is described for RHC can be illustrated by Figure 4.1. It can be concluded that the ED problem is solved by many small optimization problems with N opt forecast instead of a single large-scale problem for the total length N sim. One clear advantage of RHC compared to FH is that large power systems are generally large scale problems which include thousand of different units. Furthermore, simulations might need to be performed over whole year which corresponds to 876 steps or even more if the simulation step T 1 hour. Optimization problems with such a large length display large computational costs and might need numerous days or even weeks to be solved which practically renders them inadequate for our purposes. In addition, RHC gives the controller the opportunity to add feedback on the system. Even if forecasts are performed with maximum possible accuracy, it is evident that forecasts referring to wind & solar energy or load demand can never be exactly accurate one year earlier. Through RHC, forecasts are only considered for N opt steps and after this time period new forecasts can be applied or even new measurements x in case of an unexpected disturbance ω. For further analysis in RHC, the readers of this thesis can refer to Borreli [6]. One further and important characteristic which must be considered for an optimization problem corresponding to this thesis is related to the boundaries of each implementation. While in full horizon implementation, the results at the simulation boundaries have a negligible impact to the final solution, in RHC this situation is totally different. N opt is a small window and resulting values in boundaries play an important role to the final solution. To be more specific, in case of WH it must be known whether it was

51 4.2. FULL HORIZON VS RECEDING HORIZON OPTIMIZATION 41 Figure 4.1: Visualization of RHC concept [6]

52 42 CHAPTER 4. DISPATCH SIMULATOR on before the end of the previous N opt window otherwise it could turn on again in the beginning of the following N opt which will lead to higher consumption and unrealistic results. Therefore, boundary constraints must be added by means of transfer of all history variables that are needed from the end of the first window to the beginning of the next window. This leads also to more complex coding since further modifications should be made to all referred constraints in Chapter 3 for ensuring that all necessary information is transferred correctly. In this thesis, even if both implementations were tested up to an extent it was decided to finally apply full horizon implementation due to the following reasons: 1. Forecasts are considered to display perfect accuracy so a deterministic optimization is applied and possible stochastic disturbances are not possible. 2. The total simulation horizon is between 7 to 14 days since it is out of the scope of this thesis to perform a large scale optimization of a particular problem. The goal, though as already stated, is to assess approximating models in relation to detailed modelling in order to provide the optimal tools which will afterwards be used in large simulations. 3. In full horizon implementation, it was established that the number of constraints is significantly smaller since no restriction have to be considered in the boundaries. 4. Further analysis has shown that ED solutions become less sensitive to cost parameters for longer optimization horizon and this is a clear advantage for this thesis since it beyond of its goal to represent a single power system with a particular cost structure. As a result, for the purposes of this thesis it was concluded that full horizon implementation is regarded as the appropriate tool in order to formulate our optimization problem. 4.3 Definition of Full Horizon Implementation In Chapter 3, every asset of our total portfolio was presented in pair with its constraints, grid power injection equations, cost functions and state dynamic equations. A dispatch simulator must be derived which contains all this information based on the optimization problem in (2.9). It is reminded that full horizon implementation is going to be used so our optimization problem consists of N Power Nodes and is executed for N opt = N sim steps. For each power node i, a vector u i k is constructed which refers

53 4.3. DEFINITION OF FULL HORIZON IMPLEMENTATION 43 to step k and entails all the decision variables of power node i. To illustrate this concept the procedure is going to be followed for a single case of a power node which is selected to be the wind power node. As it was described in part 3.3, for an intermittent wind power node the basic decision variables for each time step are u k and ω k. As a result a vector Uk i as follows: ( ) U i u k = i k ωk i (4.1) k = 1,, N opt which results in a vector U i and can be described as follows: U i = u i 1 ω i 1. u i k ω i k. u i N opt ω i N opt (4.2) This process is similarly followed for each power node i and finally a vector U is constructed which entails all the continuous decision variables of the sum of our power nodes for the whole simulation. In addition, the same procedure is performed for the final vectors d and x which refer to the binary decision variables and to the state variables. In our problem only the power node of a hydro pump entails a state variable whereas the power nodes of WH/HP and conventional generation entail respectively decision variables. In the end of this procedure, the vector z is constructed and can be illustrated as follows: U z = x (4.3) d Since total vector z is constructed all constraints based on Chapter 3 must be formulated in order to construct matrices A eq, A ieq and vectors b eq, b ieq. Starting from step k = 1 all appropriate values must be found in order to formulate the rows and the columns of the aforementioned matrices. Inequality constraints refer mainly to bound and ramping constraints of all decision variables whereas equality constraints refer to state dynamics evolution combined with initial or final constraints for the state variables and finally the grid balance constraint which embodies all generations and all loads.

