WO2016131674A1

WO2016131674A1 - Method and controller for controlling a power grid

Info

Publication number: WO2016131674A1
Application number: PCT/EP2016/052702
Authority: WO
Inventors: Martin Kozek; Markus Brandstetter; Maja Miletic; Alexander Schirrer
Original assignee: Technische Universitaet Wien
Current assignee: Technische Universitaet Wien
Priority date: 2015-02-18
Filing date: 2016-02-09
Publication date: 2016-08-25
Anticipated expiration: 2017-08-18
Also published as: EP3059646A1

Abstract

The invention relates to a method for controlling power consumption in a power grid (1) having a grid controller (6) with a stored network model (7) and at least one smart consumer (4) capable of predicting its power consumption profile (Ĭ^d), wherein an iteration step is performed in the grid controller (6) including receiving a predicted power consumption profile (Ĭ^d) from each smart consumer (4), determining a proposed power consumption profile (Ĭ) for each of a selected set of smart consumers (4) by iteratively solving an opti- mization problem, and sending said proposed power consumption profiles (Ĭ) to the smart consumers (4); and repeating said iteration step for said prediction time window in a loop until an exit condition is met. The invention further relates to a grid controller (6) imple- menting this method.

Description

Method and Controller for Controlling a Power Grid

Field of the invention

The present invention relates to a method for controlling power consumption in a power grid having one or more power producers and power consumers, a network of power lines and nodes therebetween, a communication infrastructure therebetween, and a grid controller connected to the communication infrastructure. The invention further relates to a grid controller for such a power grid.

The term "power" in the present context comprises all sorts of power, be it electrical power, hydraul ic or pneumatic power, thermal power transported by fluids, water or steam in district heating grids, thermal power stored in combustible gases or fuels transported in fluid distribution grids, etc.

Therefore, while the present disclosure of the invention will be given in detail under specific reference to electrical power and an electrical power grid, the terms "power" and "power grid" as used herein comprise all these sorts of powers and grids.

Background of the Invention

While the electrical power consumption in urban power grids has been stagnating in the last decades or, in some regions, even slightly decreased, the peak-to-average ratio of the power consumption has increased considerably and represents a risk to the grid infrastructure. One of the most important goals for grid management is to reduce the electric power consumption during peak demand periods in order to assure reliability and efficiency of the electrical power grid. Moreover, the use of renewable energy has had a strong annual growth over the past decade (see, e.g., reference [1]). There are many efforts and increasing support to further develop exploitation of renewable sources motivated by climate change ( in order to reduce the emissions of greenhouse gases) and high oil prices [2]. However, the fluctuating availabil ity of electricity produced by renewable sources is an additional challenge for the performance of the electrical power supply [3]. It is desirable to accommodate these load variations by upgrading the grid's infrastructure into a ^"smart grid^", through a cohesive integration of new communication and information technologies, which would allow power uti lities to monitor and manage the power grid, and take control actions remotely, from a central location [4]. in order to reduce peak electricity demand in electrical grids, a common approach is to introduce smart pricing, which has been extensively studied since the beginning of the 1980s (e.g. see [5]). In more recent work [6, 7] a real-time pricing tariff is constructed based on spot market prices and time-series grid loads. The obtained dynamic electricity tariff is directly incorporated into the cost function of a building model predictive control (MPC) for, e.g., a building climate control. In [8] authors introduce an advanced pricing method: the so- called V ic k rey-C 1 a rke-G roves mechanism is used to determine the price charged to each user based on its declared energy demand information, as well as its minimal and maximal electrical power levels and its minimal total energy requirements. It is demonstrated that the proposed method encourages the truthfulness of the users and results in load shifting to off- peak hours. Another smart pricing mechanism has been proposed in [9], which ensures both optimality and fairness in autonomous demand response systems. I n [ 1 0] scheduling of the power consumption for the consumers in the smart grid is proposed based on the water- filling algorithm, as a response to the dynamic prices set by the utility company.

All of the abov e mentioned approaches are decentralized. Alternatively, cooperative distributed model predictiv e control ( DM C) can be used for demand management in smart grids, which in general has high communication requirements. In order to lower the communication requirements in large-scale systems, the authors of [ 1 1 ] propose a low- communication DM PC, where it is suggested that the end-users only communicate w ith their neighbours. They introduce an algorithm for two agents based on game theory and demonstrate its robustness against communication failure through simulations. In [12], dynamic prices are proposed in order to improv e overall consumption uniformity. It is studied how close the total power consumption load in the grid comes to the ideal flat profile, depending on the amount of information the consumers are willing to share between each other. Assuming all demand information is shared, a cooperative game is proposed. Otherwise, distributed stochastic strategies are recommended. Further, in [13] and [ 14] the authors formulate an energy consumption scheduling game for a giv en dynamic pricing strategy and show that the global optimal performance, with the energy costs as the objectiv e function, is achiev ed at the Nash equilibrium.

