Small Grid Research Articles

Mitigation of Short-Term Wind Power Ramps through Forecast-Based Curtailment

As the penetration of renewable energy generation in electric grids becomes more substantial, its contribution to the variability of the net load becomes more noticeable. Particularly in small or weak grids, the rate at which the output power of a wind farm decreases may become a concern to grid operators. In the present work, a novel approach, called forecast-based curtailment (FBC), is shown to be able to self-mitigate downward ramps on short time scales at a very small energy penalty, compared to conventional mitigation schemes, such as flat curtailment or up-ramp limitations. FBC allows to achieve compliance with ramp limits imposed by system operators at a very small energy cost and modest additional upfront investments.

Potential and costs of decentralized heat pumps and thermal networks in Swiss residential areas

Globally, there has been an increasing trend toward electricity production from renewables, hence promoting the opportunity for the electrification of the residential heating sector as opposed to fossil fuels' combustion. Decentralized heat pumps and district heating networks could play a key role in the Swiss future approach to heating. This study provides a detailed enviro-economic assessment of the application of heat pumps and district heating systems based on renewable sources in different Swiss building archetypes on different scales. The results show that the levelized costs of heat from individual heat pumps and large-scale district heating networks (using industrial excess heat with a lower energy price than for the other energy carriers) are 7% to 20% and 65% to 75% higher than individual fossil fuel boilers for single-family houses, respectively. However, for multifamily houses, individual heat pumps, small thermal grids, and large-scale district heating networks are competitive. The district heating network coupled with a large centralized water source heat pump is found to be the most expensive solution (0.24-0.29 CHF/kWh) for residential heating today. The sensitivity analysis suggests the economic feasibility of different heating technologies significantly overlaps with each other depending on the boundary conditions, hence justifying not only the mixed opinions by experts in Switzerland but also the disagreements over costs of these technologies in international literature. It can be concluded that, based on the levelized cost of heat, there are sufficient economically viable options for decarbonization and electrification of the residential heating sector. While this is an important finding for policymakers, the barriers, for example, high upfront cost, need to be addressed. Highlights Low-carbon heating technologies are investigated for residential heating supply. Levelized costs of heat from heat pumps and district heating networks are computed. Levelized costs of the heating technologies significantly overlap with each other.

A Grid Partition Method for Atmospheric Phase Compensation in GB-SAR

Time-series interferograms acquired on a deep pit with a Ground-Based Synthetic Aperture Radar (GB-SAR) system showed that the atmospheric phase (AP) could be complexly space variant due to rapid changes of the weather conditions and steep topography. Conventional compensation methods that simulate the AP with typical parametrical models are no longer applicable. Based on the theoretical path integral model of the AP, a grid partition (GP) method is proposed. By dividing one interferogram into a certain number of small grids, the refraction variation inside each grid is assumed to be a constant. A system of linear equations is first built based on sufficient permanent scatterers (PSs). Then, bounds and inequality constraints are set to limit the refraction variation of each grid. A constrained linear least-square problem is solved with two-step process to estimate and compensate the AP. To fully validate the feasibility of the GP method, simulated phase interferograms based on four conventional AP models and with the consideration of deformation areas and noise phase are first processed. Then, four experimental interferograms with different types of AP components are processed and made comparisons with the conventional parametrical methods. The quantitative comparisons of the simulated and experimental data sets both proved that the GP method can well reduce the AP errors.

Increasing the Physical Representation of Forest‐Snow Processes in Coarse‐Resolution Models: Lessons Learned From Upscaling Hyper‐Resolution Simulations

AbstractProcesses shaping forest snow cover evolution often vary at small spatial scales, which are not resolved by most model applications. Representing this variability at larger scales and coarser model resolutions constitutes a major challenge for model developers. In this study, we use a well‐validated hyper‐resolution forest snow model that explicitly resolves the spatial variability of canopy‐snow interactions at the meter scale to explore adequate representation of forest‐snow processes at coarser resolutions (50 m). For this purpose, we assess coarser‐resolution runs against spatially averaged results from corresponding hyper‐resolution simulations over a 150,000 m2 model domain. For the coarser‐resolution simulations, we tested alternative upscaling strategies. Our results reveal considerable discrepancies between strategies that utilize generalized canopy metrics versus strategies that apply a more detailed set of process‐specific canopy descriptors. Particularly, the inclusion of canopy descriptors that represent the various scales and perspectives relevant to the individual processes leads to accurate simulation of forest snow cover dynamics at coarse resolutions. Our results further demonstrate that a realistic representation of snow‐covered fraction in snowmelt calculations is important even for relatively small (∼50 m) grid cells. Ultimately, this work provides recommendations for modeling forest‐snow processes in large‐scale applications, which allow coarse resolution simulations to approximate spatially averaged results of corresponding hyper‐resolution simulations.