54 44 CHAPTER 4. DISPATCH SIMULATOR As a remark, for the conventional load power node since there is no flexibility for the controller (ω k = ), there is no need for any decision variable to be used. Load demand is considered an external process and can be incorporated in the grid balance equation as follows: N i=1 ( P i gen,k P i load,k) = ξ k drv, (4.4) k = 1,, N opt where ξdrv, k represents the externally driven load demand for each time step k and Pgen,k i,p load,k i the power injections for overall power nodes. Definition of the parameters of the final problem In this section all the values of the problem will be defined such as cost factors and all necessary parameters for the formulation of the final optimization such as general bounds for minimum/maximum limits for inequality constraints or the simulation length will be clearly explained. Cost Factors As already stated the general portfolio that is formulated in this work comprises of the following power nodes: conventional generation, hydro pump storage, solar generation (PV), wind generation and controllable demand units. For the conventional generation, a quadratic cost function used as it is described in (3.8) so factors a, b and c must be defined. Since goal of this thesis was not to depict a particular power system with its actual cost structure it was decided to define the parameters of the quadratic function from [7] as: a =.3 [e 2 /MW h 2 ], b = 8 [e/mw h] and c = 2 e. Factor a regards the quadratic costs while factor b represents the marginal costs. As it already has been stated, factor c refers to the fixed costs and does not influence the final solution. For the intermittent generators, it was assumed that marginal costs are since the most important contribution from fuel costs is nonexistent due to the abundance of sun and wind. However, large values of curtailment costs were defined so that the controller curbs the free intermittent energy only as a last resort. Linear cost factor a was set 1 for wind energy whereas 2 for PV generation since it a more expensive technology and it is always decided to curtail first wind energy.

55 4.3. DEFINITION OF FULL HORIZON IMPLEMENTATION 45 As far the hydro pump storage is concerned, generally it is a complex process to define marginal costs since you must know the watervalue of the power plant which refers to the value of the water from the future. Since Hydro Pump Storages are flexible you must rely on forecasts in order to anticipate when extra production will be needed in order to store water for that period. No further analysis will be provided for this domain, but the reader can refer to [13] where a detailed cost analysis is performed. Based on this, turbining was penalized with a lower, compared to the other power nodes, linear factor equal to 5 [e/mw h] while pumping is assumed that it causes no costs in order to exploit the highest possible penetration of RES. Finally, for the controllable demand units no cost factor is defined since with their optimal operation they always reduce the total costs. General Parameters For all the power nodes driven by external processes, real supply/demand profiles were taken by the transmission system operator of Germany regarding year 21. The data of this process show granularity of 1 hour time step which fits with the granularity of this thesis. However, for the RES it was decided to increase the supply artificially since special interest appears when there is a case of high RES penetration. Through testing, it was decided that all values of wind and solar energy to be scaled equally with factors: λ wind = 1 and λ pv = 15. In addition, since aggregation is not examined in conventional generation, its entire sum is lumped into a single power node. Conventional generation must be able to provide all the energy in case of no RES penetration and zero storage level of a hydro pump. As a result, the maximum power rating is equal to the maximum demand while the minimum power rating for the on state is assumed to be 1% of its maximum rating. In order to be realistic, this conventional generation refers to gas turbines which can modify their output beyond a small time frame. It is considered that based on their ramping capability, 2 hours from the off state are needed to reach the maximum output state. For the hydro pump storage, it is assumed that a storage reservoir is included that can provide 1% of the total demand load so C is fixed accordingly. The maximum outputs for both turbining and pumping is considered to be 2% of the maximum load demand. Furthermore, for the initial state it assumed that x initial =.5 which means that the storage level is on the middle in the beginning of the simulation. Since reasonable use of the water in hydro pump must be made (inflows are neglected), a final constraint is needed according to which no more than 2% of the initial water will be used every week since one year has 52 weeks and water must be available for all