In recent years, applications of hierarchical model predictiv e control ( HM PC) to energy management in electrical power grids has been presented in the literature. In papers [15,16,17] the control structure consists of three tiers: top-level controller, aggregators and consumers. In [15, 16] the authors propose a hierarchical controller with the goal to coordinate the demand of the consumers w ith the fluctuations in the power production which is typical for renewable energy sources. The control algorithm requires the information on the av ailable electrical energy in the grid on some finite horizon and exact individual power and energy constraints of the intell igent users, represented by convex polytopes. Optimization and distribution is performed recursively over the time steps of the horizon. The disadvantage of the method proposed in [ 1 ] is that the top-lev el controller aggregator need to hav e the information about the physical bounds of the energy storage levels and capacity of the consumers. Furthermore, the upstream communication demand in the grid is high. In [16] the top-lev el controller needs to hav e a storage model of the consumers and the information on the demand of the consumers. Another H M PC approach is presented in [17], where the automatic generation control is accomplished by means of a complex cascade control based on the dynamical model of the electrical grid to cov er different time scales. The grid controller integrates both renewable and traditional power producers and consumers, with focus on incorporating electrical vehicles. The disadvantage of method in [ 1 7] is that the aggregators need to hav e a battery model for each vehicle and they need to make a prediction of charging and discharging power and state of charge for each vehicle based on driver inputs and statistical data. Furthermore, in [18] the authors present a concept for an H M PC architecture to efficiently integrate activ e buildings into low voltage grids by exploiting their thermal storage capacities with the objectives to minimize total electric transmission losses and relative peak loads. The main disadvantage of al l of the above presented methods [ 15, 16, 17, 18] is that the top-level controller has a direct access to consumer behav ior, i.e. it completely dictates the consumers power consumption profile.

Object of the invention

It is an object of the invention to provide improved methods and apparatus which overcome the above-ment ioned drawbacks of the state of the art.

Summar of the Invention

I n a first aspect, the invention prov ides for a method for controll ing power consumption in a power grid hav ing one or more pow er producers and power consumers, a network of power lines and nodes therebetween, a communication i nfrastructure therebetween, and a grid control ler connected to the communication infrastructure, the grid controller having a stored model of the network, and at least one of the pow er consumers being a smart consumer capable of predicting its pow er consumption profile over a prediction time wi ndow , comprising:

for each smart consumer, determining an influence weight of its power consumption on li ne currents and node potential drops i n the network model;

for said prediction time window, performing an iteration step in the grid controller i ncl uding:

receiv ing a predicted power consumption profile for said prediction time window from each smart consumer v ia the communication infrastructure,

determining a proposed pow er consumption profile for said prediction time wi ndow for each of a selected set of smart consumers by i terat ively solving an optimization problem wherein

a measure of the differences between the to-be-determi ned proposed power consumption profiles and the received predicted power consumption profiles, respectively weighted by said in fl uence weights,

is minimized under the constraints

that l ine currents and node potential drops in the network model stay wi thin allowed ranges over the entire predict ion time w indow, and

sending said proposed power consumption profiles to the smart consumers via the communication infrastructure;

and repeati ng said iteration step for said prediction ti me w indow in a loop w hich is exited when at least one of the follow ing conditions is met:

all l ine currents and node potential drops of the netw ork, as calculated by applying the received predicted pow er consumption profiles to the netw ork model, stay withi n al low ed ranges over the entire prediction time w i ndow , or

none of the received predicted power consumption profi les has changed with respect to the preceding iteration step, or

a maximum number of iteration steps has been reached. In a second aspect, the invention provides for an electronic grid controller for a power grid which has one or more power producers and power consumers, a network of power l ines and nodes therebetween, and a communication infrastructure therebetween, the grid controller comprising:

an interface being part of the communicat ion infrastructure;

a memory hav ing a model of the network stored therein; and

a processor being configured to i mplement the methods of the invention disclosed herein.

The grid controller of the invention, being of the type of an hierarchical model predictive controller (HMPC) and also called "hierarchical grid controller^" or "hierarchical MPC" in the fol lowing, receives predicted power consumption profiles (load profiles) from end- users, e.g., smart buildings. Based on these information, it predicts the overall power consumption profile in the power grid over some finite horizon and proposes constraints on the power consumption to the consumers when necessary, in order to avoid v iolations of grid specifications at critical nodes and lines. A significant advantage of the inventive controller and controlling method is that the communication requirements w ith consumers in the grid are low. Furthermore, it can also provide for optimal satisfaction of consumer energy demand over time intervals of fixed length, so that the service quality for the consumers will not be compromised.

Another advantage of the inventive hierarchical approach disclosed herein is that it does not require information about the current state and bounds of local storage situation, which are difficult to determine, and it also does not require simplified models for the smart consumers. The smart pricing approaches as in the state of the art are decentralized methods and there each consumer makes an indiv idual response without hav ing information about the load of other consumers in the grid. On the other hand, the hierarchical approach of the method and controller of the invention manages time-varying power consumption in the grid w ithout the disadvantage of high communication requirements typical for distributed cooperative methods.

Basically, said selected set of smart consumers for which a proposed power consumption profile is determined by solving said optimization problem can include all of the smart consumers. In a further embodiment of the invention, however, said selected set can include - even only for one or more selected time step/s w ithin the prediction time w indow - only those smart consumers whose predicted power consumption profile has changed w ith respect to the preceding iteration step, wherein this change criteria can comprise any condition of the profile, thus reducing processing for some time steps of the prediction time w indow of an iteration step.