Coastal tourism planning using GIS-based system: the case of Shirud coast, Caspian Sea, Mazandaran, Iran

Coastal areas of rivers and seas are valuable and vital environmental resources, providing distinctive landscape characters of the urban/rural environment and focal zones for people’s most activities especially recreational. Since human interference often lead to changes in the natural environment, lack of comprehensive integrated planning for coastal areas, especially in estuaries where landscape characters of river and sea act integrated, would lead to degradation of coastal areas. In this research, decision-making for tourism activities allocation, which plays an important role in natural characters conservation especially in coastal sensitive areas, is done with a map-overlay method using Arc-GIS. The research tries to provide a method to reduce landscape alternations due to coastal tourism activities, suitable in both small and large scale areas. Categorizing the coastal landscape characters in two main categories including the basics of natural life of coasts and the basics of socio-cultural life of coasts and the common recreational coastal activities in two categories of nature based and utility based, the analysis is done for morphological-topographical and ecological characters in two scales. First, in the study area the “Shirud coast”, the characters were analyzed on a regional large scale, which resulted in zoning the area into nine zones. Secondly, on a smaller scale, dividing each zone into smaller grids, the sensitivity is again evaluated to increase the accuracy of results and finally, the suitable allocation of tourism activities is suggested. This research is done according to descriptive methods including desk-based studies and field surveys observation and the analyses are implemented with Arc GIS software. The results demonstrate that the areas subjected to erosion and sedimentation, the integrated transitional zones of sandy beach, estuary and the riparian buffer require the most conservation. While the estuary is labeled as the “no activity” zone, distancing from the shoreline to the backshore, the level of sensitivity is reduced where nature-based and utility-based activities are allowed.

A grid-free approach for simulating sweep and cyclic voltammetry.

We present a computational approach to simulate linear sweep and cyclic voltammetry experiments that does not require a discretized grid in space to quantify diffusion. By using a Green's function solution coupled to a standard implicit ordinary differential equation solver, we are able to simulate current and redox species concentrations using only a small grid in time. As a result, where benchmarking is possible, we find that the current method is faster than (and quantitatively identical to) established techniques. The present algorithm should help open the door for studying adsorption effects in inner sphere electrochemistry.

TO THE ISSUE OF CREATING MODELS OF PRECISE SOIL COMPACTION DETECTION FOR PRODUCTION SCALE USE

Typical approaches to test soil compaction, along with technical possibilities are assessed with a purpose of scaling best approaches in production. Modern soil works related technologies, GPS accuracy aspirations for precision agriculture are addressed. Two most popular technologies in Ukraine to measure soil compaction are «The Top Soil Mapper» and «CTS-1000», although they function differently, most commonly are used independently. Simple mathematical calculations bringing together timing needed to gather reliable data to form variable depth soil ripping machinery and scale of Ukrainian production demonstrate irrelevance of applying typical mechanical methods. At the same time, sensor technologies have their own limitations. Thus, this article introduces the benefits of the mutual use of «The Top Soil Mapper» and «CTS-1000». Soil compaction models presented in the article include: 1. Smallest possible grid approach of soil compaction data presentation and interpolation for further use to create variable depth soil works assignment proves to provide a very good set of results and can be used as a standard, but at the same time it consumes so much time that production scaling is not achievable. As an example we can state that a grower with 1000 fields would need about 3 years to test all the fields, when production needs are to test all the fields within two or three months (depending on weather and crop rotation). 2. Small, midsize and large grid approach of soil compaction data presentation and interpolation for further use to create variable depth soil works assignment proves to vary from being still time inefficient to having gaps to gather enough data as main approach of any data processing for further interpolation makes an unproven assumption that the sample location is a center of a zone. In case with a smallest possible grid, that difference would be minimal, with a larger distance difference can be significant. 3. Depending on detecting zones to test compaction based other parameters does not prove to work and lacks scientific background. Thus it has been suggested to combine use of «Top Soil Mapper» and «CTS-1000».