56 46 CHAPTER 4. DISPATCH SIMULATOR the year. This is a significant assumption that is not compatible with the watervalue definition that was referred above, however, it is satisfactory for ensuring the desired RES penetration. Finally, efficiencies for both pumping and generation are set to.85. To sum up, for the controllable demand response units, it was assumed that their total power rating is 1% of the maximum conventional load demand. The individual DR power rating u avg,i load depends on the N W H & N HP which define the number of the controllable demand power nodes that are used in the detailed modelling. In addition for the WH, T up and T horizon are the parameters which refer to the total hours that the WH are consuming energy and the time frame per which a single turn on event occurs respectively. Based on contracts, it is assumed that T up = 4 and T horizon = 24. Accordingly for the HP units, we define that T horizon = 24. In this case, T off and T on are used to show the turning off/on intervals for the HP events. Those values are not general and they depend on our several simulations so they will be defined accordingly on the following chapter. All those parameters and information can be summed in the following Table 4.1. Table 4.1: General Parameters for the formulation of the final problem PN Model Parameter Description Wind f k = 1 ω k PV f k = 2 ω k Conv. Gen. f k =.3Pgen,k 2 + 8P gen,k + 2 Pgen max = max(ξdrv load) P min =.1 Pgen max = P max Hydro gen u max gen gen /2 f k = 5 u gen,k u max gen =.2 max(ξ load u max load =.2 max(ξload x initial =.5 x Nopt.98 x initial η gen = η load =.85 C =.1 DR f k = N opt k=1 ξ load drv,k drv ) drv ) Finally, as evaluation quantities on the following chapter special emphasis in this thesis is given in total computational time, total costs and curtailed energy.

57 Chapter 5 Results and Comments This chapter presents and evaluates the results from all introduced methods of modelling in the previous chapters. Firstly, for each controllable demand unit the detailed modelling with a single power node only will be presented. Afterwards, this modelling will be extended to a larger number of power nodes and sensitivity analysis to the this variable number will be performed. Finally, approximating models will be presented followed by several comparisons between them and the detailed aggregation models by means of both computational cost of the simulation and accuracy in the total power of the controllable unit in order to access the different opportunities that exist in power system modelling through aggregation. Finally, comments and proposals are developed based on the results from the aforementioned comparisons. The chapter will be divided in two sections, describing WH and HP respectively. 5.1 Water Heaters In this section, the behaviour of multiple electric water heaters will be investigated and comparisons between the detailed modelling and the approximating models will be presented. For the electric water heaters, the simulation will last 2 weeks, starting from January 1 st to January the 14 nth 21. The ED results of the power system with a single power node of water heaters (ND W H = 1) are represented in Figure 5.1 for the first implementation of water heaters where they must consume constantly energy for T on in a given time frame T horizon. In this case, it is assumed that T on = 4 and T horizon =

58 48 CHAPTER 5. RESULTS AND COMMENTS Whole balance 3 2 Power(MWh) 1-1 conv gen hydro gen solar gen curtailed pv wind gen curtailed wind load hydro load water heater load curtailed wind curtailed pv time step(hour) 1 5 Renewable infeeds & Curtailed quantities Power(MWh) 1 external wind external solar curtailed wind curtailed pv time step(hour) 1 4 Water Heater Power Power(MWh) Time step(1 hour) Figure 5.1: Dispatch result for 1 WH Power Node with consecutive consumption

59 5.1. WATER HEATERS 49 Regarding the DR load profile, it can be observed the units do not follow a particular pattern for its on-state. There are days that the consumption takes place during the night period with low total load demand, however there are days in which they turn on during noons with extra pv activity or generally periods with high wind penetration. As a result, it can be stated that the night tariff scheduling of WH will not be the optimal way to coordinate our controllable units in case of high penetrations RES scenarios. Furthermore corresponding results are illustrated in Figure 5.2 for the second implementation of WH (3.5.2) where the WH must consume energy for exactly T on in a given time frame T horizon but without any restriction for consecutive operation. In this case, it is assumed again that T on = 4 and T horizon = 24. In order to identify more clearly the difference between the two implementations which includes the WH power for both occasions Figure 5.3 is constructed Water Heater Power 5 4 Power(MWh) 3 Free operation Consecutive operation Time step(1 hour) Figure 5.3: Comparison for 1 WH Power Node for both implementations It can be observed that for some days the behaviour is similar (days: 4 & 5), in some occasions the dispatch is just shifted some steps (days: 1 2 & 6 7) while in some other occasions, in implementation 2 the controller decides to open the WH in non-consecutive hours (days: 8 & 14). The difference from those 2 profiles lie on the way they are modelled. On the one hand, it is clear that during days where total consumption of 8 hours consecutively happens, implementation 1 outweighs implementation 2 since in case 1 the consumption can start for example on 23. of the first day and continue until 7. the next one while in case 2 the closest solution that is permitted is to turn on the boilers from 2. the first day to 4. of the second. This is a clear advantage of implementation 1, however, during other days it might be optimal to open the boilers in non-consecutive hours because of rather increased RES penetration in arbitrary hours. This is a clear asset