In a preferred practical implementation of this embodiment, to this end, for each smart consumer a flag indicating whether its predicted power consumption profile has changed from one iteration step to another - and preferably such a flag for each time step of the prediction time w indow - can be stored in a compliance matrix which defines said set and can be used in the solving of the optimization problem.

In this way, iterations in w hich new proposed power consumption profiles are negotiated between the grid controller and the smart consumers can be restricted to "compliant^" consumers, i.e. to those consumers, which still productively cooperate in the negotiation. while consumers which are or become "non-compliant^" during the process are excluded from further calculation. This on the one hand saves substantial processing power and time and on the other hand accounts for any potential non-compliance of consumers in the iteration process. The method and controller of the invention can thus handle non-compliant behaviour of consumers, while not taking the sovereignty completely away from the consumers. This can especially be of advantage under unexpected and fluctuating minimal power consumption of the users.

In a further embodiment of the invention a subset of critical lines and critical nodes can be determined from all lines and nodes in the network, and for said constraint, that l ine currents and node potential drops in the network model stay w ithin allowed ranges over the entire prediction time window, only the line currents and node potential drops of said subset are considered when solv ing the optimization problem. This embodiment yields a further reduction in processing time and power.

If a smart consumer needs a guaranteed minimum power consumption profile to assure a safe or critical operation, the method of the invention can comprise in a further embodiment the step of obtaining a required minimum power consumption profile for at least a part of said prediction time window for at least one of the smart consumers, wherein in solv ing the optimization problem the further constraint, that the to-be-determined proposed power consumption profile for said smart consumer is greater than or equal to the required minimum power consumption pro ile for any time step thereof, is considered. Said required minimum power consumption profile can, e.g., be obtained as a specified time segment of the predicted power consumption profile of said smart consumer. For example, the consumer can specify this time segment when sending its predicted power consumption profile to the grid controller, indicating that the predicted power consumption profile in this time segment is a "non-negotiable^" minimum.

In any of these embodiments of the invention, said measure hich is minimized in solv ing the optimization problem can be any cost function which is dependent on the differences between the to-be-determined proposed power consumption profiles and the received predicted power consumption profiles, weighted by said influence weights, in a particularly preferred practical embodiment of the invention, the measure is based on the cost function

w ith

Θ . being a vector of dimension n_s of the to-be-determined proposed power consumptions at time step j of the prediction time window for all smart consumers,

0^' being a vector of dimension n_s of the received predicted power consumptions at time step j of the prediction time w indow for all smart consumers, n_s being the number of smart consumers,

n_p being the number of time steps of the prediction time w indow, and

W being a positive definite matrix containing said influence weights. In particular, said measure can also be further based on the additive term

with

being a radially unbounded, positive definite real function,

s. being a slack variable for the i-th smart consumer, and

γ being a weight,

which accounts for the constraint (25) detailed below. In particular, this slack variable and constraint can ensure for a power consumer that the total power consumption (energy) over the prediction time window, as proposed by the grid controller to the consumer, is at least the total power consumption (energy ) predicted by said consumer.

Further objects, features and benefits of the method and grid controller of the invention w ill become apparent from the fol lowing detailed description of exemplary embodiments thereof.

Brief Description of the Drawings

The invention will now be explained in detail w ith reference to the appended drawings, in which:

Fig. I is a single line diagram of an exemplary power grid.

Fig. 2 is a control scheme of an exemplary smart consumer, and

Fig. 3 is an exemplary diagram of iteration results.

Detailed Description

Fig. 1 shows a simplified single line diagram of an exemplary power grid 1 hav ing one or more internal or external power producers 2 and power consumers 3, 4, a network of power nodes N; and power lines Lj_j between the i-th and j-th node, a wireless or wirebound communication infrastructure 5 connecting producers 2, consumers 3, 4 and further functional components (such as e.g. transformers) as far as necessary, and a grid controller ("HMPC") 6 connected to the communication infrastructure 5 v ia an interface. Some of the consumers 3, 4 are conventional "non-smart^" consumers 3 (depicted as arrows), whereas some other are "smart^" consumers 4 - such as exemplary predictively controlled buildings PCB 1 , PCB2 - which are capable of predicting their power consumption profile over a limited future time horizon, i.e. a prediction time window, as detailed further below .

It goes without say ing that everything which is said here about smart consumers - and in the following the example "building^" is used sy nonymously for "consumer^" - applies in the same way to smart producers and smart producer-consumers ("prosumers^"); the term "smart consumer" as used herein thus also comprises smart producers and smart prosumers and the term "power consumption" or "power production" is only a matter of sign.

The grid controller 6 has a model 7 of the network stored in its memory and a processor (not shown) accessing the memory for performing the methods disclosed herein. The network model 7 comprises impedances or admittances, respectively, of all nodes and lines (including components such as transformers etc.). as well as power generation profiles of the producers 2 and power consumption profiles of the non-smart consumers 3, as far as necessary for the methods and apparatus detailed below.