CFD modeling of liquid entrainment through vertical T-junction of fourth stage automatic depressurization system (ADS-4)

In this study, a three-dimensional (3D) transient computational fluid dynamics (CFD) model is presented to investigated liquid entrainment in gas–liquid flow through vertical T-junction of fourth stage automatic depressurization system (ADS-4) in AP1000. A specialized CFD model for the liquid entrainment phenomenon study, which is based on coupled Eulerian-Eulerian VOF (volume of fluid) formulation with suitable interfacial drag and standard κ-ε turbulence model for each phase was proposed. The entrainment rate was calculated and results were validated with ADETEL experimental data, which was built at XJTU-NuTheL. The good agreement between CFD calculations and experimental data demonstrated that the proposed CFD model could reasonably predict entrainment within the studied range. The effect of vertical to horizontal branch diameter ratio and gas mass flow rates on liquid entrainment were also studied. In addition, the liquid volume fraction distributions and velocity field were investigated to develop an understanding of the entrainment process. The relatively high demand for computational resources due to very small timestep size and small grid size to accommodate high flow velocities was found to be a challenge.

NUMERICAL IMPLEMENTATION OF A PARALLEL ALGORITHM FOR SOLVING THE PROBLEM OF POLLUTANT TRANSPORT IN A RESERVOIR ON A HIGH-PERFORMANCE COMPUTER SYSTEM

The purpose of this work is to create a software package for a distributed solution of the problem of transporting a pollutant in a reservoir with complex bathymetry and the presence of technological structures. An algorithm has been developed for the parallel solution of the problem of transporting a pollutant (pollutant) in a reservoir on a graphics accelerator controlled by the CUDA (Compute Unified Device Architecture) system; a comparative analysis of the operation of algorithms on a CPU (Central Processing Unit) and on a graphics accelerator GPU (Graphics Processing Unit) made it possible to evaluate their performance. The software implementation of the modules included in the complex is described, the main classes and implemented methods are documented. The results of numerical experiments showed that solving of pollutant transport’s problem based on the CUDA technology is ineffective for small grids (up to 100 ´ 100 computational nodes). In the case of large grids (1000 ´ 1000 computational nodes), the use of CUDA technology reduces the computation time by an order of magnitude. An analysis of the experiments carried out with the developed components of software showed that the maximum value of the ratio of the algorithm operating time that implements the set task of transferring matter in a shallow water on a GPU to the operating time of a similar algorithm on the CPU was 24.92 times, which is achieved on a grid of 1000 ´ 1000 computational nodes. Implementation of methods for decomposition of grid regions is proposed for solving computationally laborious problems of diffusion-convection, including the problem of transporting pollutants in a reservoir with complex bathymetry with technological objects that take into account the architecture and parameters of a MSC (Multiprocessor Computing System) located on the basis of the infrastructure facility of the STU (Scientific and Technological University) “Sirius” (Sochi, Russia). Consideration was made for such a property of a computing system as the time it takes to transmit and receive floating point data. An algorithm for the parallel solution of the task under the control of MPI (Message Passing Interface) technology has been developed, and its efficiency has been assessed. The acceleration values of the proposed algorithm are obtained depending on the number of involved computers (processors) and the size of the computational grid. The maximum number of computers used is 24, the maximum size of the computational grid was 10 000 ´ 10 000 computational nodes. The developed algorithm showed low efficiency for small computational grids (up to 100 ´ 100 computational nodes). In the case of large computational grids ( from 1000  1000 computational nodes), the use of MPI reduces the computation time by several times.

Study of offshore wind power penetration rate in gas turbine generator platform power grid

Because of the energy supply of the offshore oil platforms mainly relies on the supply of gas turbine, the development of renewable energy will become a rich energy supply for offshore oil platforms and a reliable way to improve the stability of the grid of the platforms. This paper developed the simulation model of offshore oil gas turbine platform power grid, simulated the hybrid power grid structure, the energy storage of frequency regulation, high wind power penetration rate, wind power and gas turbine power frequency stability and fault treatment. The method of increasing renewable energy penetration rate was proposed, and the hybrid energy coordinated control strategy of offshore oil platform was also carried out. Based on the weakly connected distributed offshore oil platform power grid project, this paper studied the influence of wind power and turbine power penetration rate on the power grid, in order to explore, determine and verify the limit of high wind power penetration rate under the condition of “large generation machine and small grid”. The simulation experiment of high wind power penetration rate under the condition of different load conditions and different wind speed limits were discussed, on the basis of research the influence degree of the stability of power grid frequency of low wind power swings, as well as to the wind power is given priority to with gas turbine mainly in both cases the advantages and disadvantages of the grid system, and attempt to put forward the corresponding switch control strategy.