60 5 CHAPTER 5. RESULTS AND COMMENTS Whole balance 3 2 Power(MWh) 1-1 conv gen hydro gen solar gen curtailed pv wind gen curtailed wind load hydro load water heater load curtailed wind curtailed pv time step(hour) 1 5 Renewable infeeds & Curtailed quantities Power(MWh) 1 external wind external solar curtailed wind curtailed pv time step(hour) 1 4 Water Heater Power Power(MWh) Time step(1 hour) Figure 5.2: Dispatch result for 1 WH Power Node without mandatory consecutive consumption

61 5.1. WATER HEATERS 51 for implementation 2 because of its flexibility to turn on the boilers just for 1 hour when needed. Given the load profile for 2 weeks and having scaled the RES, as already stated, overall costs, curtailed energy and consumed energy from the boilers can be represented on table 5.1. In our thesis, implementation 1 seems to be more economically efficient than the second one. However, this is completely based on the given externally driven processes. Table 5.1: Data from comparison of 1 Power Node WH for both implementations Quantities Implementation 1 Implementation 2 Total costs [1 6 e] Curtailed Energy [MWh 1 6 ] DR Consumption [MWh 1 6 ] It is evident that in case we had applied heterogeneous scaling in the wind/pv supply then from hour to hour there would be clear differences and possibly the controller would desire to turn on the boilers completely different hours during the day. So in case of a high penetration stochastic RES scenarios with clear differences between the hours the result would be different Sensitivity to aggregation level In this subsection, the change in dispatching when DR can be provided by several clusters will be illustrated. For that purpose, several simulation have been performed for both implementation with N DR values ranging from 1 to 35. Firstly, it will be shown for each implementation how system costs, computational time and curtailed energy are influenced by increasing the number of clusters of WH. For implementation 1 for each case of N DR = 1, 5, 1, 2, 35 the WH power can be represented in Figure 5.4. It can be noticed that for an increasing N DR the units are activated increasingly in a consecutive pattern. This is an important fact since the consumption profile of the WH in aggregation becomes more continuous which is desirable since overall consumption becomes more continuous which reduces the jumps of the conventional generation units. Ramping is reduced for conventional generators during on/off state of the WH and this leads to the reduction of the total costs. In addition, the total energy consumption is independent of the number of clusters since the average power of each cluster is fixed accordingly through a constraint. An undeniable fact is that the computational cost increases at a great extent while the number of clusters increases.

62 52 CHAPTER 5. RESULTS AND COMMENTS Water Heater Power 5 4 Power(MWh) Time step(1 hour) 1 4 Total power of water heaters Power(MWh) Time step(1 hour) 1 4 Total power of water heaters Power(MWh) Time step(1 hour) 1 4 Total power of water heaters Power(MWh) Time step(1 hour) 1 4 Total power of water heaters Power(MWh) Time step(1 hour) Figure 5.4: Total Power for N=1, 5, 1, 2 and 35 clusters of WH for consecutive consumption

63 5.1. WATER HEATERS 53 After N DR = 5 a huge amount was needed for the simulation to terminate and respectively for longer than 2 weeks simulation periods, even the case of 35 power nodes was not solved in reasonable time. Finally, based on Figure 5.4 it can be concluded that the largest change is observed from N DR = 1 to N DR = 5. Above N DR = 2, the solution is starting to converge in a stable result. Those values can be clearly shown on Table 5.2 while the dispatch results of the case N DR = 35 can be illustrated in Figure 5.5. Table 5.2: Data for 1, 5, 1, 2 and 35 PN of WH with consecutive consumption Number of Clusters Total Costs Curt.Energy DR Cons. [1 6 e] [MWh 1 6 ] [MWh 1 6 ] Figure 5.5: Dispatch results for 35 PN of WH with consecutive consumption Correspondingly, for implementation 2 without the mandatory consecutive consumption, each case of N DR = 1, 5, 1, 2, 35 the WH power can be represented in Figure 5.6.