Fig. 2 shows a control scheme within one of the smart consumers 4 (here: a smart building) used in the follow ing examples. T_seik₊₁ refers to thermal ly activated building systems (TABS) set points for a next time step k + 1, To are baseline trajectories. Q are trajectories of the heat flows to the TABS, both containing entries over n_p. T_room is the room temperature, and SBUI/SBS is the simpl ified building and building services model.

Fig. 3 shows iteratively the possible reduction of decision variables after each iteration step in the method disclosed herein, for an exemplary number of decision variables /¾_ec = 4 at the beginning, under the assumption that in each iteration step only one building (smart consumer) becomes non-compliant. When /¾_ec = 0, no decision variable is left and v iolations of the grid constraints are possible.

With reference to Figs. I to 3. the operation of the grid controller 6 and the method performed by the grid controller 6 in cooperation with the smart consumers 4 will now be described in greater detail in form of an exemplary, non-limiting embodiment of an electrical power grid I .

Whi le this embodiment refers specifically to an electrical power grid, the term "power^" in the present context comprises all sorts of power, as stated at the outset, be it electrical power, hydraul ic or pneumatic power, thermal power transported by fluids, water or steam in district heating grids, thermal power stored in combustible gases or fuels transported in fluid distribution grids, etc. Thus, the term "power^" and the "power grid^" I as used herein comprise all these sorts of powers and grids.

Similarly, the term "current^" as used herein does not only comprise electrical currents but also encompasses mass flows or thermal energy flows on physical transportation lines, and the term "potential drop^" as used herein does not only comprise voltage drops but also encompasses pressure drops, thermal energy drops et cet. occurring in physical distribution nodes. 1 System modelling

1.1 Electrical Grid

1.1.1 Smart Grid Communication

A "smart grid" represents an efficiency improvement of the e isting electrical power grid by employing the newest communication and information technologies, with the goal of reducing financial and environmental costs of power generation and consumption [10]. The smart grid 1 may employ a two-way digital communication infrastructure 5 which enables the producers 2 and consumers 3,4 to exchange information with the grid controller 6, see Fig. 1.

1.1.2 Modeling and simplification of the grid

In order for the I I PC 6 to be able to predict the line currents and voltage drops in the grid on some prediction horizon, and to be able to propose restrictions on the power consumption of the smart consumers (PCBs) 4 at critical times optimally, it is beneficial to have a simple model for the grid. This ensures that all calculations of the control algorithm can be performed in real time.

The grid I under consideration has a simple radial ( tree) structure, however, more complex mesh structures are also covered by the proposed methodology. For the parameterization of the generation, grid distribution and power consumption profile in th is exemplary embodiment, the following assumptions hav e been made:

a) Supply: In this example, the electricity producers and the transformer are not modeled by a specific dynamical model . Instead, the interaction of the grid with the electrical substation is replaced by an external grid element with constant v oltage.

b) Demand: Power consumption profiles for the passive consumers are assumed to be known. One possible way to obtain this information is to generate it on the basis of annual measured energy consumption and synthetic normal ized power consumption profiles. Furthermore, time-varying pow er consumption profiles for the PCBs are available.

c) Distribution: Network topology, characteristics of lines and transformers, and operation states of the networks are assumed to be known. For example, they may be based on specifications prov ided by the Distribution System Operator.

First off, simulations of the grid can be done to identify the technical limitations and possible v iolations of system constraints. These can be done by power system simulation software ( e.g. PowerFactory, see [24]). Based on appropriate criteria, some nodes and branches in the grid may be identified as critical . For example, based on the simulation results of the grid, the branches and nodes with the lowest loading and voltage sensitivity are identified, and those nodes and branches with highest sensitiv ity can be referred to as critical . In order to avoid simulation effort for the uncritical nodes, the electrical grid can be simplified. For simplification, the uncritical nodes are appropriately grouped and replaced by single nodes. The power consumption profiles of those consumers on the reduced nodes are summed up and located on the replacing node. Branches with PCBs are typically left unchanged to resolve grid constrains on a detailed level.

1.2 Linear Grid Mode!

For the formulation of the optimization problem solved by the grid controller 6, it is advantageous to have a linear model of the grid I . In this section the linear grid model 7 is derived, where the corresponding modeling approach is applicable to three phase electrical grids with symmetrical power consumption predictions.

Impedance and Admittance Matrices

The three-phase activ e and reactive power are mapped to a one phase model. The grid is modeled with the bus network admittance matrix Y and the bus network impedance matrix Z Y [19]. The admittance matrix Y of the simplified grid model is a complex n_nodes x n_nodes matrix generated with the aid of nodal analysis [20], with the number of nodes n_nodes. Therefore Ohm's law can be written as

YD . (1) with the node current vector I and the node voltage v ector U , containing the currents and electric potentials in all nodes of the electrical grid

where Nj denotes the -th node, i = 1, . . . , n_nodes. Equation (1) represents n_nodes linear equations. One equation in (1) is eliminated by choosing the node connected to the external producer to be the slack node, and for notational simplicity let it be denoted by Νχ. For the line currents are unknown, and = V₀ is the nominal voltage of the electrical grid.