Selective harmonic active tuning control method for hybrid active power filters

This paper proposes a selective harmonic active tuning control method for hybrid active power filters (HAPFs). A HAPF is composed of a voltage source inverter (VSI) and an LC filter. With this method, the VSI in the HAPF is controlled as a virtual capacitance that varies with the dominant harmonic frequencies. Thus, the LC filter can be forced to be tuning at these frequencies, which significantly improves the harmonic filtering characteristic of the LC filter. Meanwhile, to enhance the filtering performance in the case of a small power grid impedance or to damp the resonances between an LC filter and the grid impedance, the harmonics in the grid source current are detected and proportionally fed back to the controller, which brings about an effect equivalent to adding a series harmonic resistance on the grid source side. The proposed control method has the advantages of a simple control structure and easy implementation. Simulation and experimental results confirm the effectiveness and feasibility of the proposed method.

RETRACTED ARTICLE: Coastal atmospheric climate and decoration art design of coastal environment based on GIS

Ecotourism in coastal cities is becoming more and more popular as a new way of tourism. The experience of tourists’ comfort and the experience of environmental decoration art in coastal cities have become an important indicator affecting the quality of ecotourism. Based on the statistical analysis of climate resources in a certain area in the eastern coastal area, this paper establishes the grid calculation model of climate factors by using multiple linear regression method and uses the data of automatic meteorological observation station to check and modify the calculated climatic element values and then obtains the small grid data and its distribution map of climate resources by using GIS technology platform. At the same time, from the perspective of the comfort evaluation of tourist attractions, this paper makes a quantitative analysis on the three indicators of temperature, humidity, and light intensity. According to the evaluation method, a set of comfort evaluation system is developed by using VBA language and GIS tools. From the perspective of system application, the system can display the comfort related indicators in the form of cloud map in the form of geographic information of tourism area. The effect of comfort evaluation on the target area is achieved.

Grid Support with Wind Turbines: The Case of the 2019 Blackout in Flensburg

The work presented in this paper aims to show how modern wind turbines can help to control the frequency in a small grid which suffers from large power imbalances. It is shown for an exemplary situation, which occurred in Flensburg’s distribution grid in 2019: a major blackout, which occurred after almost two hours in islanding operation, affecting almost the entire distribution grid, which supplies approximately 55,000 households and businesses. For the analysis, a wind turbine model and a grid support controller developed at the Wind Energy Technology Institute are combined with real measurements from the day of the blackout to generate a fictional yet realistic case study for such an islanding situation. For this case study, it is assumed that wind turbines with grid support functionalities are connected to the medium voltage distribution grid of the city. It is shown to what extent wind turbines can help to operate the grid by providing grid frequency support in two ways: By supplying synthetic inertia only, where the wind turbines can help to limit the rate of change of frequency in the islanded grid directly after losing the connection to the central European grid. In combination with the primary frequency control capabilities of the wind turbines (WTs), the disconnection of one gen set in the local power station might have been avoided. Furthermore, wind turbines with primary frequency control capabilities could have restored the grid frequency to 50 Hz shortly after the islanding situation even if the aforementioned gen-set was lost. This would have allowed connecting a backup medium voltage line to the central European grid and thereby avoiding the blackout.

Study on an Improve Finite‐Control‐Set ‐Model Predictive Control (FCS‐MPC) Strategy for a T‐Type Rectifier with Direct Power Control Strategy

Model predictive control has been wildly used in the modern power electronic converter with the advantages of the multi‐objective optimization, no additional modulator is required and small grid current harmonic. However, the switching operation is not constant and higher sampling frequency must be required. A novel direct power control using model predictive control (DPC‐MPC) strategy is presented in this paper, which achieve the optimal vector prediction scheme with constant switching frequency operation. The calculation process is greatly simplified due to the optimal voltage vector generated directly, then, it applied to the control of a three‐phase three‐level T‐type grid converter. A cost function is first built to balance the neutral‐point‐potential voltage. After then, the cost function is updated by considering the delay caused by the calculation, sampling and gate driver. Finally, a comprehensive comparative study with the conventional DPC‐MPC strategy is introduced to verify the superiority of the presented DPC‐MPC. The simulation and experimental results show that the steady and the transient performance of the grid current with the proposed DPC‐MPC control strategy are effectively improved. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Can Kohonen networks delineate forest stands?