The reduced admittance matrix Y is generated by removing the row and column which corresponds to the slack bus, leading to n_nodes— 1 equations and the reduced impedance matrix Z = Y ¹. Moreover, let I and U denote the node currents and voltages, where the slack node is removed. The node voltages are derived by

U = ZI - Z ZI +V₀l_n . (3)

~AU

where l_nriod„_~~i is a vector of size n_nodes— 1 containing ones in all entries. Therefore the node voltage drop v ector All is given by:

AU = ZI. (4) 1.2.2 Obtaining the Node-Line Transformation Matrix

The line currents are sorted into a vector J_]ine = [ ne,i,j] ^T, containing all couples (i, j) such that the line between nodes Ni and N, exists. The lines connected to the slack node are omitted. The current between node Ni and node N_j is calculated by

i, j G {2, . . . , n_nodes}. Hence I \ _in. G R"^lines , where n_!ines is the total number of lines in the grid which do not connect to the slack node. Further, (3) implies:

^nodes

¾, = ¾ + V , (6)

1=2

where i G {2, . . . , n_nodes}. Equation (5) results in

with T [ .i column I of the node-line transformation matrix T_L-N to map node currents on line currents:

1.2.3 Linear System Matrices

In this subsection a static, linear state system for the simplified grid is derived, based on the above analysis. Assuming that the nodal voltage drop does not differ more than 10% from the nominal voltage V₀ (this is in general the allowed range for voltage drops, see

[ 2 1 ] ), and that the electric consumption of the passive consumers and the PCBs is known, the electric current can be determined as follows:

Re(I_Ni) = Irn(I_Ni) =— -^ ,

Vi G {2, . . . , n_nodes}, where P_eu and Q_e denote the active and reactive power summed up over all consumers at the -th node, respectively. Therefore, the node voltage drops A U G Rhodes X «P _ant the line currents /_lil , G R"^{iineE >< 7lp} for the prediction horizon [tk, tk.+rip ] can be written as:

A U = D_AVP_e] + E_AlJL passive (9)

I line -Pel +

(10) with the power consumption matrix of the passive buildings L_passive = [-P_ei,_Passive , Q_paSsive ; ^passive] ^T t R consisting of the active, reactive and apparent power on the prediction horizon, with n_passive the number of nodes where passive buildings consume power. The active power consumption of the predictively controlled buildings on the prediction horizon is given by P_ei £ M"^{B Xrip}, with n_s the number of predictively controlled buildings. The utilized matrices concerning the predictiv e controlled buildings are

Dsi =†H H i , ΛΗΤ,,,,,,! . (1 1) l_ine - ^#_/LMET_L,NT_{PRED !} ( 12) with T_PRED denoting the position of the nodes with predictive controlled buildings in the simplified electrical grid, R = Re(Z), X = Im(Z) . Furthermore Φ_ΛΓ and Φ/_ΗΗΡ are diagonal matrices such that ^υ,α = tan and Φ/, = where c¾ is the pha.se shift specific for the /-th predictiv ely controlled consumer. For example, in a building where most of the consumpt ion is caused by the heat pump, the phase shift of the heat pump is <pi. The matrices for the passive power consumption profiles are

EAV = (RT_passi_veTp + XT_passiveTq) ( 13)

Vo

E_{\ tl},. = jyT_L^T_passiveT_s ( 14) v"o

with T_PASSIVE denoting the position of the nodes at which passive consumers and the transformation matrices T , TQ and T_s select the active, reactiv e or apparent power out of the passive power consumption matri L_passive, respectively.

Note that in order to deriv e a real l inear model, the imaginary part of AU is neglected and the absolute values of I i_im. are summed up. This model simplification is justified by validation using simulation results.

13 Model predictive control in buildings

For the hierarchical grid control algorithm, it is assumed that some buildings in the power grid hav e predictive control algorithms with a global electric power consumption constraint. The MMPC respects a global constraint on the overall electric consumption at each time step k and generates predictions ov er the prediction horizon t_k+_np] - It consists of three modules as shown in Fig. 2, and is described in the following:

In Module 1 predictions ov er [t_k, t_k+Tlp] are obtained based on a nonl inear simplified building model and building services model (SBUI/SBS) consisting of the heating and cool ing system of the building and the building's thermal dynamics.

In Module 2 a linearized model predictive control optimization problem is solved over a prediction horizon [t_k, t_k+np] in order to adjust the heat flow trajectories obtained in Module I . The general optimization problem is a continuous linear programming problem with constraints on the room temperatures ,.,„„„_,;, and Thermally Activ ated Building Systems (TABS) temperatures _TABS._fc, as well as those for the ov erall heat flows at each

1 I prediction step. It uses an LTI state space model of the building defined as

x_k+1 = Ax_k + Bu_k

(15) y_k = Cx_k + Du_k,

obtained by linearization of the nonlinear simplified building and building serv ices model, with the state matri A, the input matri B, the output matrix C, the feed-through matrix D, the system states x_k = [T _ABSiJfc ^TL_m J^T _> ^the output vector y_k = [^χ P_e],_fc]^T, and the input vector u_k = [Q_\ieat,k Q,_:<>„_{\ /}J ' · The input vector u_k consists of the heating flows Q_heat-fc and the cool ing flows Q,. _{l )} . at time step k. In addition to room and TABS temperatures, the output vector y_k consists of the power consumption profile P_e\,k, for the prediction horizon n_p.