ABSTRACT Forest inventory based on airborne laser scanning produces data for small grid cells. Various segmentation methods are used to cluster the grid cells into spatially continuous larger areas that correspond to forest stands. This study examined the performance of self-organizing map (SOM) in the delineation of forest stands. The SOM is also called Kohonen map or Kohonen network, according to the developer of the method. The data consisted of stand attribute grids in 16-m resolution. Spatiality was integrated in the SOM by using the geographical coordinates as criteria in the clustering. The clusters created by the SOM were spatially distinct and formed continuous areas corresponding to traditional stands when geographical coordinates were given more weight in the clustering compared to other stand attributes such as fertility class and stand basal area. The delineation improved when the attributes of the created stands were used as the starting point of repeated SOM rounds, and removing small stands at the beginning of each round. When small within-stand variation in stand attributes was targeted, the SOM often produced irregular stands consisting of disconnected parts. These delineations could be fine-tuned by applying mode filtering to the grid of stand numbers generated by the SOM algorithm.

Resource Management of Heterogeneous Cellular Networks With Hybrid Energy Supplies: A Multi-Objective Optimization Approach

Heterogeneous cellular networks with hybrid energy supplies can relieve traffic pressure and reduce grid energy consumption. In heterogeneous cellular networks, rational resource management can help improve system performances. In general, more than one performance is expected to do well, but there can exist a trade-off among different performance metrics, thus making resource management a multi-objective problem. The existing solution usually transforms a multi-objective problem into another single-objective problem by assigning weights for various objectives. However, it is difficult to know the exact weights in advance, and different systems call for different requirements for objectives. Hence, a multi-objective optimization approach based on the gravitational search algorithm (GSA) is proposed to find a series of Pareto optimal solutions. The decision-makers can select an appropriate solution according to the system requirement. In this work, three different multi-objective GSA-based algorithms are proposed to determine user association and power control, with the goal to optimize the traffic load balancing among small base stations and grid energy consumption per unit throughput simultaneously. The complexity of the proposed algorithms is analyzed, and simulations compare the performances of the proposed algorithms and the benchmark algorithm. Experimental results reveal the feasibility and effectiveness of this approach.

Investigation of a high temperature gel system for application in saline oil and gas reservoirs for profile modification

The deeper the burial of the oil and gas reservoir, the higher the temperature of the oil and gas reservoir. When profile adjustment of water injection wells, plugging of oil production wells (gas production wells) and temporary plugging or killing of oil production wells (gas production wells) are required, the higher the temperature resistance of the plugging agent (above 150 °C) is required. In addition, high temperature resistant plugging agents should also be used in profile control or water plugging in reservoirs (120°C–200 °C) that are thermally recovered by steam injection or in situ combustion. It is of great importance to develop a high-temperature-resistant gel plugging system to fulfill the requirements of deep buried high-temperature oil or gas reservoirs or thermal recovery high-temperature oil reservoir operations.A gel system formed by the terpolymer (L-1) and a new cross-linking system (HB-1) was developed, and its properties were systematically studied in the condition of extremely high temperature. Studies have shown that the gel system could form stable continuous 3D network structures in high temperature (120°C–200 °C) and thus have an excellent long-term thermal stability.The gel system can adjust the strength by changing the concentration of the terpolymer (0.05%–1%) and the crosslinker (0.05%–1%), which can achieve different construction requirements. Compared with the conventional high-temperature-resistant gel, the cross-linking system HB-1 contains 4 hydroxyl groups (–CH2OH), it can crosslink with 4 amide groups (–CONH2) on the terpolymer. Therefore, this gel system can form a smaller grid size of 3D network structures (less than 5 μm), so it has stronger temperature resistance and better long-term thermal stability. When the temperature is 200 °C, the grid size of network structure is about 10 μm, but the strength of the gel system can still be maintained at code F-G. When the salinity of the water solution is 200,000 mg/L, the gel system on the contact surface can still maintain a small grid size of 3D network structures (about 10 μm). The gel system could maintain most of the initial viscosity and viscoelasticity, even after experiencing the mechanical shear or the porous-media shear. A gel system with strength F can still be formed after shearing by the pore media of 10 sand-packed tubes, and the plugging rate of each sand-packed tube exceeds 97%. It can be found by scanning electron microscope pictures that the gel system can be very firmly attached to the pore media, which can achieve profile adjustment of the formation. The gel system could have great potential in the application of high temperature oil or gas reservoirs.