In Module 3 set point trajectories T_set,fc+i are generated for the next time step by a mixed integer optimization to realize the heat flows Q = [u_k u_k+1 . . . Uk+_np-i] via building services sub-control loops.

The heat pumps of the building are modeled with the Coefficient of Performance (COP) being defined by

COP_fc = (16)

w ith Qup_,k the generated heat flow and the consumed electric power by the heat pump.

As seen in (16), COP depends on the prediction step t_k, however from applications it is known that it can be considered constant.

2 Hierarchical MFC

In this section the concept and the algorithm of the hierarchical model predictive grid controller 6 (HMPC) is presented. The controller 6 secures adherence to the specifications on the grid level, by ensuring that the node potential and the line currents stay in their allowed ranges at the critical nodes in the grid 1. This approach assures that power outages and disturbances, as well as permanent damage to the grid infrastructure, can be avoided.

2.1 Objectives on the grid level

First, the critical l ines and nodes influenced by the predictive controlled buildings are identified. Hence, let n_cn and n_c\ denote the corresponding number of the identified nodes and lines, respectively. The vectors AU,_:r\t, E W^tcn and J_{< i},, G R™^ci contain the voltage drops and line currents of the critical nodes and lines. Those are obtained by

J -i line, where Τ_υ E R^{Tlcn X Tlnod}'" and Tj E R^{Tlcl X Tllin}'" are the transformation matrices which take only those rows with critical nodes and l ines. The goal of the controller is to enforce the following constraints:

j = 1, n_p, for some predescribed allowed deviations <¾ , a₂ E (0, 1) from the nominal voltage VQ and line current I₀ in the grid. As already mentioned in Subsection 1.2, <¾ = 0.1 is typically chosen (see [21]) while a₂ depends on the material of the power cable (see [26]). In case when producers/prosumers are present in the grid, it becomes relevant to include lower bounds on the voltage drops and l ine currents in ( 1 ) in order to avoid over-voltage and/or negative currents.

2.2 Control algorithm

In this subsection the prerequisites for the proposed control scheme, and the controller algorithm are presented. The following assumptions have been made:

a) The controller 6 is equipped with a mathematical model 7 of the grid.

b) Smart consumers 4 in the grid 1 can predict their power consumption profile.

c) Controller 6 has an approximation for the power consumption profiles of the passive consumers 3 on the prediction horizon.

The approximation for the power consumption profile of the passive consumers 3 can be easily obtained from the recorded historic data of the grid. At some fixed time step t = t_k, it is assumed that each PCB provides a prediction of its power consumption on a prediction horizon [t_k, t_k+rip} . This information is contained in the matrix

[Pi Pi■■ · P _p] - K."°^x"^p , where p _j denotes the power consumption of the i-th building at the prediction step t = t_k+j. However, with this desired electrical power consumption of the PCBs 4, the grid constraints (18) are not necessarily satisfied. Thus, the main idea of the HMPC is to iteratively constrain the overall electrical power consumed by the PCBs at the critical time steps. For this purpose, the power consumption of the PCB is normalized: let the matrix Θ ¹ E R^{¾ X rap} be given by θ ' W\ θ \ . . . θ^_ρ] , where

' norm.i

% = l ., . . . , n_s, j = 1 , . . . , n_p. Here P_norm,i s the maximal electrical power consumption of ί-th consumer. In a building where the power consumption is mainly caused by 1 1 VAC, P_norm j can be obtained as maximal electrical power consumption of the heat pump

P_norm,i = ^§_ρ-, where COP; is the coefficient of performance of the heat pump, and QUA the maximal total heat flow, see (16). The idea is that the matrix Θ* contains the information about the consumption constraints prescribed by the grid controller. In case that no constraints at time t = t_k+j exist, then Θ* = l„_s, i.e. the vector Θ* has ones in ail entries. In case of a violation of the line current or the node voltage drop condition (18), the set of all prediction steps in which a grid violation occurs is denoted by J C {1, ... , n_p}. Then the following optimization problem is solved over ail critical prediction time steps: /_<(¾), (20)

with the constraints Uff < _aiV₀, = l,...n_cn (21) iff ≤ (l + «₂) o, i = l, ...7i_d (22)

≥ 0, (23)

Si ≥ 0, (24) ra_m ra_m

∑ ^{0 d} ≤ ∑ ⁰ + ^Si> , ¾ = !, · · · ¾ (25) and where j G 7. The weight matrix W G R"-^sX"-^s is a positive definite matrix i.e. W > 0. One possibility is to take W to be a diagonal matrix with the inverse summed normalized influences in its diagonal:

i = 1, ... , n_s. Influence values for the buildings r [rir₂... r„J are calculated as a sum of influences over the critical nodes and lines:

Hence, weights are smaller for those buildings with larger influence on the v iolated v alues. More generally, matrix W may also be chosen as a positively definite matrix to account for couplings between consumers in the grid. Moreover, the objective of the constraint (25) is that on the time interval [t_k, t_k+„_m] each building is prov ided with the amount of electrical energy which it has requested, whereby n_m < n_p is some appropriately chosen time step. The corresponding slack variables Si are added to the cost function fi with the weight 7 > 0, where fi : R— R is a positive definite, radially unbounded function. One possibility is to take fi(s)— s², for 1, ... , n_s. After solving the optimizing problem (20> (22), the constraint matrix P' jp, p.,... p^c _np] G R^ra=^Xrap is defined by li =

(26) where jf- - is the upper constraint on the consumption of building % at the prediction time t = t_k+j set by the HMPC. The constraint (23) ensures that the upper constraints remain positive. Next, the new predictions

the additional constraint:

i = 1, . . . , n_s, j = 1 , . . . , n_p. If the grid conditions (18) are not satisfied at all prediction steps with new power consumption profiles, the optimization problem ( 20) ( 25 ) is solved again. However, when the grid conditions (18) are satisfied after some iteration step, the buildings can perform their control actions and the algorithm proceeds to the next time step t_fc+i . The complete iterative algorithm is described in Algorithm 2. 1 .

Algorithm 2.1 Hierarchical grid algorithm

Initialize:

Θ = ones(n_s, n_p)

Iter = 0

while Iter < Maxlter do

Iter =

Obtain predictions for predictive controlled buildings P[ with constraints (27) Use (9) and ( 10) to calculate U_crit and I_crit

if Condition (18) holds then

Break

else

Solve the optimization problem ( 20) with constraints ( 2 1 ) ( 25 )

end if

end while

Return:

Constraints P_ej

Remark 2.1. Note that once that grid controller 6 has set consumption constraints at some prediction step t = t_k+_j, this constraint holds in all subsequent iterations as well. This ensures that in those later iterations, at prediction step t = t_k+j, grid conditions (21) and ( 22 ) hold. Therefore, the algorithm is carried out in ma imum n_p iterations, since in worst ca.se scenario the constraints are calculated for each prediction step separately. However, in order to achieve real time performance of the algorithm, the total number of iterations at the time step t = t_k may be l imited to a previously set integer Maxlter < n_p, as implemented in Line 3 of the algorithm. However, with l imitation of the iteration steps.violation of the grid constraints ( 2 1 ) ( 22 ) is possible.

Remark 2.2. The two-way communication between the grid controller 6 and the smart consumers 4 in the smart grid proceeds as follows: The information on the predicted power consumption travels from each smart consumer to the grid controller. Inversely, the information travels back from the hierarchical controller to each smart consumer in the form of proposed bounds on consumers power consumption. Note that it is not required that the grid controller has a model of the smart consumers in the grid, nor there exists a direct information exchange between individual consumers.

2.3 Feasibility of the optimization problem

In cases when the power consumption profiles of the passive consumers lead to violations of grid constraints, the optimization problem ( 20) with constraints ( 2 1 > (25 ) is not feasible (since the feasibil ity set is empty ) and there are no possible measures to avoid critical voltage drops and/or line currents. In the following discussion, it will be assumed that this is not the case. Then it can be easily seen that the optimization problem ( 20) ( 25 ) is always solvable and the solution is unique. For this purpose note that the cost function is radially unbounded (since it is quadratic), positive definite, and continuous. The feasibility set T is nonempty, because

for % = { 1, . . . , n_s}, j = { 1, . . . , n_p} lies in the feasibility set (trivially). Moreover, conditions ( 2 1 ) ( 25 ) imply that the feasibility set is closed and convex. Hence, the cost function in ( 20 ) obtains a minimum on the feasibil ity set , and the minimizer is unique (see p.7 l , [22] ).

2.4 Robust non-compliance of buildings

In this subsection the Algorithm 2. 1 is extended to scenarios in which some of the PCBs 4 in the grid do not comply with the restrictions ( 27) given by the HMPC 6. It is possible that some of the buildings 4 can not afford to lower their power consumption to meet the restrictions set by the control ler without compromising their performance. It is therefore desirable that the controller recognizes these cases and accepts the minimal required power consumption of those PCBs 4. The algorithm for the controller which respects the non-compl iance of the consumers is given in Algorithm 2.2. The clear advantage of this approach is that the consumers in the grid would be more will ing to accept the concept of a HM PC, if the controller is flexible to their needs. The set W contains the information about the compliant consumers for each prediction step. In the initialization at a new time step t = it is assumed that all consumers are compl iant at all prediction steps. In the Lines 10 16 of the algorithm, the controller identifies the non-compliant buildings for each step of the prediction and eliminates them from the set W. This procedure has a tree structure. In extreme cases the iteration adv ances to the end of the branch where no decision-variable exists and violation of the constraint is possible, see Fig. 3. Algorithm 2.2 Hierarchical grid algorithm that incorporates the non-compliance of the consumers

Initialize:

Θ = ones(n_s, n_p)

Iter = 0

V {!,..., n_s} x {l,...,n_p}

while Iter < Maxlter do

Iter =

Obtain predictions for predictive controlled buildings P[ with constraints (27) Use (9) and (10) to calculate Δ £/,.„,, and I_crit

ifCondition( 18) holds

then

Break

else

for j = 1 : n_P do

fo

end if

end for

Solve the optimization problem (20) with variables 9_itj, for {i, j} G W and with constraints (21 ) (25)

end if

end while

Return:

Constraints PS

Remark 2.3. Note that non-compliance of the buildings 4 can lead to unavoidable violations of the grid constraints, in the sense that at some prediction steps t = t_k+j the grid condition (18) is v iolated and none of the PCBs 4 are compliant, and therefore H M PC 6 faces an in feasible optimization problem. The proposed strategy of the HMPC 6 for such scenarios is to introduce the grid constraints (21 > (25) as soft constraints [25 ] in the optimization problem, making the problem feasible and uniquely solvable again. The Algorithm 2.2 can have maximally n_p x n_s iterations, since in the worst case scenario the constraints need to be introduced at each prediction step, and in each iteration of the algorithm one consumer 4 becomes non-compliant. References:

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Claims

Claims:

1 . A method for controlling power consumption in a power grid (1) having one or more power producers (2) and power consumers (3, 4). a network of power lines (L) and nodes (N) therebetween, a communication in rastructure (5) therebetween, and a grid controller (6) connected to the communication infrastructure, the grid controller (6) having a stored model (7) of the network, and at least one of the power consumers (3, 4) being a smart consumer (4) capable of predicting its power consumption profile (0^d) over a prediction time window, comprising:

for each smart consumer (4), determining an influence weight (W) of its power consumption on l ine currents (I) and node potential drops (ΔΙΙ) in the network model;

for said prediction time window, performing an iteration step in the grid controller (6) including:

receiving a predicted power consumption profile (0^d) for said prediction time window from each smart consumer (4) v ia the communication infrastructure (5), determining a proposed power consumption profile (Θ) for said prediction time window for each of a selected set of smart consumers (4) by iteratively solving an optimization problem wherein

a measure of the differences between the to-be-determined proposed power consumption profiles (Θ) and the received predicted power consumption profiles (0^d), respectively weighted by said influence weights (W), is minimized under the constraints

that line currents (I) and node potential drops (AU) in the network model (7) stay within allowed ranges over the enti e prediction time window, and sending said proposed power consumption profiles (Θ) to the smart consumers (4) v ia the communication infrastructure (5);

and repeating said iteration step for said prediction time window in a loop which is exited when at least one of the following conditions is met:

all line currents (I) and node potential drops (AU) of the network, as calculated by applying the received predicted power consumption profiles (0^d) to the network model (7), stay within allowed ranges over the entire prediction time window, or none of the received predicted power consumption profiles (0^d) has changed with respect to the preceding iteration step, or

a maximum number of iterat ion steps has been reached.

2. The method according to claim 1 , wherein said selected set includes all of the smart consumers (4).

3. The method according to claim 1 , wherein s id selected set includes, for a selected time step (j) of the prediction time window, only those smart consumers (4) whose predicted power consumption profile (0^d) has changed with respect to the preceding iteration step.

4. The method according to claim 3, wherein for each smart consumer (4) a flag indicating whether its predicted power consumption profile (0^d) has changed from one iteration step to another is stored in a compliance matrix which defines said set and is used in the solving of the optimization problem, preferably such a flag for each time step (j) of the prediction time window.

5. The method according to any one of the claims I to 4, wherein a subset of critical lines (L_cri_t) and critical nodes (N_ait) is determined from all lines (L) and nodes (N) in the network, and wherein for said constraint only the line currents and node potential drops (AU_cri_t) of said subset are considered when solv ing the optimization problem.

6. The method according to any one of the claims I to 5, comprising

obtaining a required minimum power consumption pro ile for at least a part of said prediction time window for at least one of the smart consumers (4),

wherein in solving the optimization problem the further constraint, that the to-be- detcrmincd proposed power consumption profile (Θ) for said smart consumer (4) is greater than or equal to the required minimum power consumption profile for any time step thereof, is considered.

7. The method according to claim 6, wherein the required minimum power consumption profile is obtained as a specified time segment of the predicted power consumption profile (0^d) of said smart consumer.

8. The method according to any one of the claims 1 to 7, wherein said measure is based on the cost function

with

0_j being a vector of dimension n_s of the to-be-determined proposed power consumptions at time step j of the prediction time window for all smart consumers,

0_j being a vector of dimension n_s of the receiv ed predicted power consumptions at time step j of the prediction time window for al l smart consumers, n_s being the number of smart consumers,

n being the number of time steps of the prediction time window, and

W being a positiv e definite matrix containing said influence weights.

9. The method according to claim 8, wherein said measure is further based on the additiv e term

with

being a radially unbounded, positive definite real function,

being a slack v ariable for the i-th smart consumer, and

being a weight.

10. An electronic grid controller for a power grid (1) which has one or more power producers (2) and power consumers (3, 4), a network of power lines (L) and nodes (N) therebetween, and a communication infrastructure (5) therebetween, the grid controller (6) comprising:

an interface being part of the communication infrastructure (5);

a memory having a model (7) of the network stored therein; and

a processor being configured to implement the method according to any one of the claims 1 to 9.