A multi-center quadrature scheme for the molecular continuum

A common way to evaluate electronic integrals for polyatomic molecules is to use Becke’s partitioning scheme (Becke and Chem, 1988) in conjunction with overlapping grids centered at each atomic site. The Becke scheme was designed for integrands that fall off rapidly at large distances, such as those approximating bound electronic states. When applied to states in the electronic continuum, however, Becke scheme exhibits slow convergence and it is highly redundant. Here, we present a modified version of Becke scheme that is applicable to functions of the electronic continuum, such as those involved in molecular photoionization and electron–molecule scattering, and which ensures convergence and efficiency comparable to those realized in the calculation of bound states. In this modified scheme, the atomic weights already present in Becke’s partition are smoothly switched off within a range of few bond lengths from their respective nuclei, and complemented by an asymptotically unitary weight. The atomic integrals are evaluated on small spherical grids, centered on each atom, with size commensurate to the support of the corresponding atomic weight. The residual integral of the interstitial and long-range region is evaluated with a central master grid. The accuracy of the method is demonstrated by evaluating integrals involving integrands containing Gaussian Type Orbitals and Yukawa potentials, on the atomic sites, as well as spherical Bessel functions centered on the master grid. These functions are representative of those encountered in realistic electron-scattering and photoionization calculations in polyatomic molecules.

Characterization and suppression of the hydrodynamic instability in the time domain for acoustic propagation in a lined flow duct

The gradient term suppression (GTS) method for removing the hydrodynamic instability appearing in the time-domain solutions of the linearized Euler equations (LEE) along a lined flow duct is assessed. For this, the characterization of a convective instability in the time domain, with the aid of a complementary modal analysis, is first presented. The effect of the mesh size and spatial filtering on the instability is investigated. In particular, a convergence of the instability in the time domain is achieved for a small enough grid size. The consequence of suppressing the mean flow gradient term on the modes is then investigated. It is shown that the unstable modes are indeed removed, but also that acoustic modes are significantly modified, especially for low Helmholtz numbers. The GTS method is finally applied to the NASA grazing impedance tube benchmark. It is found that tuning the weight of the mean flow gradient term within the LEE can be effective for suppressing the instability while conserving a reasonable accuracy of the acoustic component.

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number \(\gamma ^{\infty }_{all}\) of a graph, which is the minimum number of guards required to defend against an infinite sequence of attacks. This paper first continues the study of the eternal domination game on strong grids \(P_n\boxtimes P_m\). Cartesian grids \(P_n \square P_m\) have been vastly studied with tight bounds existing for small grids such as \(k\times n\) grids for \(k\in \{2,3,4,5\}\). It was recently proven that \(\gamma ^{\infty }_{all}(P_n \square P_m)=\gamma (P_n \square P_m)+O(n+m)\) where \(\gamma (P_n \square P_m)\) is the domination number of \(P_n \square P_m\) which lower bounds the eternal domination number [Lamprou et al. Eternally dominating large grids. Theoretical Computer Science, 794:27–46, 2019]. We prove that, for all \(n,m\in \mathbb {N^*}\) such that \(m\ge n\), \(\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor +\Omega (n+m)=\gamma _{all}^{\infty } (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})\) (note that \(\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil\) is the domination number of \(P_n\boxtimes P_m\)). We then generalise our technique to prove that \(\gamma _{all}^{\infty }(G)=\gamma (G)+o(\gamma (G))\) for all graphs \(G\in {\mathcal {F}}\), where \({\mathcal {F}}\) is a large family of D-dimensional grids which are supergraphs of the D-dimensional Cartesian grid and subgraphs of the D-dimensional strong grid. In particular, \({\mathcal {F}}\) includes both the D-dimensional Cartesian grid and the D-dimensional strong grid.